Geometry revealed: A Jacob's Ladder to modern higher geometry - PDF Free Download (2024)


Selected works of Marcel Berger R ESEARCH Le spectre d’une vari´et´e riemanniene (with Paul Gauduchon and Edmond Mazet), Springer, 1971 (Collaboratively under the pseudonym Arthur L. Besse) Manifolds all of whose geodesics are closed, Springer, 1978 (Collaboratively under the pseudonym Arthur L. Besse) Einstein Manifolds, Springer, 1978 P EDAGOGY AND POPULARIZATION Geometry I, II. Springer, 1987, 2009 Differential Geometry: Manifolds, Curves and Surfaces (with Bernard Gostiaux), Springer, 1987 “Peut-on d´efinir la g´eom´etrie aujourd’hui?”, in Results in Mathematics, vol. 40, pp. 37–87, 2001 A Panoramic View of Riemannian Geometry, Springer, 2003 ´ ADPF (Association pour la diffusion Cinq si`ecles de math´ematiques en France, Ed. de la pens´ee franc¸aise), 2005 Convexit´e dans le plan et dans l’espace: de la puissance et de la complexit´e d’une notion simple (with Pierre Damphousse), Ellipses, Paris, 2006 “Geometry in the 20th century”, 2002, in History of mathematics, edited by V.L. Hansen and J.J. Gray, Encyclopedia of Life Support Systems (EOLSS), Developed under the auspices of UNESCO, Eolss Publishers, Oxford, UK


Geometry Revealed A Jacob’s Ladder to Modern Higher Geometry


Author Marcel Berger ´ Insitut des Hautes Etudes Scientifiques (IHES) Bures-sur-Yvette France

Translator Lester J. Senechal Professor Emeritus Department of Mathematics Mount Holyoke College South Hadley, MA 10475 USA [emailprotected]

Springer-Verlag thanks the original publishers of the figures for permission to reprint them in this book. We have made every effort to identify the copyright owners of all illustrations included in this book in order to obtain reprint permission. Some of our requests have however remained unanswered. We have inserted all sources and owners where known.

ISBN 978-3-540-70996-1 e-ISBN 978-3-540-70997-8 DOI 10.1007/978-3-540-70997-8 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010920837 Mathematics Subject Classification: 51-01, 01-01 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Wmx design ´ Cover illustration: Le Songe de Jacob (detail), Nicolas Dipre (D’Ypres), Ecole d’Avignon, Mus´ee du c bpk, Berlin, 2009 Petit-Palais, Avignon Printed on acid-free paper Springer is part of Springer Science+Business Media (

About the Author

Marcel Berger has played a unique role in the development of geometry in France. The important exceptional holonomy is due to him, and practically all the geometry that is dear to hearts of physicists is related to it. The elegance of his theorem on the 1=4 pinching continues to attract young mathematicians to Riemannian geometry. A veritable school of geometry formed about him in the 1970s. His students, and his students’ students (now about 90 in number), form the nucleus of geometry in France. He maintains contact with the association “Arthur Besse” where he and his students have written several books: the one on the spectra of Riemannian manifolds was for a while the bible of the subject. His book on closed geodesics turned out to be a scientific adventure story because important problems were solved during the writing process. The one on Einstein manifolds became a best-seller because it appeared at an opportune moment and popularized a little-known subject whose methods were primitive. It showed great flair, for today these manifolds constitute a common area of research for theoretical physicists and mathematicians of all specialties. Last but not least, Marcel Berger recognized Mikhael Gromov’s talent and convinced him to remain in Paris, which has been a determining factor for the development of geometry in France. Marcel Berger himself had likewise been the beneficiary of the support of André Lichnerowicz, a leading figure in differential geometry




and relativistic mechanics, who gave Berger a thesis topic precisely on the holonomy groups that subsequently played an important role in his work. As a teacher, Marcel Berger has been able to share his passion for geometry from the outset that of elementary geometry in all its forms with generations of students. His book Geometry remains unequaled as a survey, disclosing and modernizing all the various points of view that comprise elementary geometry, which of course is neither simple nor easy. Marcel Berger was Professor at the Universities of Strasbourg (1953–1964), Nice (1964–1966), Paris (1966–1974), Director of Research at CNRS (1974–1985 and 1994–1996), Director of IHES (1985–1994). He was President of the French Mathematical Society 1979–1981.

Introduction Jacob had a dream: and behold, a ladder was placed upon the earth, and it reached to heaven; and behold, the angels of God ascended and descended on it; and the Lord stood above it and spoke (Genesis 28, 12–13) Jacob awoke from his sleep and said: “Truly, the Lord is in this place and I did not know it”. . . So Jacob arose early in the morning and took the stone that had served as his pillow and set it up as a pillar and poured oil upon it (Genesis 28: 16,18)

Numerous problems of geometry that are quite visual and can be presented in a very simple manner have one or more of the following properties in common: they remain unsolved, or have been solved only recently following great efforts; for being well understood and eventually completely or partly solved they require the creation of concepts and tools that vary in their degree of abstraction, which is in any case greater than what is required for stating the problem; the mathematical tools used in solving them were conceived for quite other purposes. In this work we present a whole series of such problems while showing the necessity of abstract concepts and how they enter progressively into the solution. These are conceptual notions, each built “above” the preceding and permitting an increase in abstraction, represented metaphorically by Jacob’s ladder with its rungs. This parable appears as a leitmotiv throughout this book. We don’t neglect mentioning problems that still remain open, an “openness” that may seem a priori astonishing, but less so once we understand the totality of the efforts and conceptual progress needed for solving similar problems. These classical problems are ever the object of vigorous research, all the while mathematical research is constantly suggesting new ones. And thus so-called elementary geometry is indeed very much alive and at the very heart of the work of numerous contemporary mathematicians. This book pursues another goal: to show of course the unceasingly renewed vigor of the spirit of geometry, but also to offer readers the elements of a modern geometric culture. For mathematical instruction in our time presents a disquieting paradox. On the one hand, geometry is increasingly present in daily life; we live in a civilization of images. Virtual reality, robotic vision, aerial navigation and the conquest of space require more and more specialists: engineers, aerial controllers, navigators in space, etc. But at the same time geometry in any case spatial geometry is almost completely missing from the instructional programs of schools and VII



universities. The small amount of three-dimensional geometry that nominally still belongs to those programs is in fact practically never treated. It should seem obvious that geometry is of relatively large importance in the whole of mathematics, but the reality is quite different: the language of geometry, geometrical metaphors, have taken hold throughout as an expedient in modern mathematics. The most banal of these metaphors consists of calling a set of objects of the same type a “space” and the elements “points”. This has spread to such an extent (and it’s no paradox) that mathematicians throw off pictures of the space in order to warm to the objects created by them. Just think for example of the function spaces, whose introduction has thrown light on numerous problems in analysis. “It may seem surprising that a simple change in language has brought such progress. The impact that it produces seems to come from what might be called a transfer of intuition,” writes Jean Dieudonné in an article entitled Domination universelle de la géometrie. He adds: “in breaking out of its traditional boundaries, [geometry] has revealed its hidden powers, its flexibility and its extraordinary adaptation abilities, thus becoming one of the most used and universal tools in all branches of mathematics.” It also happens that a theory breathes new life by renewing contact with geometry in a rather unexpected way, so as Antaeus revived his strengths by contact with mother Earth: it’s been the case, in recent history, with probability theory, with topology under the geometrization of Thurston and Gromov, and with that part of functional analysis which, under Alain Connes’ influence has become noncommutative geometry. For Alain Connes, “a geometer is a person with sufficient vision to be able to create sufficient mental images that permit treating varied mathematical problems.” For “what is difficult and essential in mathematics is the creation of enough mental images to allow the brain to function.” An attempt at explanation was given by Michael Atiyah in his 2000 Fields Lecture Mathematics in the twentieth century: “Vision ... uses up something like 80 or 90 percent of the cortex of the brain. There are about 17 different centers in the brain, ... some parts are concerned with vertical, some parts with horizontal, some parts with color, perspective, finally some parts are concerned with meaning and interpretation. Understanding, and making sense of, the world that we see is a very important part of our evolution. Therefore spatial intuition or spatial perception is an enormously powerful tool, and that is why geometry is actually such a powerful part of mathematics not only for things that are obviously geometrical, but even for things that are not.” There remains in the cortex a place for algebra, which Atiyah associates with time, with the succession of events, of operations. For the geometer, this onedimensionality evokes the necessity of complying with the logical rules for proof, of metaphorically projecting our intuition onto a single axis. Geometers are often tempted to reject these rules when it is perceived that they bully the intuition, but know from experience that it’s precisely these that lead to pushing our imagination



beyond its limits so as to create completely new mathematical objects, as we attempt to show in this work. As for the book’s structure, we have on the one hand grouped these problems by their nature, affinity and similarity. For the most part the chapter can be read independently. Nevertheless, together with our book Geometry [B] for details beyond the conceptual, this work can serve for a course in geometry as seen from the cultural aspect. We can in fact perceive the various chapters as extensions of [B], illuminating it with some very recent results. We have therefore used [B] as a systematic reference for “elementary” geometry. Although biased, this choice is justified since only [B] treats all the notions used here; conversely, for each particular subject there are so many books that it is hopeless to give systematic references. But more important is the fact that, in contrast to [B], the results studied are not, with rare exceptions, proved in detail. To lighten the reading, some definitions have been placed at the ends of chapters under the rubric XYZ. Only the crucial ideas and above all the abstract concepts introduced for attaining these results are elucidated. In this respect we follow in the steps inaugurated by the absolutely remarkable book Geometry and the Imagination by Hilbert & Cohn-Vossen (original German, 1932, English translation, 1952), which filled a need for modern and easily accessible cultural geometry. It is this book that we hope to emulate, for it seems to us that a modern version is now much needed. This can only be at the price of a huge increase in size, given the exponential growth of results since the appearance in German in 1932 of Hilbert’s course. In our preface to the 1996 republication (Hilbert & Cohn-Vossen, 1996) we emphasized that the work is not intended to be read from the first to the last page, but that we rather hoped that the reader would open it at random and page through it and plunge into this or that chapter with some pleasure depending on intuition and inclination. Will we likewise here be able to transmit our conviction that geometry is especially alive and that there are still innumerable ways to be explored and concepts to be created? It is important to state that we have by no means covered all the directions of contemporary geometry. Thus we have made but little room for geometric probability, very little for combinatorics and none at all for some recent extensions of the notions of space and point. For this Cartier (1998) and Chap. 3 of Gromov (1999) may be consulted. A good reference for combinatorial geometry is Pach and Argawal (1995); we also find much that is well presented in Aigner and Ziegler (1998). A good idea of several new directions in mathematics and contemporary geometry can be obtained from the recent Carbone, Gromov and Prusinkiewisz (2000). We should add that mathematics today is advancing extremely fast. We must therefore alert the reader and especially the researcher that we are certainly not completely up-to-date in all subjects treated. This book began as a course given at the University of Pennsylvania in the fall of 1994, at SUNY Stony Brook in the winter of 1995 and at ETH Zürich in 1995–1996. We extend our gratitude to those mathematics departments for making it possible for us to offer courses which diverged very noticeably from traditional instruction.



For the editing we are greatly indebted to the Catherine and André Bellaïche, who bore the overall responsibility for the French edition for Editions Cassini and carried out important work in its realization. From the scientific point of view, practically all chapters have been painstakingly reviewed and edited by outside experts. These generous colleagues are: Patrick Popescu-Pampu for Chap. VII, Daniel Meyer for Chaps. VI and VIII, Pierre Arnoux and Sylvain Gallot for Chaps. XI and XII. Their criticisms and additions, sometimes very detailed, have been essential. André Bellaïche gave the entire text a final critical reading with special attention to Chaps. I and IX, where he rewrote several passages. Donal O’Shea provided many valuable comments on the early chapters during the translation process. To these friends I add with pleasure the name of Lester Senechal, who made the present English translation for Springer-Verlag with great dispatch, considering the compass of the book, and in the course of translation offered numerous remarks, corrections and criticisms important for the completion of this work. Bibliography [B] Berger, M. (1987, 2009) Geometry I,II. Berlin/Heidelberg/New York: Springer Aigner, M., & Ziegler, G. (1998, 4th ed. 2010). Proofs from the book. Berlin/Heidelberg/New York: Springer Atiyah, M. (2002). Mathematics in the 20th century. The Bulletin of the London Mathematical Society, 34, 1–15 Carbone, A., Gromov, M., & Prusinkiewisz, P. (2000). Pattern formation in biology and dynamics. Alghero: World Scientific Cartier, P. (1998). La folle journée, de Grothendieck à Kontsevich. Bulletin of the American Mathematical Society, 38, 389–408 Dieudonné, J. (1980). The universal domination of geometry. Berkeley: International Congress of Mathematical Education IV Dieudonné, J. (1981). Domination universelle de la géométrie (traduction du précédent). IREM de Paris-Nord Gromov, M. (1999), Metric structures for Riemannian and non-Riemannian spaces. Basel: Birkhäuser Hilbert, D., & Cohn-Vossen, S. (1932, 1996). Anschauliche geometrie. Berlin/Heidelberg/New York: Springer Hilbert, D., & Cohn-Vossen, S. (1952). Geometry and the imagination (English translation). New York: Chelsea Pach, J., & Argawal, P. (1995). Combinatorial geometry. New York: Wiley

Table of Contents About the Author




Chapter I. Points and lines in the plane In which setting and in which plane are we working? And right away an utterly simple problem of Sylvester about the collinearity of points I.2. Another naive problem of Sylvester, this time on the geometric probabilities of four points : : : : : : : : : : : : : : : : : : : : : : I.3. The essence of affine geometry and the fundamental theorem : : : : : I.4. Three configurations of the affine plane and what has happened to them: Pappus, Desargues and Perles : : : : : : : : : : : : : : : : : : : : : I.5. The irresistible necessity of projective geometry and the construction of the projective plane : : : : : : : : : : : : : : : : : : : : : : : : : I.6. Intermezzo: the projective line and the cross ratio : : : : : : : : : : : I.7. Return to the projective plane: continuation and conclusion : : : : : : I.8. The complex case and, better still, Sylvester in the complex case: Serre’s conjecture : : : : : : : : : : : : : : : : : : : : : : : : : : : I.9. Three configurations of space (of three dimensions): Reye, Möbius and Schläfli : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : I.10. Arrangements of hyperplanes : : : : : : : : : : : : : : : : : : : : : I. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :



Chapter II. Circles and spheres II.1. II.2. II.3. II.4. II.5.

II.6. II.7. II.8.

1 6 12 17 23 28 31 40 43 47 48 57 61

Introduction and Borsuk’s conjecture : : : : : : : : : : : : : : : : : A choice of circle configurations and a critical view of them : : : : : A solitary inversion and what can be done with it : : : : : : : : : : How do we compose inversions? First solution: the conformal group on the disk and the geometry of the hyperbolic plane : : : : : : : : : Second solution: the conformal group of the sphere, first seen algebraically, then geometrically, with inversions in dimension 3 (and three-dimensional hyperbolic geometry). Historical appearance of the first fractals : : : : : : : : : : : : : : : : : : : : : : : : : : Inversion in space: the sextuple and its generalization thanks to the sphere of dimension 3 : : : : : : : : : : : : : : : : : : : : : Higher up the ladder: the global geometry of circles and spheres : : : Hexagonal packings of circles and conformal representation : : : : : XI

61 66 78 82

87 91 96 103



II.9. Circles of Apollonius : : : : : : : : : : : : : : : : : : : : : : : : : II. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

113 116 137

Chapter III. The sphere by itself: can we distribute points on it evenly?


III.1. The metric of the sphere and spherical trigonometry : : : : : : : : : III.2. The Möbius group: applications : : : : : : : : : : : : : : : : : : : : III.3. Mission impossible: to uniformly distribute points on the sphere S2 : ozone, electrons, enemy dictators, golf balls, virology, physics of condensed matter : : : : : : : : : : : : : : : : : : : : : : : : : : : III.4. The kissing number of S2 , alias the hard problem of the thirteenth sphere III.5. Four open problems for the sphere S3 : : : : : : : : : : : : : : : : : III.6. A problem of Banach–Ruziewicz: the uniqueness of canonical measure III.7. A conceptual approach for the kissing number in arbitrary dimension III. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

141 147

149 170 172 174 175 177 178

Chapter IV. Conics and quadrics


Motivations, a definition parachuted from the ladder, and why : : : Before Descartes: the real Euclidean conics. Definition and some classical properties : : : : : : : : : : : : : : : : : : : : : : : : : : IV.3. The coming of Descartes and the birth of algebraic geometry : : : : IV.4. Real projective theory of conics; duality : : : : : : : : : : : : : : : IV.5. Klein’s philosophy comes quite naturally : : : : : : : : : : : : : : IV.6. Playing with two conics, necessitating once again complexification : IV.7. Complex projective conics and the space of all conics : : : : : : : : IV.8. The most beautiful theorem on conics: the Poncelet polygons : : : : IV.9. The most difficult theorem on the conics: the 3264 conics of Chasles IV.10. The quadrics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : IV. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

181 183 198 200 205 208 212 216 226 232 242 245

Chapter V. Plane curves


IV.1. IV.2.

V.1. V.2. V.3. V.4. V.5. V.6.

Plain curves and the person in the street: the Jordan curve theorem, the turning tangent theorem and the isoperimetric inequality : : : : : : : What is a curve? Geometric curves and kinematic curves : : : : : : : The classification of geometric curves and the degree of mappings of the circle onto itself : : : : : : : : : : : : : : : : : : : : : : : : The Jordan theorem : : : : : : : : : : : : : : : : : : : : : : : : : : The turning tangent theorem and global convexity : : : : : : : : : : Euclidean invariants: length (theorem of the peripheral boulevard) and curvature (scalar and algebraic): Winding number : : : : : : : :

249 254 257 259 260 263




The algebraic curvature is a characteristic invariant: manufacture of rulers, control by the curvature : : : : : : : : : : : : : : : : : : V.8. The four vertex theorem and its converse; an application to physics V.9. Generalizations of the four vertex theorem: Arnold I : : : : : : : : V.10. Toward a classification of closed curves: Whitney and Arnold II : : V.11. Isoperimetric inequality: Steiner’s attempts : : : : : : : : : : : : : V.12. The isoperimetric inequality: proofs on all rungs : : : : : : : : : : V.13. Plane algebraic curves: generalities : : : : : : : : : : : : : : : : : V.14. The cubics, their addition law and abstract elliptic curves : : : : : V.15. Real and Euclidean algebraic curves : : : : : : : : : : : : : : : : V.16. Finite order geometry : : : : : : : : : : : : : : : : : : : : : : : : V. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : : : : : : : : : :

Chapter VI. Smooth surfaces

269 271 278 281 295 298 305 308 320 328 331 336 341


Which objects are involved and why? Classification of compact surfaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : VI.2. The intrinsic metric and the problem of the shortest path : : : : : : VI.3. The geodesics, the cut locus and the recalcitrant ellipsoids : : : : : VI.4. An indispensable abstract concept: Riemannian surfaces : : : : : : VI.5. Problems of isometries: abstract surfaces versus surfaces of E3 : : : VI.6. Local shape of surfaces: the second fundamental form, total curvature and mean curvature, their geometric interpretation, the theorema egregium, the manufacture of precise balls : : : : : : : : : : : : : VI.7. What is known about the total curvature (of Gauss) : : : : : : : : : VI.8. What we know how to do with the mean curvature, all about soap bubbles and lead balls : : : : : : : : : : : : : : : : : : : : : : : : VI.9. What we don’t entirely know how to do for surfaces : : : : : : : : VI.10. Surfaces and genericity : : : : : : : : : : : : : : : : : : : : : : : VI.11. The isoperimetric inequality for surfaces : : : : : : : : : : : : : : VI. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

380 386 391 397 399 403

Chapter VII. Convexity and convex sets


VII.1. VII.2. VII.3. VII.4. VII.5. VII.6. VII.7.

History and introduction : : : : : : : : : : : : : : : : : : : : Convex functions, examples and first applications : : : : : : : Convex functions of several variables, an important example : : Examples of convex sets : : : : : : : : : : : : : : : : : : : : Three essential operations on convex sets : : : : : : : : : : : : Volume and area of (compacts) convex sets, classical volumes: Can the volume be calculated in polynomial time? : : : : : : : Volume, area, diameter and symmetrizations: first proof of the isoperimetric inequality and other applications : : : : : :

: : : : :

341 345 347 357 361

364 373

: : : : :

409 412 415 417 420

: :


: :





Volume and Minkowski addition: the Brunn-Minkowski theorem and a second proof of the isoperimetric inequality : : : : : : : : VII.9. Volume and polarity : : : : : : : : : : : : : : : : : : : : : : : VII.10. The appearance of convex sets, their degree of badness : : : : : : VII.11. Volumes of slices of convex sets : : : : : : : : : : : : : : : : : VII.12. Sections of low dimension: the concentration phenomenon and the Dvoretsky theorem on the existence of almost spherical sections : : : : : : : : : : : : : : : : : : : : : : : : : VII.13. Miscellany : : : : : : : : : : : : : : : : : : : : : : : : : : : : VII.14. Intermezzo: can we dispose of the isoperimetric inequality? : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

: : : :

439 444 446 459

: : : :

470 477 493 499

Chapter VIII. Polygons, polyhedra, polytopes Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Basic notions : : : : : : : : : : : : : : : : : : : : : : : : : : : Polygons : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Polyhedra: combinatorics : : : : : : : : : : : : : : : : : : : : Regular Euclidean polyhedra : : : : : : : : : : : : : : : : : : Euclidean polyhedra: Cauchy rigidity and Alexandrov existence Isoperimetry for Euclidean polyhedra : : : : : : : : : : : : : : Inscribability properties of Euclidean polyhedra; how to encage a sphere (an egg) and the connection with packings of circles : : VIII.9. Polyhedra: rationality : : : : : : : : : : : : : : : : : : : : : : VIII.10. Polytopes (d > 4): combinatorics I : : : : : : : : : : : : : : : VIII.11. Regular polytopes (d > 4) : : : : : : : : : : : : : : : : : : : : VIII.12. Polytopes (d > 4): rationality, combinatorics II : : : : : : : : : VIII.13. Brief allusions to subjects not really touched on : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : VIII.1. VIII.2. VIII.3. VIII.4. VIII.5. VIII.6. VIII.7. VIII.8.

505 : : : : : : :

505 506 508 513 518 524 530

: : : : : : :

532 537 539 544 550 555 558

Chapter IX. Lattices, packings and tilings in the plane IX.1. IX.2. IX.3. IX.4. IX.5. IX.6. IX.7. IX.8.

Lattices, a line in the standard lattice Z2 and the theory of continued fractions, an immensity of applications : : : : : : : : : : : : : : : Three ways of counting the points Z2 in various domains: pick and Ehrhart formulas, circle problem : : : : : : : : : : : : : : : : Points of Z2 and of other lattices in certain convex sets: Minkowski’s theorem and geometric number theory : : : : : : : : : : : : : : : : Lattices in the Euclidean plane: classification, density, Fourier analysis on lattices, spectra and duality : : : : : : : : : : : : : : : : : : : : Packing circles (disks) of the same radius, finite or infinite in number, in the plane (notion of density). Other criteria : : : : : : : : : : : : Packing of squares, (flat) storage boxes, the grid (or beehive) problem Tiling the plane with a group (crystallography). Valences, earthquakes Tilings in higher dimensions : : : : : : : : : : : : : : : : : : : : :

563 563 567 573 576 586 593 596 603




Algorithmics and plane tilings: aperiodic tilings and decidability, classification of Penrose tilings : : : : : : : : : : : : : : : : : : : IX.10. Hyperbolic tilings and Riemann surfaces : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

607 617 620

Chapter X. Lattices and packings in higher dimensions


X.1. Lattices and packings associated with dimension 3 : : : : : : : : : X.2. Optimal packing of balls in dimension 3, Kepler’s conjecture at last resolved : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : X.3. A bit of risky epistemology: the four color problem and the Kepler conjecture : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : X.4. Lattices in arbitrary dimension: examples : : : : : : : : : : : : : : X.5. Lattices in arbitrary dimension: density, laminations : : : : : : : : X.6. Packings in arbitrary dimension: various options for optimality : : : X.7. Error correcting codes : : : : : : : : : : : : : : : : : : : : : : : : X.8. Duality, theta functions, spectra and isospectrality in lattices : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :





: : : : : : :

639 641 648 654 659 667 673

Chapter XI. Geometry and dynamics I: billiards



Introduction and motivation: description of the motion of two particles of equal mass on the interior of an interval : : : : : : : : : : : : : XI.2. Playing billiards in a square : : : : : : : : : : : : : : : : : : : : : XI.3. Particles with different masses: rational and irrational polygons : : : XI.4. Results in the case of rational polygons: first rung : : : : : : : : : : XI.5. Results in the rational case: several rungs higher on the ladder : : : XI.6. Results in the case of irrational polygons : : : : : : : : : : : : : : XI.7. Return to the case of two masses: summary : : : : : : : : : : : : : XI.8. Concave billiards, hyperbolic billiards : : : : : : : : : : : : : : : : XI.9. Circles and ellipses : : : : : : : : : : : : : : : : : : : : : : : : : XI.10. General convex billiards : : : : : : : : : : : : : : : : : : : : : : : XI.11. Billiards in higher dimensions : : : : : : : : : : : : : : : : : : : : XI.XYZ Concepts and language of dynamical systems : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

675 679 689 692 696 705 710 710 713 717 728 730 735

Chapter XII. Geometry and dynamics II: geodesic flow on a surface


Introduction : : : : : : : : : : : : : : : : : : : : : : : : : Geodesic flow on a surface: problems : : : : : : : : : : : : Some examples for sensing the difficulty of the problem : : Existence of a periodic trajectory : : : : : : : : : : : : : : Existence of more than one, of many periodic trajectories; and can we count them? : : : : : : : : : : : : : : : : : : : XII.6. What behavior can be expected for other trajectories? Ergodicity, entropies : : : : : : : : : : : : : : : : : : : : : XII.1. XII.2. XII.3. XII.4. XII.5.

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739 741 743 751

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XII.7. Do the mechanics determine the metric? XII.8. Recapitulation and open questions : : : XII.9. Higher dimensions : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : :


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779 781 781 782

Selected Abbreviations for Journal Titles


Name Index


Subject Index


Symbol Index


Chapter I

Points and lines in the plane I.1. In which setting and in which plane are we working? And right away an utterly simple problem of Sylvester about the collinearity of points We first work in the coordinate plane, which is familiar to everyone, with its points and lines. As is usual in the “elementary” geometry of school instruction, this has to do with Euclidean geometry, where there are distances (lengths), angles, circles, etc. This will also be the setting of the next chapter, but even in this first chapter we will see that we can already do many subtle and difficult things and even find open questions with only the so-called “affine plane”. Affine geometry is a weaker structure than Euclidean geometry. Simply put: we won’t be working with anything but points and lines; the mathematical definition is given in Sect. I.XYZ at the end of the chapter. Here we need only recall: two distinct points uniquely determine a line that contains them, along with a segment that joins them; two distinct lines intersect in a single point, with the sole exception of parallel lines. Regarding these, through each point exterior to a given line there passes a unique parallel to that line. Finally, there is a supplementary affine notion, more subtle than the merely set-theoretic ones of point and of line, which is the affine invariant attached to three collinear points: if a, b, c are collinear, there exists a real number (and one only) denoted ab . It indicates a ratio (that can be negative, although negative numbers had ac been long forbidden in geometry, even by Poncelet and d’Alembert, until Chasles actually gave them rights of citizenship), a ratio obtainable by parameterizing the line considered, but which does not depend on this parameterization. We can thus speak about the midpoint of a segment, the third-way point, etc. See the necessary details in Sects. I.XYZ and I.3 below. The precise mathematical language is that of the real affine plane. If we adjoin a metric which we permit ourselves occasionally, even in this chapter we then speak of a Euclidean plane. An important remark about language: we can speak of “the”, rather than “an”, affine plane. For in fact any two affine planes are necessarily isomorphic, just as are two real vector spaces of dimension 2. The same remark applies to Euclidean spaces of any dimension. But in this introductory chapter we will see that it is practically impossible to remain in the affine setting: to comprehend and unify certain things, by Sect. I.4 we will need to climb the ladder, know how “to go to infinity” and not interject the Euclidean plane but more subtly define the “projective plane”. The degree of subtlety can be seen historically: projective geometry wasn’t defined until Desargues in the 1650s and then only heuristically. The sound algebraic construction, following

M. Berger, Geometry Revealed, DOI 10.1007/978-3-540-70997-8_1, c Springer-Verlag Berlin Heidelberg 2010




the synthetic attempts of Poncelet and Chasles in the years 1820–1840, was made in the 1850s by the German school: Plücker, von Staudt and Grassmann, whereas Euclid dates from 300 B.C. In 1893 Sylvester posed the following problem: (I.1.1) Let E be a finite set of points in the plane that has the following property: for an arbitrary pair of distinct points of E there exists, on the line joining them, a third point of E. Show that this is impossible, with the obvious exception of the case where E consists of collinear points on a single line. Some readers may prefer the equivalent formulation: (I.1.2) If E is a finite set of points in the plane not composed of points belonging to a single line, then there exists at least one line that contains only two of its points.

Fig. I.1.1.

As an exercise we can attempt to convince ourselves of Sylvester’s conjecture by making some sketches: we quickly see that we are forced into constructing sets having an infinite number of points. But in spite of this easily won insight, it was not until 1932 that there was a proof of this conjecture, found by Gallai. We owe to Kelly in (Kelly, 1948) a proof that uses the following Euclidean argument: if the points are not all collinear, there is a triple of non-collinear points a, b, c of E forming a true triangle such that the distance from a to the line bc is minimum among all such triples. We already have a contradiction if b and c are of the same side of the altitude from a, for then the distance from b to ac, or else that of c to ab, is less than that of a to bc. So b and c must be on opposite sides of the base of the altitude. But there exists by hypothesis a third point d of E on bc, and we are led to a new contradiction by considering either the triangle abd or the triangle acd . But this proof leaves us with a bad taste if we are at all purist: the problem is strictly affine and we should be able to prove the conjecture in a purely affine manner, without the aid of Euclidean geometry. The purely affine proof of Gallai



Fig. I.1.2.

is found on p. 181 of Coxeter (1989) . Courageous readers may attempt to find one of their own; but it is important to note that none of the proofs cited so far is combinatorial in the sense that in a combinatorial proof we compute the number of points on this or that line, how many lines joining two points of the set pass through a given point, etc., hoping to find relations that contradict the initial hypothesis. On the contrary, Gallai’s proof uses the fact that a line in the plane divides it into two distinct connected regions; we can’t pass from one to the other without intersecting the line. Apart from that, Gallai’s proof doesn’t introduce any new concept. Where then is Jacob’s ladder? We will climb it in two different ways, but reluctant readers may skip immediately to the next section and the second problem of Sylvester. The first way of ascending the ladder provides a conceptual and combinatorial proof of Sylvester’s conjecture, due to Melchior in 1940; details can be found in Chaps. 8 and 10 of Aigner and Ziegler (1998). We now use some tools whose motivation will be given subsequently: we extend (see Sects. I.7 and I.XYZ) the real affine plane under consideration to a real projective plane P. There we consider not the finite set E of points satisfying Sylvester’s condition, but its dual, i.e. the (necessarily finite) set of lines D dual to E under a duality of P .



A duality consists of two mappings: the first associates with each point a of P a line denoted by a ; the second associates with each line d a point denoted by d . The fundamental properties of a duality are the following: – the mappings a ! a and d ! d are inverse to each other; – if the line d passes through the point a, then the line a passes through the point d . For example, corresponding to two points a and b lying on d , there are the two lines a and b intersecting in the point d . For a complete definition, see Sect. I.7.

Fig. I.1.3. Duality between points and lines in the projective plane

For the configuration of points and lines provided by D we then obtain combinatorial relations between the two sequences of integers fpr g and ftr g defined as follows: pr is the number of polygons of r sides that are found in the cellular decomposition of P that D defines, while tr is the number of points that lie on r lines of D. We have the following relations, where f0 , f1 , f2 denote the respective numbers P P of vertices, edges and polygons of the cellular decomposition: f0 D tr , f2 D pr , P P rtr D 12 rpr . But algebraic topology (see the combinatorics of polyhef1 D dra in Sect. VIII.4) tells us that, for the surface P , the Euler-Poincaré characteristic f0 f1 C f2 equals 1. To prove Sylvester’s conjecture, we need to prove that t2 > 1 (which implies that in the configuration defined by D there is one point lying on two lines and, in that defined by E, one line that contains only two points). Suppose to the contrary that we only have tr > 0 when r > 3. The Euler-Poincaré formula yields on the one hand X X X X tr C pr D 1 C rtr > 1 C 3 tr and, on the other, X

tr C


pr D 1 C

1X 3X rpr > 1 C pr : 2 2



Upon multiplying the first relation by 13 , the second by 23 and adding, we obtain a contradiction. In Aigner and Ziegler (1998) or Aigner and Ziegler (2003) there is a variant of the proof by central projection, due to Steenrod, using graph theory and spherical geometry:

c G. M. Ziegler Fig. I.1.4. Aigner, Ziegler (1998)

The second ascent entails introducing concepts of the complex affine plane and the planar cubic. We will see in Sect. V.14 that a generic cubic in the complex plane possesses nine distinct inflection points and, most importantly, that each two inflection points have the property that the line that joins them intersects the cubic once again in an inflection point. Sylvester’s conjecture is thus false in the complex plane. This isn’t so surprising, for we can’t apply reasoning “a la Gallai” for the reason that a line in the complex plane only determines a single connected region. With some planar algebraic geometry, as in Sect. V.14, it is also easy to see that each complex planar configuration of nine points satisfying Sylvester’s condition is equivalent to the one described above. However, there does exist an extension of Sylvester’s result to complex affine geometry, necessarily of dimension higher than two, as will be seen in Sect. I.8, that requires a very high ascent on Jacob’s ladder. Finally, a result such as (I.1.2) will not completely satisfy a mathematical intellect, requiring as it does for the set E merely the existence of at least one line that contains only two of its points. A few sketches will convince readers that we might prove a stronger result, of a sort such as this: we will say that a line associated with a finite set of points is ordinary if it contains but two points of the set. We denote by t .n/ the minimum number of ordinary lines of a set E of n noncollinear points. Theorem (I.1.2) states that we always have t .n/ > 1 for each integer n, but we might suppose that t .n/ may be rather large with increasing n. The question isn’t yet settled. Here briefly is the present state of affairs; for more details and references see Problem F12 in Croft, Falconer and Guy (1991), Chap. 8 of Aigner and Ziegler (2003), and the second part of Pach and Agarwal (1995) which is more conceptual and also the Introduction, p. 679, of Vol. II of Hirzebruch (1997). The exact general value of t .n/ is unknown; the best we know presently is that



we always have t > Œn=2 (integer part of n=2), which is due to Hansen, but we don’t have an optimal answer. Moreover the proof of this result of Hansen doesn’t at the moment seem to bring with it any new concept. Nevertheless, knowledge of the combinatorics of arrangements of lines in the real plane has recently increased considerably, see the reference Pach and Agarwal (1995). Finally, for the complex case, see Sect. I.8. For their aesthetic aspect and their naturalness, the configurations called Sylvester-Gallai remain much studied; see for example Bokowski and Richter-Gebert (1992). The name Erdös deserves special mention. Beyond his numerous results and his innumerable lectures, he was known first for having a rather long waiting line of researchers at the end of his lectures. Each in his turn would say: “Professor Erdös, I don’t know how to settle this or that question”. Almost invariably the response would be: “Here’s how to do it. Write the article, we’ll sign it jointly”. Given then the innumerable articles written jointly with him, practically every mathematician of a certain age appears as a connected component of Erdös and even possesses an Erdös number defined thus: it’s the minimum number of elements in a chain of several articles which ends with an article written jointly with Erdös. Your humble author didn’t escape either; his Erdös number equals 3, via Aryeh Dvoretsky (who has seven articles jointly with Erdös if we want to compute an Erdös valence) and Eugenio Calabi. Another of Erdös’s striking traits was his ease in making conjectures. For many of them he actually offered compensation (which he always paid) up to five thousand dollars, and he and his purse might thank the deity that he rarely deceived himself regarding their difficulty. I.2. Another naive problem of Sylvester, this time on the geometric probabilities of four points The second (still purely affine) problem of Sylvester from 1865 treats the arrangement of a quadruple of points in the affine plane: only two arrangements are possible (in the generic case, where three points are never collinear), either they form a convex quadrilateral or one of the points lies in the interior of the triangle formed by the other three. Then there is the natural question: (I.2.1) If four points are thrown randomly at the plane, what are the probabilities for obtaining one or the other of the possible configurations?

Fig. I.2.1.



There are really only two cases to consider; the degenerate ones have probability zero. But, for the question to make sense, i.e. in order to have a good notion of probability, we take as our target a planar domain D that is bounded and everywhere convex. The theoretical answer is then quite simple, the probability of obtaining four points such that one of them lies in the triangle formed by the other three is given by the triple integral Z Z Z 4 .I:2:2/ Sylv.D/ D Area.x1 ; x2 ; x3 / dx1 dx2 dx3 Area4 .D/ D D D where we integrate over all triples of points of D (i.e. over all the triangles contained in D) and where Area.x1 ; x2 ; x3 / denotes the area of the triangle with vertices x1 , x2 , x3 . The proof is very simple: the probability that the first three points fall respectively in x1 Cdx1 , x2 Cdx2 , x3 Cdx3 is dx1 dx2 dx3 = Area3 .D/. Knowing this, the probability that the fourth point is in the interior of the triangle formed by the first three is Area.x1 ; x2 ; x3 /=Area.D/. From this we get the formula by observing that the event considered is the union of four mutually exclusive events of equal probability.

Fig. I.2.2.

The probability of having four points that form a convex quadrilateral is then simply equal to 1 Sylv.D/. The value of Sylv.D/ depends on the “shape” of the 35 domain D considered; we have Sylv.D/ D 13 for an arbitrary triangle and 12 2 for an arbitrary ellipse. These results should give us much to think about. First, the value is the same for all triangles and for all ellipses. The reason is simple: in affine geometry, all triangles are “the same”, all ellipses are “the same”. We will return to all this amply in Sect. I.3, where we introduce notions that permit us to clarify what we mean by “the same”. It will be noted that Sylvester’s condition is purely affine. We now observe that, in Euclidean geometry, we clearly no longer have such equivalences for similarly shaped domains. The two values above show in any case that the probability of having a quadrilateral is significantly lower for triangles than for ellipses. This is intuitive enough: when we take three points, each of which is close to a vertex of the triangle, there remains but little space for the fourth point outside the new triangle thus formed. In contrast, near the boundary of a round domain, we have more space. It is important



to go further, since up to this point we know nothing about other domains. This problem was settled by Blaschke in 1917: We always have

35 1 6 Sylv.D/ 6 2 12 3

for any domain whatsoever.

And surely our curiosity won’t be completely satisfied until we know that Blaschke also showed that equality isn’t attained for the lower and upper bounds except by triangles and ellipses, respectively: a nice characterization of triangles and ellipses! See Note I.4.5 of Santalo (1976) and Klee (1969) and Sect. 5.2 of Gruber and Wills (1993). We give here the two-fold idea of Blaschke. For the left inequality we use the Steiner symmetrization which we will encounter several times in Chap. VII (beginning in Sect. VII.5.A), but why not make quick use of it right off? It is described on the diagram: with each convex domain D and each linear direction is associated the symmetrization sym .D/ of D for the direction .

Fig. I.2.3.

Blaschke shows that each symmetrization can only diminish the integral (I.2.2). This is easy enough to perceive intuitively, for a few sketches quickly convince us that the symmetrization of a triangle interior to D often becomes a quadrilateral in sym .D/. Furthermore the diminution is strict provided that the convex set is not symmetric with respect to the direction considered. Knowing this, we effect some symmetrizations about well chosen lines (the directions alone matter), for example, by taking lines with inclinations that are irrational multiples of . For inequality in the reverse sense, Blaschke introduces the notion of cosymmetrization, which we haven’t encountered anywhere else in the geometric literature. It is easy to see that cosymmetrization, conversely, strictly increases the integral (I.2.2). We approximate D by polygons and thus obtains a reduction to the polygonal case. For the polygons for which one direction is orthogonal to a line



Fig. I.2.4.

joining two nonconsecutive sides, the cosymmetrization always has at least one vertex fewer than the initial polygon and we end up with a triangle, Q.E.D. Problem (I.2.1) seems well in hand, but in fact we have cheated a bit in requiring that the domain D be convex and bounded in order for the notion of probability to make sense. In truth it suffices for D to have finite area, which doesn’t preclude “passage to infinity”. Note that such a domain, extending to infinity, cannot except for some very special cases be convex and that Sylvester’s second problem, cited frequently only for convex sets, continues to make sense for all sets of finite area. The four points may be in the domain, but the quadrilateral they determine may emerge from it, which actually needn’t trouble us; it suffices to replace, in formula (I.2.2), Area.x1 ; x2 ; x3 / by Area.Triangle.x1 ; x2 ; x3 / \ D/. This more general non compact study was undertaken only very recently and is not yet well understood. Here is what we know, a recent reference being (Scheinerman and Wilf, 1994): on the one hand, the shape that yields the lower bound p 1 Sylv.D/ over all D isn’t known or even conjectured precisely. On the other hand this work provides a result that is amazing at first glance: even though we don’t know the exact value of the optimal probability, it is possible to show that it coincides with another number, also unknown and extensively studied in combinatorial geometry, see Pach and Agarwal (1995), for it is related to planar realizations of the complete graph Kn with n vertices (a graph is complete when every pair of vertices is joined by an edge, and a result of Fany states that we only need use segments for joining vertices). Let .Kn / be the minimum number of points of intersection of the edges in an arbitrary planar realization of a complete graph Kn . By putting all the points on a circle, we get C4n intersections. Readers may find a smaller number with other examples, but a classical result states that n4 is the right order of magnitude. More precisely, there exists a positive real number (finite) such that .Kn / D : n!1 C4 n lim

The amazing result is that D p.



Fig. I.2.5.

Fig. I.2.6. A very nice complete graph with 11 vertices, due to H. Jensen

For the proof, the connection between the two concepts is achieved thus: we choose n points in D at random and in a probabilistically independent manner. It is necessary to prove two opposite inequalities. In the one direction, we start with an optimal complete graph and surround each of its vertices with a small disk of radius ". We can choose " sufficiently small so that random points taken in this collection of disks yield another optimal graph. We then study Sylvester’s probability, choosing for D the union of these disks; for " sufficiently small we obtain the required inequality. Roughly speaking, 1 Sylv.D/ is equal to the probability that the four points chosen at random form a convex set, i.e. the probability that among the three possible groupings of edges ab cd , ad bc, ac bd , one of them gives n/ rise to an intersection is .K . Thus we have roughly: C4 n

p D minD .1 Sylv.D// 6

.Kn / : C4n

More precisely, it is necessary to take into account the circ*mstance that the four points chosen may have the bad sense not to fall into four different small disks; but the asymptotic behavior of this bad case is in total of the order of O. n1 / and thus goes to zero as n goes to infinity.



In the reverse direction, we start with any domain D in which we choose n points fpi g at random and probabilistically independently, and we assume that these n points are the vertices of a complete linear graph Kn . Then the number c of crossings of this graph is a random variable whose value is always at least .Kn /. Moreover, consider the random variable X XD 1fpa ; pb ; pc ; pd g ; a;b;c;d

where the sum is taken over all quadruples of f1; : : : ; ng and where 1f;;;g is a random indicator that equals 1 if the convex envelope of fpa ; pb ; pc ; pd g is a convex quadrilateral, and equals 0 otherwise. Since a random graph can’t have more crossings than the mean, we have .Kn / 6 E.X/ for the mathematical expectation of X. The desired result is obtained by letting n go to infinity. The optimal shape of D isn’t known, as already mentioned. We haven’t yet finished with the problem (I.2.1), which violates the strict rules of the game: staying in the plane, in dimension two. (I.2.2) We randomly throw 5 points at a bounded region D of three dimensional space; what is the probability that they form a true polyhedron with five vertices? And the same problem with n C 2 points in the space of n dimensions. As before we compute the complementary probability to find the probability that the fifth point is in the interior of the tetrahedron formed by the four others. The formula is the strict generalization of that given above for an arbitrary dimension, which in dimension three will be: Z Z Z Z 5 Sylv.D/ D Volume.x1 ; x2 ; x3 ; x4 / dx1 dx2 dx3 dx4 ; Area5 .D/ D D D D where Volume.x1 ; x2 ; x3 ; x4 / denotes the volume of the tetrahedron with vertices x1 , x2 , x3 , x4 . Here half the problems remain open at present; we only know what happens on one side of the conceivable inequalities. First, the value is known for ellipsoids (here again, there is but one ellipsoid in affine geometry, in which we continue to be situated, given the nature of the problem (I.2.2)); this is due to Klingman in 1969. In each dimension d he finds for the ellipsoid E d (the binomial coefficients have their usual sense when d is odd and we use the gamma function to define the p necessary factorials when d is even; the gamma function provides a factor ): d C1 . C1/=2 .d C1/2 =2 C : Sylv.E d / D 2d C.d d C1 .d C1/2 In 1973, Grömer showed conversely that this value is attained only for ellipsoids. The value in question is thus a rational number when d is odd, and a rational multiple of d when d is even. The method of proof is again the Steiner symmetrization, which is viable in all dimensions and which we will continue to encounter in



Sects. V.11 and VII.8. A recent reference is Sect. 5.2 of Gruber and Wills (1993) where there is a nice conceptual treatment. On the other hand, for the maximum value, three problems remain open. Is it attained for tetrahedrons (in dimensions greater than three we say simplex)? Does it characterize the tetrahedrons? But above all, how can we calculate the above integral for tetrahedrons? Readers may find such ignorance surprising for so simple and ordinary a geometric object as the tetrahedron. In Sect. III.6 we will encounter two other unresolved problems on the volumes of tetrahedrons in the three dimensional sphere S3 . Readers may also try to see why Blaschke’s cosymmetrization method doesn’t work in dimension 3 or greater. We will encounter the P.D/ in a remarkable way in Sect. VII.10. Many important results in this field have appeared quite recently; see a synthesis in Bárány (2008). In dimension 3 or more we will not be satisfied with only an estimate of P.D/ in the case of ellipsoids. The problem is to estimate .I:2:3/; Z Z Z nC2 P.D/ D Volume.x1 ; x2 ; : : : ; xnC1 / dx1 dx2 : : : dxnC1 ; AreanC2 .D/ D D D as a function of invariants attached to the convex set D, where we are dealing with the volume of the simplex generated by the n C 1 points x1 , x2 , . . . , xnC1 . We will find a partial answer in Sect. VII.10.F. A final comment: we have just seen for the first time an interaction between geometry and probability. Historically the original problem is that of Buffon’s needle; see the elementary exposition in Santalo (1976) and, for a contemporary treatment, Sect. 5.2 of Gruber and Wills (1993), already mentioned above. Recent directions in geometric research, in particular the Gromov’s approach with mm-spaces (see Sect. I.XYZ), seem to indicate that the notion of measure to which the notion of probability is equivalent is every bit as important in geometry as that of distance, of metric. We will encounter other uses of geometric probability in Chaps. VII, XI and XII. I.3. The essence of affine geometry and the fundamental theorem We will attempt as always without too much formalism to enter further into a vision of the real affine plane. If we want to characterize affine geometry according to the philosophy of Klein at the turn of the twentieth century, it is necessary to study its automorphisms, by which we mean the bijections that map the affine plane onto itself and preserve its structure: lines, collinearity of points, intersections of lines, etc. In the modern definition given in Sect. I.XYZ these are the linear transformations combined with translations and thus the transformations that can be written, in arbitrary coordinates: .x; y/ 7! .ax C by C c; a0 x C b 0 y C c 0 /, with the sole condition ab 0 a0 b 6D 0 for the six real numbers a, a0 , b, b 0 , c, c 0 . Before returning



to a purely geometric characterization of these automorphisms we will identify the affine invariants, that is to say the numbers, the situations, that are “respectable” and respected by all affine transformations. In any case, we must remember that affine transformations preserve lines (i.e. collinearity of points) and send parallel lines to parallel lines. We begin with points. Two points do not give rise to any invariant since there always exists an affine transformation taking an arbitrary pair of points to another arbitrary pair; and it’s the same for three points, which explains the fact noted above: all triangles in the affine plane are the same, are indistinguishable. This is furthermore plausible although this is not a proof because a set of three points depends on exactly 3 2 D 6 parameters and the affine transformations also depend on the 6 parameters written above: a, b, c, a0 , b 0 , c 0 . But if the three points considered are collinear we come upon the first affine invariant: for three collinear points a, b, c the real number denoted by ac is a characteristic invariant, i.e. it is preserved by ab every affine transformation, and two collinear triples a, b, c and a0 , b 0 , c 0 are transformable into each other if and only if the corresponding invariants are equal. This invariant may be defined thus: ac is the value of the (unique) coordinate of the point ab c on the line defined by a, b, c in a coordinate system where a is the origin and b is the point with coordinate equal to 1.

Fig. I.3.1.

As ac traverses the interval Œ0; 1, the point c traverses the segment Œa; b defined ab by a and b. The notion of segment is thus affine, as is that of midpoint: the midpoint of the segment Œa; b is the point c such that ac D 12 . Observe that this invariant ab is not Euclidean, but that if there is an additional Euclidean structure on our affine plane, then we may always compute it with the distances ab, ac (with assignment of the usual signs). Exercise: find conditions under which two sets of four (arbitrary) points can be transformed into one another. Passing now to lines, all lines are first of all the same; then, two pairs of lines are indistinguishable, under the obvious condition that they are simultaneously incident or simultaneously parallel. For three concurrent lines, there isn’t any invariant: two triples of concurrent lines can always be taken into each other by an appropriate affine transformation. But it isn’t the same for four concurrent lines Di (i D 1; 2; 3; 4): we can attach an invariant to them in a canonical manner, i.e. their cross ratio ŒD1 ; D2 ; D3 ; D4 . This is a characteristic invariant: we can define it in an



affine manner. But we won’t do this, for it is in fact a projective invariant as will be shown in an entirely natural and simple manner in Sect. I.6.

Fig. I.3.2.

Again, we can ask numerous questions on the subject of lines. Here is one of them: given two lines, three lines, or more, what is the number of possible configurations? For two or three, it’s easy. For two: either they are concurrent or parallel. Difficulties begin with three and we encourage readers to sketch, to scribble: the lines may be concurrent or form a true triangle. But we must not forget the possibility of parallels, whence two other configurations: three parallel lines or two parallels and a third that intersects them. We see that for four and more, things become difficult; in particular we begin to get frustrated by the parallels. Here we find an additional incentive for projective geometry: parallelism doesn’t exist!

Fig. I.3.3.

The affine transformations map each line into a line, but if we want to completely capture the essence of affine geometry, e.g. by a purely axiomatic definition



(without a vector space, etc.), we will want to be sure that there don’t exist other transformations beyond the affine ones defined above of the affine plane to itself that transform each line into a line, i.e. that preserve the collinearity of points. We have a completely satisfactory answer to this question: (I.3.1) (Fundamental theorem of affine geometry) Each bijection of the affine plane to itself that takes lines into lines is an affine transformation. It is impossible to pass over the idea of the proof in silence, as much for its beauty and conceptual importance as for its allowing us to imagine what will happen in affine geometries over fields other than the reals complex numbers, quaternions, etc. that will be encountered in Sect. I.8. A detailed proof is found in 2.6 of [B]. We mention only this much: according to what has been said above we may suppose that our bijection f leaves three noncollinear points fixed, that we will use to define an origin and coordinates x, y; we then only need show that f is in fact the identity transformation. The fundamental remark is that parallel lines are transformed into parallels, since parallelism can be defined in a set-theoretic manner and f is bijective. Thus, in particular, parallelograms are transformed into parallelograms and it suffices to show that f acts identically on the first coordinate axis. To do this it will certainly be necessary to depart from this line, for any bijection of a line preserves that line, whether it acts identically or not. To define affine geometry we identify our line, the x axis, with the field R of real numbers. The figures below, based solely on parallelism, show that the restriction of f to R is an automorphism: f . C / D f ./ C f ./ and f ./ D f ./f ./.

λ 1 0








Fig. I.3.4. Above: a bijective mapping that preserves lines preserves parallelism. Below: construction of the abscissa points C and on the x axis

It is a classical exercise to show that only the identity is an automorphism of R, but be careful not to use continuity, which has no reason to exist here; we have never required that f be continuous or even suggested that such a notion can make sense in the absence of distance!



We now pose a whole series of natural questions. First, the extension of (I.3.1) to all dimensions (> 2) is trivial; in contrast, (I.3.1) is false over the complex numbers, even adding continuity; see Sect. I.XYZ. A still more subversive question (a bit off the ladder) is to ponder the local and the global. But do we need the entire affine plane for our result? Certainly we do for the proof above, where parallelism is the key. But could we do without it? The answer is no. We will see definitively in Sect. I.5 that a bounded set in the affine plane admits plenty of other bijective transformations that preserve collinearity; these are the projective transformations. Thus for a deep knowledge of local affine geometry we need to climb at least one rung. In Sect. I.XYZ we will see that a good understanding of (I.3.1) in a general context and in good rapport with the axiomatics of the nineteenth century wasn’t really achieved until 1950. But back to the elaboration: what happens if we no longer require bijectivity or globality, or again if we study mappings between spaces of different dimensions? In the local but bijective case, readers will see, with the aid of passage to the infinite in the spirit of Sect. I.6, that the question is easily answered by reverting to the local affine case, but with full preservation of parallelism. To finish our discussion of the essence of affine geometry, we pose two more questions. The first is that of incomplete duality: two distinct points determine a unique line, but in contrast two lines determine a point only if they are not parallel. Projective geometry will be the appropriate context (see Sects. I.5 and I.7) for having a duality without exception. A second question concerns topology: what is the topology of the set D of all the lines of the affine plane? What is its “shape”? The answer is that the topology of D is that of an open (no boundary) Möbius strip. We can convince ourselves with the sketches below. We puncture the plane at a fixed origin. With the exception of the lines that pass through the origin, the lines of the plane are associated in a one-to-one manner with the points of the punctured plane (take an auxiliary Euclidean structure and project the origin onto the line in question) and it only remains to “glue” (or sew) the punctured plane to the circle of lines that pass through the origin (caution! this is not the unit circle but is obtained by identifying antipodal points). The segments of the Möbius strip correspond to parallel lines. This operation, which consists of replacing a point by the set of lines that pass through it, is called the blowing up at the point; it is used in an essential way in algebraic geometry. More precisely, looking at the figure, we trace a disk about the blowing up point and replace it by a Möbius strip, while gluing the circle which bounds the disk to the circle bounding the Möbius strip. In this operation, the point is replaced by the median circle of the strip. Analytically, the fact that the topology of the set of lines of the plane is not that of R2 is easily seen: it is not possible to obtain all lines with a single type of equation. For example, the two-parameter expression y D ax C b allows the vertical lines with equation x D c to escape. If we opt for the equation ax C by C 1 D 0, we lose the lines passing through the origin. We are thus forced to consider all the equations ax C by C c D 0; but then the triple .a; b; c/ and the triple .ka; kb; kc/, for k 6D 0,



represent the same line. We are forced to pass to the quotient and to equivalence classes: this is precisely what we do in constructing projective geometry in Sect. I.5.


Fig. I.3.5. Correspondence between lines not passing through O and points of the punctured plane

to be added: a circle representing the lines passing through O

0 glue

π 2π


Fig. I.3.6. Blowing up at the origin. The half lines emanating from the origin O (and not containing the origin) are glued onto a circle of length , two opposite half lines being glued to the same point of the circle. The punctured disk (or plane) thus becomes a Möbius strip

I.4. Three configurations of the affine plane and what has happened to them: Pappus, Desargues and Perles We consider the three figures below, the first two are very old, the third dates from 1965. They seem innocent enough, but they are going to give rise, each in its turn, to very different phenomena. There are surely plenty of other plane affine configurations, but our choice has been dictated by the extensions for which the first two have given rise and the surprising consequences of the third. Readers will be able to guess the significance of .93 ; 93 / and .103 ; 103 / and .103 ; 103 / or otherwise refer to Sect. I.XYZ or to Sect. I.9. The first configuration is that of Pappus (fourth century): given six points situated three apiece on each of two lines, then the three other points that can be derived from them, as indicated on the figure, are again collinear. In the second, Desargues’ theorem (circa 1630), we have two triangles called hom*ological (here abc and a0 b 0 c 0 ), which means that the lines joining corresponding vertices are concurrent. The conclusion is that the three points x, y, z indicated on the figure (the points of intersection of the hom*ological sides) are again collinear. Finally, in the third, the conclusion is that the quotient of the affine invariants (see above) 12 = 42 is forced, by the allignments drawn, to be 13 43 p equal to 12 .3 5/.



Fig. I.4.1.

There are at least three things to mention regarding Pappus’s theorem. The first, very briefly: when we have six points on two lines, we have a particular case of six points on a single conic because the pair of lines may be considered as a degenerate conic; see Chap. IV. In this more general case, the indicated collinearity still holds: this is the famous theorem of Pascal; see Sect. IV.2. μ a







to infinity b′



a′ μ

a′ c′

Fig. I.4.2.

b′ λ



We now speak about Pappus’s proofs. The good proof, illuminating for the sequel, is one that uses projective geometry, considered amply in the next section. Suppose that two of the three points of intersection constructed are “at infinity”; see the figure. Then, as we will see, two pairs of lines that otherwise would intersect are parallel. It is required to show that the third pair is also made up of parallels. We pass from ac 0 to ca0 by a hom*othety of ratio , and we pass from ba0 to ab 0 by a hom*othety of ratio (all these hom*otheties have center O). Thus we pass from b to c by a hom*othety of ratio and from c 0 to b 0 by a hom*othety of ratio . But since D , the proof is complete. Alerted by what has been said in regard to the fundamental theorem of affine geometry, readers may ask what happens with the theorem for affine geometry over the other fields and thus deduce that Pappus’s theorem is true for complex affine geometry, but not for quaternion affine geometry, since the quaternion field isn’t commutative. It is a consequence of an axiomatic study of affine geometry that the commutativity of the underlying field can be characterized by the validity of the configurations of Pappus. All this dates from the time indicated above in Sect. I.3; see for example Artin (1957) or Baer (1952). Recently Schwartz (1993) has given Pappus a second look. Here, very briefly, is what it’s about, see the original text for more details. The starting point is this naive remark: to every pair of triples of collinear points, Pappus associates a third such triple; we then have an operation on such triples. Whence two questions: what is the algebraic nature of this operation? What happens if we iterate it a few or many times, or even indefinitely? In Schwartz (1993) these two questions are resolved and each is placed on an appropriate rung of the ladder; see also Berger (2005).

Fig. I.4.3.

The fundamental remark is that the operation “two triples produce a third” can be inverted: we can go backwards. The reversal is illustrated by the figure on the right. To study the iteration of this operation (after having composed two triples T and T0 to obtain T00 , we may compose T and T00 , or T0 and T00 , and so forth), Schwartz introduced what he called “labeled boxes”, consisting of two triples in the box labeled ..a; b; c/; .a0 ; b 0 ; c 0 //, we have that abb 0 a0 is the box and c, c 0 are the points labeled on the sides “above” and “below” along with the transformations W .a; b; c/; .a0 ; b 0 ; c 0 / 7! .a; b; c/; .a00 ; b 00 ; c 00 /




W .a; b; c/; .a0 ; b 0 ; c 0 / 7! .a0 ; b 0 ; c 0 /; .a00 ; b 00 ; c 00 / :

It is easy to see that these two operations are related by only two conditions: 2 D identity, 3 D identity. The group they generate is none other than the famous modular group, i.e. the group denoted SL.2; Z/, defined as the group of ab with integer entries and determinant ad bc D 1. It is interestmatrices cd ing to encounter in connection with Pappus this group that governs a good part of mathematics and is the most important after R and C. We find it in number theory, complex analysis, Riemann surfaces and algebraic geometry, i.e. for elliptic curves; see Sect. V.14. We will encounter it again in connection with polygonal billiards in Chap. XI.

c IHES Fig. I.4.4. Schwartz (1993)

Now Schwartz has studied the figure obtained by applying the operations of this group to an initial box. It is drawn in Fig. I.4.4, to which we in fact need to add a whole complement (in order to go backwards), that turns out to be a Möbius strip (not drawn: this would be difficult). Schwartz shows that the discrete set of points marked by all the triples thus obtained can be extended by continuity to a closed continuous curve. If we start with one box that is an harmonic quadrilateral and only in this case all the marked points lie on a single line. In every other case the curve is fractal, but with an exceptional additional property: at each of its points, the line of support of the triple passing through this point intersects it in exactly one point, the one considered. This isn’t the case for most fractals, where either there are plenty of lines that don’t intersect the curve, e.g. the snowflake, or at the other extreme every line passing through this point intersects it amidst other points, e.g. a fractal curve that spirals. It seems that the only other known comparable example



is that of the graph of Brownian motion in one dimension: at each of its points it behaves like the graph of the function x 7! x 1=2 . It is the moment to suggest that readers develop one or more purely affine proofs of Pappus’s theorem, if only to appreciate projective geometry and in spite of the fact that they will need to climb a bit up the ladder. We can also use projective geometry for a proof of the Desargues configuration by letting two of the collinear points go to infinity. We then only need use a hom*othety with center O. Thus the commutativity of R isn’t needed, but the complete calculation will show readers that we use the associativity of R: ./ D ./ for all , , . This is important in the axiomatic theory of affine and projective spaces: we can replace the associativity of the object which must play the role of the underlying field by the requirement that Desargues’ theorem hold. Interested readers will verify by calculation that to ascertain that two nonintersecting lines are parallel in an affine plane over an arbitrary field we need to use its associativity, it being understood that a line is a set defined by an equation ax C by C c D 0 and that two lines are parallel if and only if they are obtainable from each other by translation.

Fig. I.4.5. Proof of Desargues theorem. We can assume that x and y , the respective points of intersection of bc and b 0 c 0 and of ac and a0 c 0 are at infinity, i.e. that bc and b 0 c 0 , and ac and a0 c 0 , are parallel. It is then just a matter of showing that ab and a0 b 0 are parallel. But these hypotheses bring with them the existence of a hom*othety with center O that sends a, b , c to a0 , b 0 , c 0 , respectively. Hence the result

But there exists another proof that will subsequently appear less artificial. We embed the affine plane in the affine space of three dimensions and consider the figure obtained as the projection into dimension two of the figure below, where the three lines defining the projection between the two aren’t coplanar. The result is then trivial: the three points x, y, z are collinear since they belong to the intersection of two planes, which is always a line. The preceding explains why, in the axiomatic theory of affine or projective geometry, the situation in dimension two is completely different from the general case: affine or projective planes are hardly categorical. A typical example: there exists a




a c

b a′

c′ b′

Fig. I.4.6. A figure necessarily drawn in the plane, but where we nonetheless see the perspective representation of a figure in space s

a c

b a′

c′ b′

Fig. I.4.7. The same figure deprived of what allows us to see it “in space”

quasi-field, the Cayley octonions, denoted by Ca, where there is no longer associativity; see Sect. I.XYZ. Although a projective plane, denoted by CaP2 , can be well defined over Ca, we can never define CaPn for any n > 3; see Besse (1978). The reason for this is precisely that Desargues’ theorem would be valid there according to the above figure; but we know that this would imply the associativity of the algebraic object, the Cayley octonions. We mention in passing that CaP2 is for us one of the most beautiful of all geometric objects and that we could call it the panda of geometry. But in spite of this exceptional beauty, it is difficult to construct and extremely few authors construct it in detail; an exception can be found in 3.G of Besse (1978).



Finally, for a dynamic study of Desargues’ configuration like that for Pappus, and by the same author, see Schwartz (1998). For another approach to iterations of geometric theorems, see Smith (2000), also cited at the end of Sect. II.1. The philosophy of Perles’s example is as follows: the configuration can never be realized in the rational affine plane, i.e. the subset of the affine plane made up of all points whose two coordinates are rational numbers in a given coordinate system p (modulo which we always have isomorphic objects); the reason is simply that 5 is irrational. The existence of irrational affine configurations was known before Perles, see for example the notion of accessible point on p.126 of Coxeter (1964). For computer enthusiasts this means that such configurations are not, in an exact sense, visible on the screen. On thep other hand we can inject the irrationals in a formal way, especially a number such as 5, which can be defined for example by the equation x 2 5 D 0. But the precise Perles configuration has a much deeper interest: it allowed him to show the existence of polytopes in dimension 8, that can never be realized with the same combinatoric and with vertices having rational (or, equivalently, integer) coordinates. We will return to this question amply in Sect. VIII.12. I.5. The irresistible necessity of projective geometry and the construction of the projective plane We have had reason to be unhappy on several occasions above: first, while Pappus’s theorem like Desargues’ in the purely affine context presents several variants because of possibilities of parallelism. We have an even simpler question, encountered at the end of Sect. I.3: into how many regions do two, three, four, etc. lines divide the plane?

Fig. I.5.1.

Even though its formal definition in algebraic language may seem unproblematic, it requires a bit of time to begin to feel at ease with projective geometry and we thus beg readers to be patient and not to become discouraged. As further evidence of this difficulty it should suffice to remark that, even though introduced by Desargues



at the beginning of the seventeenth century, projective geometry wasn’t firmly established until the second half of the nineteenth century. Desargues’ naive definition is as follows: the projective plane P associated with the affine plane P extending P is nothing other than P itself to which a line P1 of points at infinity is adjoined, the elements of P1 (the line “at infinity” of P) being the set of directions of lines of P: P D P [ P1 . We then say that two distinct parallel lines intersect precisely at the point at infinity that corresponds to their common direction. As for a line D of P and the line at infinity, they intersect precisely at the point of P1 corresponding to the direction of D. Finally, for lines joining two distinct points of P : if one is in P and the other in P1 , the line joining them is the one that passes through the first point with the direction given by the second; the line joining two points at infinity is the line at infinity. Thus for two lines in P just as for two points we can make existence statements without exception, without fear of parallelism.

Fig. I.5.2.

But this construction is abstract. It demands an act of faith and furthermore doesn’t give us a basis for calculation, for which coordinates are needed. For finding a concrete geometric construction of P , we are inspired by the proof of Desargues’s theorem obtained by embedding P in a space Q of three dimensions and taking an arbitrary point O of Q not in P. We have climbed a rung! With each point of P is associated a unique line of Q which passes through O. We will call lines through O “O-lines” for short and O the “origin” of Q; an O-plane of Q is likewise a plane through the origin. Among the lines passing through O precisely those are missing that are parallel to the plane P of Q; but we see that these are associated in a biunique fashion with the directions of the lines of P. We only need add that through a point at infinity corresponding to a direction of a set of parallel lines of P there is an O-line that has that direction. We thus define P concretely as the set of all the lines passing through the origin of Q. The lines of P will be the planes (always passing through the origin) of Q. The intersection axioms are now evident: two distinct O-lines uniquely determine an O-plane of Q; thus a line of P and two distinct O-planes Q intersect in a well determined O-line of Q. So now we no longer have any exception or particular case, just as we have wanted. Even though it isn’t really necessary (and has no significance for projective geometry over an arbitrary field), we can make this construction of P still more



Fig. I.5.3.

plausible as follows: if a point m regresses to infinity along a line D of P, the line OD tends toward the directed line parallel to D. Historically this construction of projective space simply reflects the need that painters have for representing a portion of space in a picture. The point O above is nothing other than the eye of the painter (the observation point) and the plane P the picture (the picture plane). The “empirical” rules for geometric constructions employed in the arts are consequences of projective geometry. We can now calculate in P since we have the vectorial calculus in Q at our disposal: the points of P are none other than those of Q within multiplication by a scalar. Let us quickly see how things work. For an arbitrary coordinate system .x; y; z/ in Q, the points of P will thus be triples of reals, not all zero, modulo an equivalence relation: the triple .x; y; z/ is equivalent to the triple .kx; ky; kz/ for all nonzero real k, a triple of hom*ogeneous coordinates for the same point. Most important is the case where the coordinate system is such that the plane P is defined in Q by the equation z D 1. Then the points of P have for hom*ogeneous coordinates the triples .x; y; z/ with z 6D 0: the point .x; y/ of P will have hom*ogeneous coordinates .x; y; 1/ and all associated triples. Conversely, the triple .x; y; z/ associated with .x=z; y=z; 1/ will be a triple of hom*ogeneous coordinates of the point .x=z; y=z/ of P. Thus the points of the line of the equation ax C by C c D 0 satisfy, in hom*ogeneous coordinates, the equation ax C by C cz D 0. The passage from the first equation to the second is called hom*ogenization. In contrast, the points at infinity are those of type .x; y; 0/, and the point at infinity of a line satisfies the hom*ogeneous equation of this line. It is convenient to use the notation .x W y W z/ to represent the set of all triples of hom*ogeneous coordinates that can be obtained from the triple .x; y; z/ by scalar multiplication. We then have .x W y W z/ D .x 0 W y 0 W z 0 / if and only if there is a nonzero scalar k such that x 0 D kx, y 0 D ky, z 0 D kz, i.e. .x W y W z/ and .x 0 W y 0 W z 0 / represent the same point of P .



As an example of significance for us in the spirit of Sect. I.1, see the expression of Hesse’s configuration in hom*ogeneous coordinates in Sect. I.8. On the other hand, what we see globally is the projective plane, a quite different story that we will touch on later. The human mind doesn’t like objects obtained through an equivalence relation that can’t be embedded in any ordinary space. This introduction of projective spaces may seem a bit artificial, but is in fact an essential tool for many problems where we have to consider things “within a scalar”. We will see examples of this in II.6 and IV.7 for the space of all circles, or that of all spheres or of all conics. An additional property of projective spaces is that they are compact, which is essential for certain problems; they are truly “round” (there are no longer points at infinity, they have been tamed): everything is “at a finite distance”. To respond to a whole array of natural questions we now need to study projective geometry (planar here, but see Sect. I.XYZ) from the points of view of geometry, algebra (group of transformations) and topology (topology of the projective plane). This study must be done for the structure itself, initially independent of its being an extension of affine geometry. But of course we will want to know subsequently how to return to the affine plane. A (the) projective plane P is defined a priori as the set of vectorial lines (one-dimensional subspaces) of a (the) real vector space P of dimension 3, the lines of this projective plane being the vectorial planes (twodimensional subspaces). For the algebraist this will be the quotient of Pn0 modulo the equivalence: v v 0 if there exists a real k such that v 0 D kv. What are the good transformations of P ? In the spirit of (I.3.1) it is now easy for us to find biunique transformations of an affine plane that preserve lines, but only locally: simply consider the figure below and the projection starting at the origin of the space of three dimensions Q, where we have embedded two copies P and P0 of the affine plane. We shouldn’t fail to mention that we have a injective transformation from all of P onto all of P0 , with just one line of P and one line of P0 removed. Note that, in the projective coordinates of P and P0 obtained starting with systems of ordinary coordinates .x; y; z/ and .x 0 ; y 0 ; z 0 / in Q in which P and P0 are respectively the planes z D 1 and z 0 D 1, these transformations are expressed in a linear fashion. These are the transformations we need to apply if we want to assemble different aerial photos in order to compose a single map. Whence the following definition: the projective transformations of P are the (invertible) linear mappings of Q applied to (vectorial) lines. So much for geometry, but for the algebraist we consider linear transformations within a scalar. For example, in a coordinate system, we deal with all 3 3 invertible matrices modulo the multiplication of all their terms by a single nonzero scalar. More conceptually: the group of projective transformations of Q is the quotient of the linear group of Q by nonzero multiples of the identity.





O Fig. I.5.4.

In hom*ogeneous coordinates a projective transformation will always have the form: x 0 D ax C by C cz;

y 0 D dx C ey C f z;

z 0 D gx C hy C iz

or else, in affine coordinates: x0 D

ax C by C c ; gx C hy C i

y0 D

dx C ey C f : gx C hy C i

From this we deduce many things, in particular the important possibility of finding, for each quadruple of noncollinear points, coordinates written as .1; 0; 0/;

.0; 1; 0/;

.0; 0; 1/;

.1; 1; 1/;

which is called a projective frame. For the transformations of a projective line, see the following section. The preceding shows that a perspective (i.e. a central projection) of one line onto another is a hom*ography (defined on the next page) and (see below) preserves the cross ratio: see Fig. 1.6.2.

Fig. I.5.5.

Now in the spirit of Sect. I.3 it’s a rather easy exercise to show that, given two quadruples of non collinear points of P , there exists a unique projective transformation taking one into the other. In the axiomatic theories this result is difficult, but essential; it is thus called the second fundamental theorem of projective



geometry; see, in addition to Artin (1957) and Baer (1952), the classic (Veblen and Young, 1910–1918). For us, in the vectorial context the result proceeds from the following fact (left to readers): for each quadruple a, b, c, d of noncollinear points, we can find a system of hom*ogeneous coordinates such that a D .1; 0; 0/, b D .0; 1; 0/, c D .0; 0; 1/, d D .1; 1; 1/. The theorem then follows at once. But now we have to answer the question for four collinear points. For collinear triples, we have of course transitivity. But for four, by definition of the projective plane, we need to know what happens for four vectorial lines of a vectorial plane, a question that we left open in Sect. I.3. In fact this opens an abyss under our very feet: we have completely forgotten to speak of the projective line! Otherwise expressed: what are the lines of P ? What is their geometry, assuming they have one? I.6. Intermezzo: the projective line and the cross ratio A (the) projective line is thus the set of lines of a vectorial plane, a set that we will denote by RP1 , in agreement with Sect. I.XYZ. The topologist is quickly satisfied here; the figure below shows that this set is in bijection with a (the) circle. This isn’t astonishing, the construction of the projective line consists of completing the affine line by appending a single point 1 at infinity; everything then closes up in a circle. It is well to emphasize that for the line there aren’t two points at infinity, but one only. As in the affine plane it matters little in which sense we pass to infinity; we end up with the same point. This compactification of the line into a circle by a point at infinity is a particular case of a more general construction. Readers should be aware that there exist other types of compactification; we encounter some of these in Sect. II.3.

Fig. I.6.1.

Algebraically, a projective line D, in particular RP1 , can always be written as the set of pairs .x; y/ of real numbers, not all zero, within equivalence: .kx; ky/ is equivalent to .x; y/ for each nonzero real k. As in the case of triples, we can use the notation .x W y/ to designate the pairs taken within multiplication by a scalar.



We recover the affine line as the set of pairs where y is nonzero: .x; y/ .x=y; 1/, which provides an embedding t 7! .t; 1/ of the affine line into the projective line. The projective transformations, called hom*ographies of the projective line, are the mappings .x; y/ 7! .ax C by; cx C dy/. Interpreted for the affine line, these are at Cb the mappings t 7! ct which thus extend onto the projective line by dc 7! Cd 1 and 1 7! ac , consistent with the notion of limit, to comfort us once more if need be. The projective group of the line the group of projective transformations, hom*ographies has three parameters, permitting us to uniquely map each triple of points into a given triple. It clearly does not preserve, when restricted to the affine line, the invariant ac encountered in Sect. I.3. On the other hand, there does exist an ab invariant for four (distinct) points fmi g i D 1; 2; 3; 4, called the cross ratio of these four points and denoted Œmi D Œm1 ; m2 ; m3 ; m4 . Two quadruples of points are in projective correspondence if and only if their cross ratios are equal. In an arbitrary 1 t4 t1 coordinate system, for points mi D .ti ; 1/, this cross ratio equals: tt33 t = . After t2 t4 t2 a moment’s reflection its value is no longer surprising, it being the quotient of two affine invariants associated in a natural way with the quadruple considered. On a projective line, likewise for four distinct points, it is necessary that the cross ratio accept the value 1, e.g. for all x we have x D Œ0; 1; x; 1. We note that the fact that the mapping m 7! Œa; b; m; c; establishes a bijection between a projective line and RP1 is equivalent to the fact that we are able to take .a; b; c/ as the “projective frame”, and is important for our being able to speak of “harmonic conjugation”. Note that the cross ratio can be defined on an affine line; its invariance carries over by the fact that it is preserved by the point projection of the figure below, which furthermore allows it to be calculated for a quadruple of concurrent lines in the affine plane: ŒŒap; Œaq; Œar; Œas D Œp; q; r; sD I see Fig. I.6.2. We have the interpretation: the lines passing through the point a form a projective line, which answers finally the question posed in Sect. I.3. The above formula plays an essential role in certain geometric constructions. The case where the cross ratio equals 1 is particularly important; we say then that

Fig. I.6.2.



the four points are in harmonic division. A purely geometric construction is given in Sect. I.7; note its systematic usage in Sect. IV.4. The cross ratio is not invariant when we permute the points considered, but its behavior is simple and most interesting; see 6.3 of [B] for a detailed study. Direct calculation shows that Œb; a; c; d D Œa; b; c; d 1 and Œa; b; c; d C Œa; c; b; d D 1, which allows us to calculate what happens for all the other permutations. But keep in mind for later (see Sect. V.14) that the simplest cross ratio which is invariant for all 2 3 permutations of the four points is .2C1/ . We find in pp. 43–51 (Darboux, 1917) .1/2 the calculation providing this invariance of for the four roots of an equation of fourth degree, as a function of the coefficients of this equation. The real projective line and its group of transformations is not an object that has been artificially concocted by geometers for their exclusive enjoyment. First of at Cb all, the “hom*ographic” functions t 7! ct are encountered everywhere; they Cd are quotients of affine functions and are very important in the complex case. An important physical application of the notion of projective line is found in the theory of centered systems in optics. Lenses, mirrors, etc., are arranged in some way on a line that is their common axis; zoom lenses of the most sophisticated variety are of this type. Then the correspondence between a point of the axis and its image is always a hom*ography. To convince ourselves of this it suffices to study the case of a mirror or of a single lens; we succeed since the hom*ographies form a group. Readers will surely remember the following formula from school: 1 1 1 C 0 D ; x x f where x is the abscissa of the object, x 0 that of its image, and f the focal length, positive or negative. Note that in optics infinity is essential; here it provides the focal point (image of the point at infinity) and the focal objective (reciprocal image of the point at infinity).

Fig. I.6.3.

The hom*ographies, real (as here) or complex, are of primary importance in geometry; we will see this very soon in Sects. II.3 and II.4. Their classification by their fixed points among other ways, especially as involutions, i.e. the hom*ographies whose square equals the identity is fundamental, but we will have but little to do with it (at the end of Sect. IV.6); see Chap. 6 of [B]. Finally, note that



the complex projective line, in its role as a topological object, is nothing other than the sphere S2 , for it is obtained by appending a point to the complex line C (which is the real plane R2 ): C D C [ 1 D S2 ; we will see this again in II.4. I.7. Return to the projective plane: continuation and conclusion We haven’t yet finished with the projective plane. We first note that the cross ratio allows us to recognize when two quintuples of points are projectively equivalent; compare with Sect. I.3 for the affine case and its invariant. We now study in depth the relation between affine geometry and projective geometry, if only to make rigorous the proofs of the theorems of Pappus and Desargues that were outlined in Sect. I.3. Starting with P, we constructed P , which contains the line at infinity. The essential things is that in P D P [ P1 and above all in any projective plane P whatever we can choose a line D and make it the line at infinity of the complement PnD of D in P . That is to say, in the construction of Fig. I.5.3, we replace the plane z D 0 by the plane defined by the origin and the desired line taken in the plane z D 1. The affine space so defined is “the” plane parallel to this new plane. For example the affine invariant ac of three collinear points on a line F equals the cross ratio Œc; b; a; 1F , ab where 1F denotes the point at infinity of the line F, i.e. 1F D D \ F. To say for instance that c is the midpoint of ab is equivalent to saying that c; b; a; 1F form a harmonic division. In summary, we can accomplish this: in an arbitrary affine plane, completed to form a projective plane, we can alter the line at infinity, i.e. stay in the projective plane with all its advantages, all its properties, but decide on “new parallels”. We can also speak of the “transfer to infinity of one or more collinear points”: we let the new line at infinity pass through these points (or we can use a projective transformation sending the given line to the line at infinity). All this is certainly a rung up Jacob’s ladder, where we can manage things a bit better. Fundamental is the fact that the cross ratio is conserved under these “transfers”.

Fig. I.7.1.



The preceding technique was used to prove Pappus and Desargues in Sect. I.4. We now use it to demonstrate the classical property of the configuration of the complete quadrilateral:

Fig. I.7.2.

In this figure the four points a, c, x, y are in harmonic division. To see this, it suffices to transfer the two points f and g to infinity. Then a; b; c; d becomes a parallelogram and our result simply translates the fact that the diagonals of a parallelogram intersect at their midpoints. Despite its simplicity, the configuration .62 ; 43 / of the complete quadrilateral may be seen as a geometric rendering of the fact that the solution of an equation of fourth degree may be reduced to that of a third degree equation. Indeed, this configuration associates in a canonical way a triple of points with a quadruple (find this in the figure). See Sect. I.XYZ for an entirely projective proof. We now attack the question of duality, used in Sect. I.1 and imperfect in the affine context: there points and lines played similar, but not identical, roles. Furthermore, the space of all lines had a topology different from that of the points (the affine plane), among other reasons because we could not find a good one-to-one correspondence between these two sets (see Sect. I.3).

Fig. I.7.3.

In P the duality is perfect with regard to the line joining two points and to the intersection of two lines. However, we would like to obtain a one-to-one correspondence between P and the set D of all its lines: but this is utterly simple, since P is the set of lines through the origin of a vector space Q of dimension three and D



the set of vectorial planes: we only need put a Euclidean structure on Q. With the directed line D we associate the perpendicular plane denoted by Dperp . There is a single defect: this bijection depends on the Euclidean structure chosen. The algebraist might prefer an alternative, but equivalent, presentation: let us choose some representation in projective coordinates, i.e. a projective frame of Q. Then the desired bijection consists of associating with the point .a; b; c/ the line with equation ax C by C cz D 0. As for “modern” algebraists, they will observe that if P is the projective space associated with Q, then D is identified naturally with the vector space Q of Q. But, just as there doesn’t exist a natural isomorphism between Q and Q , there doesn’t exist a natural canonical isomorphism between P and D. For more on geometric dualities, the correlations, see 14.8.12 of [B] or p.260 of Frenkel (1973) for the general case, and Sect. I.8 below and Sect. IV.4 for the very particular case of Möbius tetrahedra. Duality will be unavoidable in a large part of Chap. VII. Furthermore this duality is completely geometric: given two points, the point of intersection of their two image lines has for an image precisely the line that joins the initial two points. This allows us to systematically obtain twice as many theorems, or to relate a desired theorem to another, perhaps simpler, theorem. In the sequel we will encounter examples in various contexts; see Sects. IV.4 and VIII.8 (conics, Pascal and Brianchon, inscribability of polyhedra). Right away readers can look for the duals of the theorems of Pappus and Desargues (see Sect. IV.4 as needed). Note that the mapping D ! D of the left part of Fig. I.7.3 is imperfect: the origin doesn’t have a dual; it is in fact the line at infinity. Attentive readers will not have missed noticing that, even though P and D are now in good bijection and have the same topology, this doesn’t at all divulge the nature of the topology of P . The first thing to observe is that P is compact, and this is also true for all the more general projective geometries of Sect. I.XYZ. In fact, if P is the set of points of Qn0 considered within multiplication by a nonzero scalar, this is also the set of points of the unit sphere of Q (a Euclidean structure is chosen for Q) modulo multiplication by ˙1, in other words the set obtained by identifying antipodal points of this sphere. Nonetheless, the “shape” of P is not simple, and for an essential reason: if P is clearly a surface, moreover compact, it can nonetheless never be realized as a surface that is embedded in three dimensional space, this since P isn’t orientable. The trick that was used for the projective line in Sect. I.6 doesn’t work anymore. For

Fig. I.7.4.



on the oriented sphere, which must replace the oriented circle (compare Fig. I.6.1 with Fig. I.7.4), it is necessary to append all the points at infinity, and not just a single point; and in order to do that, cause the intervention of a “blowing up” (see Sect. I.3 and Fig. I.3.6). A better way of understanding the topology of P is to see that not only can we obtain P by identifying antipodal points of the sphere, but that we can also be content to let this identification operate just on a hemisphere (boundary included): we need then only identify antipodal points of the equator. We can still choose to keep a band about the equator, it still being required that we identify antipodal points in this band. We obtain in this way a Möbius strip and P then appears as the union of a Möbius strip and a spherical cap, i.e. a disk sewn together without ambiguity.

Fig. I.7.5. Ways of seeing the projective plane. At left: identify antipodal points. Middle, identify antipodal points of the equator. At right, identify antipodal points of a band (which comes down to preserving only the middle line of the band, while identifying ab and b 0 a0 ; pay attention to the direction of travel)

It is because P contains a Möbius strip that it is not an orientable surface; and it is for this reason that P is not embeddable in R3 : if fact, to embed P in R3 would be to define a transformation of P into R3 that is continuous and injective; the geometer says without double point or without self-intersection. In view of compactness such a transformation would automatically realize a homeomorphism of P onto its image. However, a result from topology states that each compact surface in R3 without boundary possesses an interior and an exterior and is, for this reason, orientable. A more direct proof of the fact that P is not embeddable in three dimensional space amounts to observing that a Möbius strip cannot be glued, following a common boundary, to a topological disk without the band and the disk intersecting. The boundary of the Möbius strip is indeed a circle (moreover unknotted), but a circle intertwined with the strip:

Fig. I.7.6.



We can nonetheless realize P as a surface in R3 , but this surface must intersect itself. A first realization effectively consists, starting with a hemisphere, of identifying the antipodal points of the equator while cutting the boundary of the hemisphere into four arcs and gluing opposite arcs in an appropriate way:

Fig. I.7.7.

Fig. I.7.8. Two views of Boy’s surface. At left, a window has been opened to see the triple point

The figure obtained bears the name cross-cap. It possesses one ugly singularity at its vertex and a second at the other extremity of the curve of self-intersection, whereas Boy’s surface, another realization of P , only presents transverse selfintersections analogous to the intersection of two planes, with a single triple point. (What is called a singularity of a surface depends on its mode of definition. If, as here, the surface in R3 under consideration is presented as the image of another surface, the self-intersections are not considered singularities, provided that the sheets that intersect are regular open images of the plane. The singularities correspond to pinchings of the surface. These notions generalize those of a double point and of a cusp of a parameterized curve. Technically, a point is regular if at this point the rank of the differential of the parameterizing transformation is equal to 2; it is singular otherwise.) Returning once again to the difficulty of visualizing the real projective plane, let us quote Léon Brunschvicg: “what we see is in space, but we don’t see the space.” and “space and its source in experience are attained in reason.” On the other hand, we assert that there exists a very beautifulpalgebraic of P p embedding p in R6 given by the mapping .x; y; z/ 7! .x 2 ; y 2 ; z 2 ; 2yz; 2zx; 2xy/. More



precisely, let us consider the sphere S2 as the set of .x; y; z/ with x 2 C y 2 C z 2 D 1. As we have just seen, each point .x; y; z/ of this sphere has an image in P ; in contrast a point of P originates from exactly two points of S2 , for the antipodes .x; y; z/ and .x; y; z/ define the same point of P . We say that this transformation S2 ! P is a “covering by two sheets”. But once having chosen coordinates, we have a Euclidean structure on the space R3 and of course induce a metric a notion of distance on the sphere; but also a metric on P by defining the distance between two points of P , regarded as lines of the space, to be equal to the angle (between 0 and =2) that these two lines form. Now the transformation of S2 ! R6 defined above also defines a transformation P ! R6 , according to the sign rule. This last transformation is injective, in contrast to the mapping of the sphere. The surface image, both of the sphere and the projective plane, is called the Veronese surface. It is found in many other situations, for example in the theorem on five conics of Chasles in Sect. IV.9. It’s a justifiably superb geometric object, for numerous reasons. First, the embedding is isometric; it preserves distances in P and in R6 . Moreover, it is equivariant, which is to say that the isometries of P are realized via this embedding, as induced by the global isometries of R6 (or S5 ). Better still, it is algebraic in nature. Finally, it is rigid, for Kuiper has shown that, modulo an affine transformation of R6 , it is the only surface whose topology is that of P and such that, for each direction of R6 , it has only three perpendicular tangent planes in that direction. Knowledgeable readers are aware that there must always be at least three, according to Morse theory; for this theory see 4.2.24 of Berger and Gostiaux (1988), Bott (1988) and Bott and Tu (1986). The fact that there aren’t more characterizes it. We can caricature this by saying that the Veronese surface is an extremely susceptible creature: the merest touch causes it to have more tangent planes in all directions; it becomes dented. For all this and for a second characterization by the condition that each tangent plane cuts it into exactly two pieces, see Kuiper (1984). Readers who like the intrinsic can replace the transformation given in coordinates by the one given by y 7! .x y/2 of R3 into the space of dimension 6 of quadratic forms on R3 . Here we arrive at the forms of rank 1; see also Sect. IV.9. The Veronese surface exists over all fields and in all dimensions; we will encounter it in Sect. IV.9 and in studying Borsuk’s conjecture in Sect. II.0. The transformation that defines the Veronese surface is easily extended to all projective spaces of arbitrary dimension and over any field. Moreover we can work with polynomials of arbitrary degree, not just of degree 2. A third realization of P is Steiner’s Roman surface. Like Boy’s surface, it possesses a triple point and three lines of self-intersection, which are segments; but it also has singularities pinchings at the extremities of the lines of selfintersection, thus six in total; in return it possesses a double infinity of ellipses; see toward the end of Sect. II.6. It’s the only surface, except for the ellipsoids, with this property. The fact that this surface has lots of ellipses arises simply from its being a projection in three dimensions of the Veronese surface (it’s the image of the transformation .x; y; z/ 7! .yz; zx; xy/ of S2 into R3 ). The term “Roman” comes from



the fact that Steiner discovered it when he was in Rome in 1844. We will encounter the Veronese surface in Sects. II.0 and V.9.

c G. Fischer Fig. I.7.9. Steiner’s Roman surface. Fischer (1986a)

Note that the complement in P of a projective line is connected, in contrast to the affine case, infinity serving as the connection bond. The same thing is true for the median line of the Möbius strip. How do we see that we have the same phenomenon? By considering, in the projective plane, a band containing a given line D. The band situated between two lines parallel to D won’t do, since it contains only a single point at infinity, but the region contained between the two branches of a hyperbola (situated on both sides of D) contains a whole segment of points at infinity, and it clearly has the topology of a Möbius strip, since it is obtained by identifying, in a rectangle, two opposite sides traversed in opposite senses. This fact explains why a curve located on one side of its asymptote, when it tends toward infinity, always returns from infinity in the opposite direction. The curve is in fact tangent to its asymptote at its point at infinity, and if the contact is ordinary the curve does not cross its tangent. In particular this is the case for the point at



Fig. I.7.10. A neighborhood of a line in the projective plane has the topology of a Möbius strip

infinity of a hyperbola: the hom*ography or hom*ogeneous coordinate transformation, .x; y; z/ 7! .z; y; x/, or .x; y/ 7! .1=x; y=x/ in affine coordinates, transforms the hyperbola with equation xy D z 2 (xy D 1 in affine coordinates) into the parabola with equation zy D x 2 (y D x 2 in affine coordinates), tangent to the x axis at the origin. The curve does not cross its asymptote; it’s the plane that makes a half turn like a Möbius strip. As for the connectivity property indicated above, it explains the well-known trick of cutting a paper Möbius strip along its center curve and then continuing to cut along new median curves. It is left to readers to carry out the necessary experiments.

Fig. I.7.11.

We have seen that there exists a canonical metric structure on P D RP2, derivative from that for the sphere: the distance between the two points p, q 2 RP2 is the angle, between 0 and =2, between the two lines of R3 which give rise to p and q. This geometry is called elliptic; it must be seen as a generalization of Euclidean geometry, for any two projective lines intersect in a single point (which is not the case for the great circles of spherical geometry). Here there are never any parallels, whereas in hyperbolic geometry in contrast there is an infinity of parallels for any given line. For more details on elliptic geometry, see Chap. 19 of [B]. But here is an example to which we should pay attention. It has to do with studying the cases of the equality of triangles: are two triangles for which two sides are the same equal, in particular are their three angles the same? An initial remark: two points in P are joined by a unique shortest path (projection of an arc of a circle onto the sphere) if their distance apart is less than =2, otherwise there are exactly two shortest paths; but we will only discuss triangles with distances between vertices all less than =2.



Fig. I.7.12. The three lines .D; E; F/ and the three lines .D0 ; E0 ; F0 / form equilateral triangles of side=3 in P . But these triangles are completely different: in the second the three angles equal , in the first the three angles equal A, where cos A D 13 (apply the fundamental formula of spherical trigonometry)

Fig. I.7.13. The two types of triangles in projective space, distinguished by their rise

We return to the equality of triangles, the exemplary case being, viewed as sketched in R3 , that of a trihedron with three equal angles of =3 and of a degenerate trihedron formed by three lines of a plane which makes among them the equal angles =3. What can happen here, seeing that everything goes well in spherical geometry? To understand this, it is natural to go back to the sphere; but just one point of P provides two different points (antipodes) of the sphere. A curve of P , here a side of a triangle, once a vertex has been lifted onto S2 , is lifted without ambiguity into S2 because the projection of the sphere onto the projective plane is bijective and bicontinuous when restricted to a sufficiently small open set of S2 , typically the open hemisphere (spherical cap of aperture ) centered at a given point. Continuing in like manner for the two remaining sides we obtain a curve formed by three arcs of circles in S2 , but for which the terminal point is either the chosen point of departure or its antipode. These two cases are exactly those of the two triangles in =3 considered. There are thus two types, I and II, of triangles in P , but note that the type II will only be encountered if the sum of the sides is greater or equal to . Readers will easily show, by deftly applying the case of equal spherical angles, that



the equal angle case holds if, besides the equality of the respective sides, the two triangles considered are of the same type.

Fig. I.7.14. Lifting into S2 of the two “exemplary” triangles of Fig. I.7.13

With the canonical metric structure of P , the associated duality of Sect. I.7 is expressed thus: the projective line that is dual to a point p is made up of points of P located at a distance =2 from p. For this associated geometry, perpendicular bisectors, etc., see [B]. It is interesting to note that elliptic geometry, which furnishes a trivial counterexample to Euclid’s parallel postulate, was not known until well after hyperbolic geometry was discovered. This is due, among other reasons, to the difficulty of “seeing” the real projective plane. Finally, we indicate why the only transformations that preserve lines are the projective transformations: the proof is achieved by fixing any line at infinity and applying the fundamental theorem of affine geometry to the complement. This result is often called the second fundamental theorem of projective geometry; see Sect. I.XYZ for the “first fundamental theorem”.

I.8. The complex case and, better still, Sylvester in the complex case: Serre’s conjecture In Sect. I.1 we briefly alluded to affine geometry over the field of complex numbers, and even over the quaternions. The definition of the affine plane (over the reals), in which we have worked until now (see Sect. I.XYZ), extends trivially to the case of an arbitrary base field, not only to number fields, in particular the complex numbers C (commutative) and the quaternions H (non commutative), but also to all other fields, in particular the finite fields, the most simple among them being the field of two elements Z2 D Z=2Z. But there is no reason at all to restrict ourselves to dimension 2, since everything is constructed using only the algebraic theory of vector spaces (if necessary, refer to Sect. I.XYZ). Beyond what will be said in this section, see p. 9 of the introduction of Orlik and Terao (1992). For example, there is no strict notion of cross ratio in the non commutative case, typically over the quaternions, but only of conjugate classes, which take its place.



Let us repeat that Sylvester’s theorem of Sect. I.1 is false over the complex numbers, the simplest numerical example is written in projective coordinates for the complex projective plane CP2 ; specifically, the nine points with projective coordinates .0; 1; 1/; .1; 0; 1/; .1; 1; 0/; .0; 1; !/; .1; 0; !/; .1; !; 0/; .0; 1; ! 2 /; .1; 0; ! 2 /; .1; ! 2 ; 0/: p Here ! denotes a cubic root of unity other than 1, e.g. .1 C i 3/=2. These nine points are the inflection points of the cubic (projective) equation x 3 C y 3 C z 3 3axyz D 0 (a 6D 0). We verify by hand, without having need of the theory of planar cubics, that on each line joining two of these nine points there is always a third. Algebraically the condition for the collinearity of three points is translated by the fact that the determinant of their nine coordinates is zero. This configuration .94 ; 123 / is called Hesse’s configuration. Readers who like simple calculations but hate projective coordinates will be able, with the aid of an appropriate projective transformation, to write the coordinates for a system of nine points of this type in the complex affine plane. There are too many zeros in the three possible places in the above array to allow us to proceed solely by division of the same coordinate. We will have observed that collineation here is complex, but we can regard the condition by taking a complex vector space as a real vector of twice the dimension; then Sylvester’s condition will be that we always have a third point on the complex line generated by two given points; but of course this complex line is in effect a real plane.

Fig. I.8.1. Hesse’s configuration

In 1966 Jean-Pierre Serre announced the following conjecture: “Let there be given, in a complex affine space of arbitrary dimension, a finite system of points satisfying Sylvester’s condition: show then that this set of points is necessarily contained in a (complex) plane V”. We emphasize that the proof, if there is one, cannot be purely combinatorial; and that if Sylvester was wrong in the complex case, it’s because a line D does not separate the complex plane C 2 into two regions: C 2 n D is



connected, we can “circle about D”. Readers will have noted that the complex plane is of real dimension 4 and that lines are of real dimension 2 and thus “surfaces”. The answer is positive and is found in Kelly (1986); or see p. 802 of Hirzebruch (1987). But it is extremely difficult and requires climbing many rungs up the ladder. We indicate the principal steps, each of which constitutes at least one rung. We can place ourselves in dimension 3, for if the result holds there it will be true in any dimension, just as Sylvester’s result for the plane immediately implies the same result in any dimension. Let therefore P be a system in CP3 satisfying Sylvester’s condition. From a well chosen point in CP3 , take the projection of the initial configuration onto a complex projective plane CP2 ; the projection is also clearly a Sylvester configuration. We construct all lines that join pairs of its points and let tr denote the number of such lines that contain exactly r points. Let n be the total number of points; we have tn D 0 (unless all the points are collinear) and, on the other hand, t2 D 0. With this set of lines we create, as in Hirzebruch (1987b), an abstract algebraic surface in roughly the following way: we turn a certain integral numbers of times about the lines to obtain a covering of CP2, as is done in C about the origin with the transformation z 7! z k . Finally, we must desingularize the places where the lines intersect each other. The miracle is that this surface possesses two invariants, its Chern classes c1 .S/ and c2 .S/. Since 1977 it has been known that these satisfy the inequality .c1 .S//2 6 c2 .S/, a relation that is very difficult to prove. This was first accomplished in 1977 as a corollary to a theorem of Yau. This theorem, together with others, earned Yau the Fields medal; he showed that on certain algebraic surfaces Kähler metrics can be found for which the curvature is very special (so called Einstein varieties). When the Chern classes c1 and c2 are calculated there by precisely the same integral formulas obtained by Chern for “his” classes, using the curvature, the particularity of the curvature implies the required inequality. Yau’s theorem is an ultra difficult global theorem about partial differential equations (PDE’s), conjectured by Calabi in 1954 and which had defied all the experts in PDE’s during those 23 years. This inequality between c1 and c2 forces, for our configuration, some combinatorial relations that contradict, finally, the finiteness of p. These can be found in Hirzebruch; they bring with them the consequence that, with the notations of Sect. I.1, the two conditions t2 D tn D 0 imply t3 6D 0. We then fabricate a kind of translation that engenders an infinite number of points in the configuration, which is the contradiction that was sought. We recall that Sylvester’s condition alone does not imply any such nontrivial combinatorial condition. Thus the conjecture of Serre has been settled, and even though its statement is elementary, it came only at the cost of a sizable portion of the modern theory of algebraic surfaces. For those who are curious and who like open problems, we can ask what happens with the question of Sylvester-Serre in the quaternion spaces . To our knowledge the question has never been studied. The conclusion might be, by analogy with the real and complex cases, that the set of points must ultimately be in a quaternion subspace of quaternion dimension equal to 4. Readers will perhaps not be surprised that the trinity of the three fields fR; C; Hg is encountered in other parts of geometry and elsewhere in mathematics.



For a delightful presentation of a synthesis of this prevalence of fR; C; Hg, as well as for other trinities (e.g. the three kinds of conics to be seen in Sect. IV.1), read Arnold (1999) and Arnold (2000). I.9. Three configurations of space (of three dimensions): Reye, Möbius and Schläfli We will see in Sect. I.XYZ that a systematic study of configurations is not our objective. On the other hand, just as in Sect. I.3, we present here some configurations of the real space of dimension 3 that necessitate, for their understanding or even for showing their existence, a climb rather far up the ladder. In the space of three dimensions, a configuration of type .pq ; rs / is a set of p points and of r planes such that each point belongs to exactly q planes and each plane contains s points.

c Springer Fig. I.9.1. A .124 ; 163 / configuration. Hilbert, Cohn-Vossen (1996)

The easiest is Reye’s configuration; it is of type .126 ; 126 / and its existence is elementary. Figure I.9.2 gives it with a cube and some points at infinity in the direction of the arrows: we only need verify that this actually works. But this configuration arises more naturally when we consider four spheres in space; the configuration is the one formed by their centers of hom*othety (see Chap. II). For three circles in the plane, we find a complete quadrilateral. In space our four spheres have, two at a time, two centers of hom*othety, i.e. 12 altogether for the 6 pairs of spheres. The sought-for planes are more difficult to locate, except for the four planes formed by the centers of triples of spheres. We leave this to readers. A reference for this configuration and for that of Schläfli is Hilbert and Cohn-Vossen (1952); see also below. The Möbius tetrahedra are pairs of tetrahedra in real three dimensional space such that each vertex of either one belongs to a face of the other, giving the configuration .84 ; 84 /. They can be found “by hand” easily enough, mainly by sending points to



Fig. I.9.2. At left, Reye’s configuration. At right, the configuration formed by the centers of hom*othety of three circles (below, recall the properties of the centers of hoc Springer mothety of two circles). Hilbert, Cohn-Vossen (1996)

infinity. But why can’t we do the same thing for triangles? It is easy to convince ourselves of this impossibility “by hand”, but that would not really get to the heart of the matter. We must climb the ladder, which we do in recalling the end of Sect. I.6, which turns out to be profitable in all dimensions. There we constructed a bijection between the points and lines of the plane, which with each triangle then associates a new triangle; but we can’t require that this bijection have the property that the line associated with a point pass through that point. However, this is possible in a projective space (projective, so as to avoid exceptions) Q of three dimensions: there exists a bijective transformation f of Q onto the set Q of all its planes such that p 2 f .p/ for each point p of Q. Of course we require further that the properties of collinearity and intersection be preserved. Thus any tetrahedron defines a pair of Möbius type by adding to it the tetrahedral image under f . And here is the “parachuting” of such an f : we set up projective coordinates in some way (any choice will do) and define f ..a; b; c; d // to be the plane with equation bx C ay dz C ct D 0. The membership condition is obviously satisfied and, things being linear, all properties of intersection, collinearity, etc. are preserved as we would like.

Fig. I.9.3. Möbius tetrahedra. I [B] Géométrie. Nathan (1977, 1990) réimp. Cassini. c Nathan Édition (2009)



In fact, we have indeed climbed the ladder somewhat. First, we see that we will be able do the same thing for every space of uneven dimension, the number of projective coordinates being even in this case. But above all the theory of all this was accomplished geometrically and very laboriously at the end of the nineteenth century. Now, algebraically, linear and multilinear algebra permit us to resolve completely all the questions that can be posed, and this in arbitrary dimension and over an arbitrary field. The essential problem is knowing what are the bijections between a space and its dual (the space of its hyperplanes) that preserve intersections and collinearity. The answer is that there are two possible types, ones that we have encountered: the type given by a Euclidean structure, i.e. a quadratic form (corresponding to a symmetric bilinear form) of maximum rank, and the type called symplectic, i.e. given by an antisymmetric bilinear form of maximum rank, it being understood that a symmetric or antisymmetric bilinear form of maximum rank on a vector space E defines a bijection of E onto its dual E and, by passage to the quotient, a bijection of the projective space P .E/ onto the space of its hyperplanes. In the second type, we have the Möbius property that each point belongs to its dual; and it is trivial that these structures cannot exist in even dimensions (odd dimension for the vector space). Since the quadratic forms of maximum rank at least in the real case are categorical, as are the symplectic forms of maximum rank, we now know everything. The proof of this fundamental result is Exercise 14.8.12 of [B], a more complete reference is Frenkel (1973).

c Springer Fig. I.9.4. Schläfli’s double six. Hilbert, Cohn-Vossen (1996)

Schläfli’s double six is given in the above figure. It’s a configuration of type .302 ; 125 /, but the notation here indicates 30 points and 12 lines (no planes). The



proof of its existence isn’t elementary; it makes essential use of algebraic geometry, specifically that a cubic surface (of degree 3) in three dimensional space and without singularities contains exactly 27 lines in the complex case, and in the real case 27, 15, 7 or 3 lines. An “elementary” exposition is contained in §25 of Hilbert and Cohn-Vossen (1952). The point of departure is to take four lines in space in general position. There exist two lines that intersect them all, which is seen using what is found in Sect. IV.10: the set comprised by the lines intersecting three given lines in general position is a quadric surface, so that the desired lines, based on four lines, are obtained by looking for the points of intersection of the fourth line with the quadric surface defined by the first three. In general there are two such points, our four lines being “generic”. The construction of the double six begins thus: we start with a line D1 , then construct at random five lines E2 , E3 , E4 , E5 , E6 intersecting D1 . Let D6 be the next line, other than D1 , intersecting the four lines E2 , E3 , E4 , E5 . Then

c G. Fischer Fig. I.9.5. Clebsch’s diagonal surface. Fischer (1986a)



D2 , which in addition to intersecting E3 , E4 , E5 , E6 , likewise intersects D3 , D4 , D5 . It remains to define E1 as the line likewise intersecting D2 , D3 , D4 , D5 . Now, we have numerous other intersections that form, by construction, a configuration of type .302 ; 125 /. But the proof is difficult and, as already indicated, requires the use of algebraic geometry; see the book cited above for an heuristic proof. We will encounter this configuration in Sect. V.16. There indeed exist real cubic surfaces containing 27 lines, all without singularities, e.g. Clebsch’s diagonal surface. The theoretic considerations are found in Fischer (1986b), they depend on Fig. I.9.5. I.10. Arrangements of hyperplanes We have seen that there is a difficulty in studying the finite sets of lines in the real affine plane, because of the possibility of parallels. Nonetheless, the question of the maximum number of connected components of the complement of n lines was resolved before the twentieth century, and in fact by Schläfli for the complement of n hyperplanes in arbitrary dimension. In order to have the maximum number, it is necessary to be in the generic case; what happens when more than two lines intersect, and for the parallels, dates from 1889. This does not give a classification; such a study leads in fact to problems in topology, algebraic geometry, convexity, combinatorics, number theory, (arithmetic) analysis and geometry without any obvious relations. We will not speak further here of the real case; we refer readers to its introduction in Orlik and Terao (1992) or pp. 679–706 of volume II (Hirzebruch, 1987b), not to forget the commentaries of pp. 802–804. The complex case has an even greater richness, and provides much pleasure when connections are found between seemingly different things. The book cited above (Orlik and Terao, 1992) is entirely devoted to it; it’s a subject that ascends quite high on the ladder and has a very recent development. We can’t speak about it in detail, or even partially. We have seen a first approach in Sect. I.7 above. In a second approach, we first remark that, if the complement of a hyperplane in a complex space is connected, contrary to the real case, in return it isn’t simply connected: if we “make the tour” of a hyperplane, we obtain a loop (closed curve) that isn’t contractible to a point. More generally, with a finite set of hyperplanes we associate the fundamental group of its complement, i.e. the group generated by the closed curves through a fixed point (in the complement, of course) that “turns” about the hyperplanes of the arrangement under consideration. We obtain this group by considering two closed curves to be equivalent if they are deformable each into the other, in a continuous fashion, without of course intersecting the figure about which they are turning. The nature of this group and more generally the topology of this complementary set are much studied nowadays and yield surprising relationships. The simplest case is the topology of the complement of two intersecting lines of C 2 : it is easily seen that this set can be continuously deformed onto the product of



two circles, i.e. a torus. In particular, the fundamental group of this complement is that of the torus, thus Z2 . I. XYZ For the mathematical objects described above and for proofs, a general and complete reference is [B], to which we add Hilbert and Cohn-Vossen (1952) for the configurations. Before mathematicians, curious and practical minds would ask how to make lines straight, planes “flat” all this both in drawing instruments and in industrial practice of a more or less high degree of precision; and finally for the highest degree of precision, that of metrology. We will give some brief indications about this problem in Sect. II.XYZ. The real affine plane is defined as follows: we consider a real vector space P of dimension 2, or, what amounts to the same thing modulo isomorphism the set R2 of pairs .x; y/ of real numbers. The points are thus the elements of R2 ; the lines are all the subsets that are translations of a vectorial line of R2 . We might say that our affine plane is a real vectorial plane for which the origin has been forgotten; there is no longer an origin, no special point. The properties enjoyed by lines and planes, and their relationships, are direct consequences of the axioms for vector spaces. This definition will not satisfy very abstract minds; there remains a mental trace of an origin. In [B] there is a more axiomatic presentation. Here we will retain our point of departure: with each pair .p; q/ of points of the affine space there is associated the vector (“free” in the old language) that may be denoted q p, ! which belongs to the vector space, giving rise to the affine space or often also pq, under consideration. An important element, which serves as the foundation for the more conceptual construction mentioned above (i.e. not favoring an origin, or even the ghost of one), is the barycentric calculus. Two points p, q cannot be added in an intrinsic manner (we can only subtract them, but the result is a vector and not a point of the space considered. On the other hand, the midpoint of the two points can be written pCq , 2 and more generally we can divide a segment by a given ratio; see the affine invariant introduced in Sect. I.3. What matters in pCq is that it can be written 12 p C 12 q. 2 More generally we can define in an intrinsic and practically trivial fashion any fiP nite sum p such that the sum of the coefficients (called barycentric) satisfies i i P i D 1. Two facts are essential: the associativity of this operation and the uniqueness of the coefficients for the sums for three points forming a true triangle. This furnishes “purely affine” coordinates for the plane, coordinates called barycentric; for all the details see Chap. 3 of [B]. Briefly and in any dimension, since no changes are needed: if fpi g (i D 1; : : : ; d C 1) denotes a set of d C 1 points in real affine space of dimension d forming a simplex which means that none of them belongs to the hyperplane defined by the remaining d then for each point x ofPthe space there exists a unique expression (a “barycentric sum of the fpi g”) x D i .x/pi



P such that i .x/ D 1. A simplex is: on a line, two distinct points; in the plane, a true triangle; and in three dimensional space a tetrahedron. The full simplex for a system of points is the set of all points for which the barycentric coordinates of the points of the system are nonnegative.

Fig. I.XYZ.1.

These coordinates fi g are sometimes essential; see the spheres of the “combles” in Sect. II.1. They are also much used in the study of convexity; see Chap. VII. It can in fact be shown that if, in addition, the space is Euclidean, V is the total volume of the simplex, a1 the volume (a distance in the plane, an area in space) of the face H1 opposite the vertex p1 of the simplex under consideration (i.e. formed by the d points p2 ; : : : ; pd C1 ), then the absolute value j1 .x/j is given by the formula j1 .x/j D da1V distance.x; Hi /. But in fact, instead of R2 , we can consider an arbitrary field K and an arbitrary dimension n: the affine space of dimension n over K is simply the vector space Kn , whose elements are points; the affine subspaces of dimension k are nothing other than translations of vector subspaces of dimension k of Kn . In dimension 1 we speak of lines, in dimension 2 of planes and in dimension n 1 of hyperplanes. In every affine space we have hom*otheties; the vectorial hom*otheties are the transformations x 7! x, where is an arbitrary nonzero element of the base field. The affine hom*otheties are obtained by translating these last. More precisely: the hom*othety with center a and ratio is the transformation m 7! a C .m a/. A frequent case is where the field is finite (frequently encountered in combinatorial geometry) and thus necessarily commutative. The most important field, after the field of real numbers, is the field C of complex numbers, to which we may add the (noncommutative) field H of quaternions; see 8.1 and the index of [B]. But also, in the spirit of the present book, see their appearance in polyhedra in Coxeter (1974) and, for the associated geometric groups, the important (Du Val, 1964).



Quaternion geometry is less developed, the non-commutativity of the field is a significant complication. We find that Pappus’s theorem is false, nor is there a strict notion of cross ratio; the cross ratio is simply a conjugate class in H under the inner automorphisms. The three fields R, C, H form a trinity, for which there are many associated trinities. It can lead to discovery by analogy. For such a mindset and a list of trilogies, readers should look at the engaging text (Arnold, 2000). We take from it the Table I.XYZ.1 below, which is discussed in the text just mentioned.

Table I.XYZ.1.

In the case of the Cayley octaves, also called octonions and denoted Ca, we have to do with an object that is almost a field, but which lacks associativity. The octonions form a vector space of dimension 8 over R. They are defined with a basis



consisting of the identity element 1 and seven other generators fei g .i D 1; :::; 7/ for which the multiplication table is furnished by the triangles inscribed in a heptagon as in Fig. I.XYZ.2.

Fig. I.XYZ.2.

More precisely, we consider all triples of the form fei ; eiC1 ; eiC3 g for i going from 1 to 7 and where the additions in the subscripts are computed modulo 7. Each of these triples obeys the same laws as the triple fi; j; kg from the definition of the quaternions: ei eiC1 D eiC3 , eiC1 ei D eiC3 . Even though this law is not ultimately associative, we can nonetheless define a reasonable geometry over Ca, not just in dimension 2, but also in dimension 3 and beyond. For references on this topic, which is especially subtle in the case of the construction of the “panda” encountered in Sect. I.4, the projective plane over Ca, see 4.8.3 of [B] and the references given there. See above all the entirety of Chap. XIV of (Porteous, 1969). A recent reference on octonions is Baez (2002). We must pay strict attention to the tricks intuition can play, as soon as we are working over the complex field instead of over the reals. A plane over the complex numbers has real dimension 4. A typical example is this: what is the topology of the complement of a point in C? It is that of a cylinder, which may be contracted to a circle, whereas in R it amounts to two intervals, which contract to two points; in each instance the complement in R is not connected, which is no longer the case in C. And for the complement of two intersecting lines in the plane C 2 ? In the case of the real plane, we find four connected components. But here the complement is connected; if we contract it onto the intersection of C 2 with the unit sphere S3 in R4 , it turns out, according to Sect. II.5, that the complement has the same topology as the complement in S3 of two orthogonal circles, i.e. of a torus T2 . Another motivation for geometry over the complex numbers is the recent notion of quantum computers; these are objects that truly operate in C. Another caveat is that, when we work in higher dimensions, our intuition loses practically all its rights, all its effectiveness. We will see numerous manifestations of this



in Chap. VII. But high dimensions are absolutely necessary in numerous mathematical studies, both theoretical and applied, especially since about 1950. Loading an image on a computer requires working in spaces of, say, dimension 10 000. Think also about credit cards, of cryptography, of biology and the error correcting codes that are studied in Chap. X. All of linear programming involves a potentially large number of “consumers”, and hence enormous dimensions. Another need is that of functional analysis, which leads to the study of convex sets in very high dimensions; see Chap. VII. Our intuition will be put to the test there, starting with the volume of spheres of radius 1, for which the volume tends ultra-fast to zero as the dimension becomes large. Still another need is that of physics, where statistical mechanics treats sets of particles with numbers of order 1023 , which thus lives in spaces having such dimensions. Also interesting is the remark due to Pierre Cartier that, in fact, we scarcely comprehend more than dimension 1. The reason isn’t merely that our brain works in a linear fashion, sequentially, for in fact (as far as is known today) it works rather like a parallel computer. It seems that this is simply for the crude reason “that it’s simpler”. For in fact dimension 2 isn’t yet well understood; pattern recognition, for example, is in its infancy. Dimension 3, however indispensable for all the objects of space, harbors numerous open problems, as we shall see. Some people think that our difficulties in dimension 3 are due, at least in part, to the fact that the group of rotations in space is not commutative. But all these reflections on the profound nature of mathematics and the functioning of our brain are still in limbo. To paraphrase Pierre Cartier, in (Cartier, 1991): “The very subjective, if not to say blurred, character of art criticism is without doubt due to this characteristic. In our epoch of intensive use of computers and of computer-aided creativity, there is a regrettable gap.” Making a leap from dimension 1 to dimensions 2 and 3 is one of the present obstacles to progress in mathematics. Similar obstacles have led to the creation of non-Euclidean geometry and to the discovery of Gödel’s theorem. The fundamental theory of affine geometry, seen for the real plane in Sect. I.3, is proved by the same technique for all dimensions and all fields. But there are two very important modifications that we give here; for more, see 2.6 of [B]. We proceed first to the obvious exclusions (the miracle is that there aren’t others). First the case of dimension 1, where the collinearity condition is vacuous: every bijection preserves lines, since there is but one line here! But things get more complicated according to the qualities of the field K considered. It is necessary at first to completely exclude the field Z2 of two elements, for then the affine line contains only two points and the collinearity condition is again vacuous. But above all the proof of I.3.1 goes through thanks to the fact that only the identity is an automorphism of the real number field.



It’s not entirely the same for other fields, e.g. the field of complex numbers admits the automorphism that transforms z into its complex conjugate z: N conjugation preserves sums and products. A semi-affine transformation of an affine space into a field K is, by definition, modulo a translation, a semi-linear transformation (more generally we can speak of a semi-affine transformation of one affine space into another), i.e. a transformation f such that f .x C y/ D ./f .x/ C ./f .y/, where W K ! K is any automorphism of the field K. We point out that C admits lots of other automorphisms besides the identity and the conjugation z 7! z, N but if we further require continuity, then there remain only those two. The fields of characteristic k different from 0 admit the celebrated Frobenius automorphism x 7! x k , whose importance in mathematics should not be underestimated, e.g. it enters Frobenius, Georg into Deligne’s proofs of Ramanujan’s conjectures; see Sect. III.3 (and Sect. III.6). On the other hand, the quaternion automorphisms are easy to classify; see 8.12.11 of [B]. And so these “refined” mappings are never so frightening as they can be in the case of the complex numbers. We now define a projective space of dimension n over the field K as the set of “vectorial lines” (one-dimensional subspaces) of a vector space of dimension n C 1 over K. All these spaces are in fact the same, so we can speak of the projective space of dimension n over K; we denote it by KPn . Algebraically, it’s the quotient of KnC1 n0 modulo the equivalence relation such that w v if and only if there exists k 2 K (necessarily nonzero) for which w D kv. The (projective) lines, planes, hyperplanes of KPn correspond to vector subspaces of KnC1 of respective dimensions 2, 3 and n. The case of the projective line is the object of study of Chap. 6 of [B]. We introduced the projective spaces essentially for the purpose of completing affine geometry so that the intersection theorems could be presented without exceptional cases in their statements; but apart from the miracles mentioned below, the idea of considering the elements of a vector space “within a scalar factor” is very natural, if not to say indispensable. We will see two examples in this book, the first is the space of circles and spheres in Sect. II.6; the second is that of conics and quadrics (see Sect. IV.7). It isn’t possible to really thoroughly understand circles, spheres, conics and quadrics without introducing the projective space formed by their equations. This is very much in the spirit of Jacob’s ladder. Here now, very briefly, are the essential properties of projective geometries. The complex projective spaces are among the most important objects of algebraic geometry. The projective transformations are (bijective) linear mappings of KnC1 n0 retracted onto KPn ; they are also called hom*ographies. They obviously form a group, called the projective group, denoted by GP.n I K/, whose structure is that of the quotient of the linear group GL.n I K/ by the group of multiples of the identity. In terms of coordinates, it is the multiplicative group of the .n C 1/ .n C 1/ matrices with elements in K, modulo the group of multiples of the identity matrix. The projective



spaces over finite fields are encountered in combinatorial geometry. Here is a typical example in the spirit of Sylvester’s problem of Sect. I.1: whatever its dimension, the projective space Z2 Pn over the field Z2 of two elements is a finite set such that, for each pair of points, there is a third point on the line that joins them. In fact, all projective lines over Z2 have three elements: two affine and their point “at infinity”. The hom*ographies of the projective line are classified and studied in detail in Chap. 6 of [B]. The involutions (hom*ographies which when squared yield the identity) play a particularly important role. If a hom*ography has two distinct fixed points a and b, then the cross ratio Œa; b; m; f .m/ is constant; conversely, the relation Œa; b; m; f .m/ D k defines a hom*ography, an involution when k D 1. As indicated above in Sect. I.7, there is a second fundamental theorem of projective geometry valid in any dimension and over any field; but as in the affine case it is necessary at first to make the obvious exclusion of dimension 1. But subsequently there will be no need to exclude the field of two elements; for a projective line, in contrast to an affine line, always contains at least three points. The conclusion is that we find only semi-projective transformations, i.e. projective transformations modified if need be by an automorphism of the field; see as needed 5.5.8 of [B] and the references mentioned there. The first fundamental theorem of projective geometry states that the transformations in projective dimension n are sufficiently abundant in order to transform two arbitrary .n C 2/-tuples of points into one another; see as necessary 4.5.10 of [B]. The result is trivial using linear algebra, although it bears a ponderous name, which comes from the fact that it is difficult to prove and is much more concealed if we pursue the axiomatic theory of projective geometry. Here we have parachuted the projective spaces with linear algebra, while axiomatic projective theory constructs them (more or less completely) with axioms bearing on points, lines, etc., and properties required for their various intersections, properties that are trivial in linear algebra. Readers interested in the axiomatic theory can consult the two basic books that exist Artin (1957) and Baer (1952). The theory remains difficult and, in our opinion, of mediocre elegance for the case of projective planes, since there is no Desargues’ theorem seen in Sect. I.4 at our disposal; readers may be able to form an opinion with a cultural text such as (Lorimer, 1983). In complex geometry it is very important to know that, while topologically the real projective line is the circle S1 , the complex projective line is the sphere S2 ; see the end of Sect. I.6. Note that the real plane as a real object is projectified into RP2 by appending a line at infinity, topologically a circle. But the same real plane, when seen as a complex line, is projectified with a single point at infinity, which yields the sphere S2 topologically. Thus we have two quite different compactifications of R2 . In Sect. II.3 we will see a third, also utterly essential, where compactification provides the topology of a disk; the interior is R2 but the compactification adds the boundary circle. For more on the topology of projective spaces, which are elements of constructions essential in algebraic topology, there are some references in Chap. 4 of [B]. We mention only that RPn is orientable for all odd



n, nonorientable otherwise. Let us add here, however, that the complex projective spaces are the generating elements of the cobordism ring in algebraic topology. The theory of cobordism, which dates from the 1950s, is a classification of compact difazCb ferential manifolds; see Husemoller (1975). As for hom*ographies z 7! czCd , which dominate a large part of mathematics, we will encounter them in Sects. II.3 and II.4. For a modern proof of the fundamental theorem of geometry, affine or projective, see Faure (2002). A configuration of an affine or projective plane is simply a specification of a set of points and the lines joining those points. Such a configuration evidently does not have any interest unless we require particular properties. We say that a configuration is of type .pq ; rs /, where p; q; r; s are integers, if there are p points and r lines, such that for each point of there pass exactly q lines of and if every line of contains exactly s points of . An accounting implies the relation pq D rs. The proof of the existence of a configuration of a given type may be trivial, but also more or less difficult. The complete quadrilateral is a configuration .43 ; 62 /; existence is trivial but nevertheless interesting to interpret in Euclidean geometry as that formed by the six centers of hom*othety of three circles; this has served us above for the Reye configuration. In Sect. I.7 we saw its very useful property of harmonic conjugation, proved by “transfer to infinity”. Here we give a purely projective proof “in situ”. The complete quadrilateral is thus the figure formed by four lines in general position (the sides) and their six points of intersection (the vertices). These points of intersection are joined pairwise by three diagonals. The property of harmonic conjugation mentioned above is the following: two vertices a and b not located on the same side (and thus located on one of the diagonals) having been chosen, and i being the point of intersection of the two other diagonals, the sides issuing from a divide the segment bi harmonically. This means that, if we let x and y denote the points of intersection of the sides passing through a with the line bi , we have .b; i; x; y/ D 1. a


e x c

α i

d y

β f

Fig. I.XYZ.3.

In Sect. I.7, this property was proved by transferring the points a and b to infinity, and thus in making a parallelogram out of the quadrilateral. The “classical”



projective proof is based on the fact that the perspectivity, or central projection, of one line onto another is a hom*ography and preserves the cross ratio. By considering the perspectivity with center a, we obtain Œb; ˛; e; d D Œb; ˇ; c; f . From the perspectivity with center i, we obtain Œb; ˛; e; d D Œb; ˇ; f; c. Thus Œb; ˛; e; d D Œb; ˛; e; d 1 , i.e. Œb; ˛; e; d D 1, the value C1 being excluded. We have finally Œb; i; x; y D Œb; ˛; e; d . With the configuration of the complete quadrilateral is associated the notion of polar of a point with respect to two lines. We say that two points m and n are conjugate with respect to the two lines D and D0 if they are conjugate harmonics with respect to the points of intersection with D and D0 of the line that joins them. Otherwise expressed, denoting these points of intersection by p and p 0 , we have Œm; n; p; p 0 D 1 on the line mn. The set of conjugates with respect to two lines D and D0 of a point a not located on D or D0 is a line passing through the point of intersection of D and D0 , called the polar of the point a in relation to the lines D and D0 . To see this, it suffices to take D D be and D0 D bd in Fig. I.XYZ.3. The point i is conjugate to the point a with respect to D and D0 , and the property of preservation of the cross ratio by perspectivity implies that a point of the plane is conjugate to a if and only if it is collinear with b and i . The polar of a is thus the line bi . The point b represents a special case: we agree that it belongs to the polar and agree once for all that if a and b are two distinct points of a line, then Œa; b; b; b D 1, a convention that is justified by passage to the limit. In Sect. I.3, Pappus’s theorem established the existence of configurations of type .93 ; 93 /; Desargues’ theorem proves the existence of configurations of type .103 ; 103 /. The Hesse configuration, in the complex case, encountered as a counterexample to Sylvester, is of type .94 ; 123 / and establishing its existence was not trivial. Problems concerning the existence and classification of configurations have been very fashionable for some time. Personally we find the subject less agreeable when it is made systematic; see the figures on pp. 109–111 of Hilbert and Cohn-Vossen (1952). In contrast, a particular configuration can be fascinating, even critical, when it is related to geometric properties that are more or less profound. We have seen this above; we now mention some more examples. The projective spaces over finite fields furnish important configurations because of their combinatorial consequences, the simplest example being the projective plane over the field Z2 of two elements; it contains 7 points and 7 lines and in its totality thus forms a configuration of type .73 ; 73 /. The figure below includes some real (and continuous) lines (and a circle); they have no reality in Z2 P2 , they might actually be any curves; the fact that there is a curve through three points merely conveys the information that they are on the same projective line. Readers will be able to show that such a configuration is impossible in real geometry; this is why we have drawn, in desperation, a circle in the above figure. What’s important are the combinatorics of the figure, in the sense that it associates, by incidence relations, three objects of the second type with every object of the first type. The projective spaces over finite fields have abundant application: to geometry in



Fig. I.XYZ.4. The projective plane over Z2

the spirit of this chapter (“finite geometries”), to combinatorics and to error detecting codes. We treat these codes in Sect. X.4. As for combinatorics, they are already present in Sylvester’s problem in Sect. I.1. Combinatorics is a very recent discipline and is establishing a presence in more and more mathematical domains. For more on projective spaces over finite fields, see Chap. 9 of Lidl and Niederreiter (1983) and the entire book (Hirschfeld, 1979). For more on combinatorial geometry, see all of Pach and Agarwal (1995); for a view of some very nice particular cases, see the corresponding part of Aigner and Ziegler (1998). The title Proofs from the Book, indicates this: it’s a work containing a goodly number of proofs that are thought to be “perfect”. Their collection is what Erdös called “The Book”, saying: “you don’t necessarily have to believe in God but, as a mathematician, you have to believe in The Book”; see its introduction. Regarding problems on geometric probability, of which the first historical case is that of Buffon’s needle, various references are: first the classic and easy (Santalo, 1976), then the more difficult and recent Part 5 of Gruber and Wills (1993). Finally the mm-spaces of Gromov, which are the subject of Chap. 3 12 of Gromov (1999a), are presented in an informal way in Berger (2000a); and, for a somewhat different perspective, see Talagrand (1995). Bibliography [B] Berger, M. (1987, 2009). Geometry I,II. Berlin/Heidelberg/New York: Springer [BG] Berger, M., & Gostiaux, B. (1988). Differential geometry: Manifolds, curves and surfaces. Berlin/Heidelberg/New York: Springer Aigner, M., & Ziegler, G. (1998). Proofs from THE BOOK. Berlin/Heidelberg/New York: Springer Aigner, M., & Ziegler, G. (2003). Raisonnements divins. Berlin/Heidelberg/New York: Springer Arnold, V. (1999). Symplectization, complexification and mathematical trinities. In E. Bierstone, B. Khesin, A. Khovanskii, & J. E. Marsden (Eds.), The Arnoldfest (pp. 23–28). Providence, RI: American Mathematical Society Arnold, V. (2000). Polymathematics: Is mathematics a single science or a set of arts? In V. Arnold, M. Atiyah, P. Lax, & B. Mazur (Eds.), Mathematics: Frontiers and perspectives (pp. 403–416). Providence, RI: American Mathematical Society Artin, E. (1957). Geometric algebra. New York: Interscience Baer, R. (1952). Linear algebra and projective geometry. New York: Academic Press



Baez, J. (2002). The octonions. Bulletin of the American Mathematical Society, 39(2), 145–205 Bárány, I. (2008). Random points and lattice points in convex bodies. Bulletin of the American Mathematical Society, 45, 339–365 Berger, M. (2000a). Encounter with a geometer I, II. Notices of the American Mathematical Society, 47(2), 47(3), 183–194, 326–340 Berger, M. (2005). Dynamiser la géométrie élémentaire: introduction à des travaux de Richard Schwartz. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni. Accad. Naz. Lincei, Rome, Ser. 25, 127–153 Besse, A. (1978). Manifolds all of whose geodesics are closed. Berlin/Heidelberg/New York: Springer Bokowski, J., & Richter-Gebert, J. (1992). A new Sylvester-Gallai configuration representing the 13-point projective plane in ‡ 4 . Journal of Combinatorial Theory B, 54, 161–165 Bott, R. (1988). Morse theory indomitable. Publications mathématiques de líInstitut des hautes études scientifiques, 68, 99–114 Bott, R., & Tu, L. (1986). Differential forms in algebraic topology. Berlin/Heidelberg/New York: Springer Cartier, P. (1991, octobre). “Le calcul des structures à deux ou trois dimensions est un défi pour les mathématiciens”. Pour la Science, 168, 8–10 Coxeter, H. (1964). Projective geometry. New York: Blaisdell Coxeter, H. (1974). Regular complex polytopes. Cambridge: Cambridge University Press Coxeter, H. S. M. (1989). Introduction to geometry. New York: Wiley Croft, H., Falconer, K., & Guy, R. (1991). Unsolved problems in geometry. Berlin/Heidelberg/ New York: Springer Darboux, G. (1917). Principes de géométrie analytique. Paris: Gauthier-Villars Du Val, P. (1964). hom*ographies, quaternions and rotations. Oxford: Oxford University Press Faure, C.-A. (2002). An elementary proof of the fundamental theorem of projective geometry. Geometriae Dedicata, 90, 145–151 Fischer, G. (1986a). Mathematische Modelle [Mathematical models]. Braunschweig: Vieweg Fischer, G. (1986b). Mathematical models: Commentary. Braunschweig: Vieweg Frenkel, J. (1973). Géométrie pour l’élève professeur. Paris: Hermann Gromov, M. (1999). Metric structures for Riemannian and non-Riemannian manifolds. In J. Lafontaine & P. Pansu (Eds.). Basel: Birkhäuser Gruber, P., & Wills, J. (Ed.). (1993). Handbook of convex geometry. Amsterdam: North-Holland Hilbert, D., & Cohn-Vossen, S. (1952). Geometry and the imagination. New York: Chelsea Hilbert, D., & Cohn-Vossen, S. (1996). Anschauliche Geometrie. Berlin/Heidelberg/New York: Springer Hirschfeld, J. (1979). Projective geometry over finite fields. Oxford: Clarendon Press Hirzebruch, F. (1987a). Selecta. Berlin/Heidelberg/New York: Springer Hirzebruch, F. (1987b). Collected papers. Berlin/Heidelberg/New York: Springer Husemoller, D. (1975). Fibre bundles. Berlin/Heidelberg/New York: Springer Kelly, L. (1948). The neglected synthetic approach. The American Mathematical Monthly, 55, 24–26. (Kelly’s solution of Sylvester’s problem can be found at the end of an article by H.S.M. Coxeter in the same issue.) Kelly, L. (1986). A resolution of the Sylvester-Gallai problem of J.-P. Serre. Discrete & Computational Geometry, 1, 101–104 Klee, V. (1969). What is the expected volume of a simplex whose vertices are chosen at random from a given convex body? The American Mathematical Monthly, 76, 286–288 Kuiper, N. (1984). Geometry in total absolute curvature theory. In W. Jäger, J. Moser, & R. Remmert (Eds.), Perspectives in mathematics: Anniversary of oberwolfach (pp. 377–393). Basel: Birkhäuser Lidl, R., & Niederreiter, H. (1983). Finite fields. Cambridge: Cambridge University Press Lorimer, P. (1983). Some of the finite projective planes. The Mathematical Intelligencer, 5, 41–50 Orlik, P., & Terao, H. (1992). Arrangements of hyperplanes. Berlin/Heidelberg/New York: Springer



Pach, J., & Agarwal, P. (1995). Combinatorial geometry. New York: Wiley Porteous, I. (1969). Topological geometry. London: Van Nostrand-Reinhold Santalo, L. (1976). Integral geometry and geometric probability. New York: Addison-Wesley Scheinerman, E., & Wilf, H. (1994). The rectilinear crossing number of a complete graph and Sylvester’s “four point problem” of geometric probability. The American Mathematical Monthly, 101, 939–943 Schwartz, R. (1993). Pappus’s theorem and the modular group. Publications mathématiques de líInstitut des hautes études scientifiques, 78, 187–206 Schwartz, R. (2001). Desargues theorem, dynamics and hyperplane arrangements. Geometriae Dedicata, 87, 261–283 Smith, A. (2000). Infinite regular sequences of hexagons. Experimental Mathematics, 9, 397–406 Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publications mathématiques de líInstitut des hautes études scientifiques, 81, 73–205 Veblen, O., & Young, J. (1910–1918). Projective geometry. Boston, MA: Ginn and Co.

Chapter II

Circles and spheres II.1. Introduction and Borsuk’s conjecture If the first chapter was essentially about affine and projective geometry, we now want to enter the Euclidean realm, i.e. we will now have a metric at our disposal, a notion of distance between points, with subsidiary notions such as circles and spheres. The basic reference for circles and spheres, completely authoritative at the time of its publication, is Coolidge (1916). We have made a critical selection from the enormity of classical results; see the very beginning of Sect. II.2. But of course above all we have chosen to talk about recent results, all the more if they require a climb up the ladder. Borsuk’s conjecture. In the spirit of this book and before touching on problems leading us to configurations that are natural but rather sophisticated, we must speak about Borsuk’s conjecture. Its statement is trivial, except that it deals with arbitrary dimension. It is one of the simplest assertions in all of Euclidean geometry, for it doesn’t involve anything but distance. Here it is formulated as a question, where Ed denotes d -dimensional Euclidean space without reference to any particular coordinatization (in contrast, Rd is d-dimensional Euclidean space with canonical coordinates): (II.1.1) In the Euclidean space Ed , can we decompose any bounded part E into d C 1 parts of diameter strictly less than the diameter of E?

Fig. II.1.1.

The figures above seem to show that the problem is indeed trivial. Recall that the diameter of a bounded set E of an arbitrary metric space is the supremum of the distance between points of E: diam.E/ D supfd.x; y/ W x; y 2 Eg. The conjecture is certainly true for d D 1, and it is also evident that d C 1 is necessary; that d doesn’t suffice is left to the readers. We are going to describe in a bit of detail some

M. Berger, Geometry Revealed, DOI 10.1007/978-3-540-70997-8_2, c Springer-Verlag Berlin Heidelberg 2010




of the history of this conjecture, whose various aspects seem enriching. For the algebraic topologist, we need first say that in Borsuk (1933) that among other things an important result was proved that had been conjectured by Ulam, specifically that each continuous mapping of the sphere Sd into Ed sends at least one pair of antipodal points onto the same point, which is intuitive enough for d D 1 or d D 2. At the end of the text, Borsuk conjectured (II.1.1), or rather he merely stated the problem, for he was much too good a mathematician to make a conjecture without knowing a lot more. In fact his theorem just quoted implicitly suggests that we can’t ever decompose the sphere Sd into only d pieces of smaller diameter, whence the idea for the problem. Borsuk’s proof of the quoted theorem was a bit complicated, but H. Hopf pointed out to him that it is really instantaneous thanks to degree theory; see Chap. 7 of [BG].

Fig. II.1.2. Borsuk’s theorem for d D 1 and d D 2

Borsuk’s conjecture didn’t lack for dedicated investigators; but even for the plane, the matter isn’t quite as trivial as it perhaps seems. For this and for the more recent history, the reference is 19.3 of Grünbaum (1993). We only mention that, for the plane, we can proceed thus: we inscribe E in a convex set E0 of constant width (see the end of Sect. VI.9) equal to the diameter of E. This is easily done by taking the intersection of all the circular disks containing E. Next the continuity principle (turn around!) shows that there exists a regular hexagon circ*mscribed about this new body whose opposite sides are a distance apart equal to our diameter. We cut p 3 this hexagon into three (non regular) hexagons of diameter equal to 2 , supposing diam.E/ D 1; this cuts E a fortiori in the same way.pWe have thus succeeded, moreover with a gain of a factor always equal to at least 23 . For three dimensional space, it was necessary to await (Eggleston, 1957) for the first proof. No proof is really simple. The best gain presently known is q very p 0:9987:::; whereas 0:888::: D .3 C 3/=6 has been conjectured. The fact that it is much closer to 1 than for the plane allows us to predict a difficult prospect for dimension 4. Here again the proof consists of inscribing E in an “adapted” polytope, specifically and for the moment a regular octahedron whose parallel faces are a distance apart equal to the diameter in question and, after having trimmed it down at the vertices, subjecting it to some subtle dissections (see Fig. II.1.4). Later proofs of a combinatorial nature were found using finite point sets. The properties of their



Fig. II.1.3. Circ*mscribe about our body equilateral triangles in all directions; by continuity at least two of these triangles are congruent. Moreover, as these two triangles have their parallel sides at a constant distance, their intersection must be a regular hexagon

c Springer and H. Martini Fig. II.1.4. Boltyanski, Martini, Soltan (1997)

mutual distances are studied; see e.g. 13.15 of Pach and Agarwal (1995), Chap. 1 of Zong (1996) and Boltyanski, Martini, and Soltan (1997). Meanwhile, for higher dimensions, it was necessary to make do with results for sets E having special properties. First, the solid balls in all dimensions are easily cut up see the figure above e.g. by inscription in a regular simplex. Next, every E having a center of symmetry is cut up using the following elementary observation: when two points p; q of E attain the diameter of E, then E is contained between the two parallel hyperplanes that pass through p and q and are orthogonal to the line pq. If there is a center of symmetry for the cut (and there must be, since E has a center of symmetry), this can’t be anything other than the midpoint O of pq; and thus E is entirely contained in the ball whose center is O and whose boundary sphere passes through p; q. We now need only cut this ball as above or in some other way.



We owe to Hadwiger an extremely elegant proof of the fact that (II.1.1) is valid assuming E is a set whose boundary @E is a smooth hypersurface (a differentiable submanifold of dimension d 1 in Rd ). To do this we consider the Gauss-Rodrigues mapping (see as needed Sect. VI.6), i.e. the mapping that sends each point of @E to the point of the unit sphere Sd 1 of Rd that defines the unit exterior normal at this point. We then cut Sd 1 as above and consider the parts of E that are the inverse images of the pieces obtained. By the elementary observation above, all the points of such an inverse image are a distance apart strictly less than the diameter of E since they never result from antipodal points of Sd 1 . As these parts are compact, their diameter is strictly less than that of E.

Fig. II.1.5. The Gauss mapping, discovered in fact also by Rodrigues, associates with each point of the smooth surface its unit normal vector

Thus, if we have doubts about Borsuk’s conjecture and seek a counterexample, it won’t do to use either a set that is symmetric with respect to a point or a set whose boundary is a (smooth) hypersurface. The conjecture was effectively resolved negatively for the first time in Kahn and Kalai (1993). We introduce, for each dimension d , the Borsuk number Bors.d / as being the least integer such that each bounded subset E of Ed can be decomposed into Bors.d / subsets, each of diameter strictly less than that of E. Following Schramm, p by techniques called illumination of convex sets, we know that Bors.d / 6 . 3=2 C o.1//d=2 for d sufficiently large. It is a purely combinatoric result; for this and what follows see Chap. 15 of Aigner and Ziegler (1998) and Chap. 13 of Pach and Agarwal (1995). On the other hand, this apparently enormous bound isn’t so ridiculous, since we now know that p d Bors.d / > .1;25/ for d sufficiently large. This shows that the conjecture is false starting with 964 (use a pocket calculator), but refinements of the technique of Kahn and Kalai now permit a descent down to 561. The counterexamples are of the following type: we start off from vertices of the discrete cube f1; C1gn for which the first coordinate equals C1, and we construct 2 a subset E of Rn whose n2 coordinates are the products two at a time of those of the point in question. Some combinatorics on the mutual distances of points of E, along



with some arithmetic, permit the conclusion. But the critical dimension, between 3 and 561, is not known at present. Needless to say, the conjecture is open for dimension 4; and dimension 4 is the object of many other open problems, in particular the problems concerning the sphere S3 that we will see in Sect. III.5. Here now is a thought, a reflection of Gromov (unpublished): a good candidate for refuting “Borsuk’s conjecture”, for “pure” geometry, is the Veronese manifold that generalizes the surface encountered in Sect. I.7 (see also Sect. IV.9), that is an embedding in R6 of in the real projective plane. The real projective plane is characteristically asymmetric since the central symmetry of the sphere has been destroyed precisely in forming the quotient by this symmetry. Note that the discrete version of the projective spaces is precisely the set of vertices of the cube f1; C1gn for which the first coordinate is C1. This Veronese manifold is given in “coordinate free” fashion by the quotient mapping of the mapping y 7! .x y/2 of Sd 1 into the space Rd.d C1/=2 of quadratic forms on Rd (y and y have the same image). In coordinates, this mapping is nothing other than .: : : ; xi ; : : :/ 7! .: : : ; xi2 ; : : : ; : : : ; xi xj ; : : :/. The result of Gromov is: R =2 R =4 (II.1.2) If 0 sind 1 t dt = 0 sind 1 t dt > d.d C 1/=2, then the Veronese manifold in Rd.d C1/=2 cannot be decomposed into d.d2C2/ C1 connected pieces of smaller diameter. We thus obtain a counterexample, but under the restriction of cutting into connected subsets. In return, the dimension that refutes this (weak) conjecture is much lower, i.e. d D 55. Use a pocket calculator to find 10 starting with the inequality of (II.1.2), but 55 for .10 11/=2.

Fig. II.1.6. A symmetrization of the sphere Sd 1 : apply Fubini’s theorem; moreover the diameter can only diminish

The idea behind the proof of (II.1.2) is to think about Sect. VII.12.B, whose essence is that the caps of the spheres have a very small volume so long as we haven’t reached the equator, i.e. the radius hasn’t reached =2. See as needed Sects. VII.6 and VII.12 for the volumes of spherical caps. Here we work in the



real projective space RPN , which is realized isometrically by the Veronese manifold above. Let A be a connected portion of RPN with diameter less than =2. We can then lift it within SN into A0 , which will be contained in a hemisphere. Now in SN we have an isodiametric inequality which generalizes that of the penultimate subsection of Sect. VII.7: Among all the domains of SN contained in a hemisphere and of given volume, the smallest diameter is attained exactly by the spherical caps. The proof proceeds by the Steiner symmetrization (see Sect. VII.5) for this method extends without difficulty onto the sphere while paying attention to the convexity. It’s for this reason that we need to be in a hemisphere from the outset. The diameter here being less than =2, the greatest volume will be obtained by a cap of radius strictly smaller than =4. We then finish with a calculation of volumes; the disjoint parts partitioning A0 cannot be greater in number than the quotient of the total area of the hemisphere with that of a cap of radius =4. A recent text on Borsuk’s conjecture is Raigorodskii (2004). For more on Borsuk’s conjecture, as well as for other “strange phenomena” in geometry, see the book that is entirely devoted to them: Zong (1996), especially Sects. VII.11, VII.12, VII.13.D and the slicing conjecture in Sect. X.6. II.2. A choice of circle configurations and a critical view of them We present some figures formed by circle configurations (see Sect. II.XYZ and in particular “a scandal to repair”) in the Euclidean plane E2 . We will comment on them later, restricting our attention to those that arise from the idea underlying this work. Specifically, we study geometric situations that can be stated quite simply, but which have led, and possibly still lead, to more intensely conceptual developments, to rungs up Jacob’s ladder. We comment on what exists for finding proofs (entire or partial) or for understanding them deeply or placing them in a more general context, etc. We hope that readers will take the trouble to contemplate these figures at length, to compare them and decide which are interesting and which, to the contrary, seem unaesthetic or otherwise unappealing. Chapter I was set in the affine context; we are now entering into the metric context. Readers destined to want to embrace everything, i.e. to climb the ladder in order to unify their vision, will find in Sect. II.XYZ how to connect the metric to the complex projective setting by projectifying, and then complexifying, the Euclidean plane. For those who like constructions and practical matters, we should point out that circles are constructed with compasses, whereas linear rulers and also graduated goniometers (protractors) for measuring angles are always produced in a “physical” and thus approximate way; see Sect. II.3 below, Sect. V.7 and above all Sect. II.XYZ of this chapter for more on this.



And then there is the question of critical perspective: an historical example is that of “triangle geometry”, a discipline which has seen a disproportionate flowering, but which to our knowledge has contributed absolutely nothing in the way of Jacob’s ladders. The analysis (Davis, 1995) is very interesting, even if we don’t completely share the conclusions. References on the subject can be found there of course, among them the classic (Lalesco, 1952).

Fig. II.2.1. The butterfly theorem

Fig. II.2.2. The four circles tangent to the three sides of a triangle



Fig. II.2.3. The Wallace-Simson line

To our knowledge, the butterfly (Fig. II.3.1) has only limited interest: the correctness of the middle of this completely naive figure isn’t easy to show and we leave conviction to readers. In fact it’s a trick problem: the result is formulated in a metric (Euclidean) fashion, although it is really a theorem of projective geometry. In fact, let us introduce the polar D of the point a with respect to the circle C and consider the (harmonic) hom*ology defined by the pair .a; D/; it’s an involutive projective transformation (see Sect. I.7). By construction, this hom*ology preserves C and D and thus interchanges the two points p and q. But on the line D it is a hom*ography that preserves the point at infinity, thus an affine transformation. Now, it preserves the midpoint a of pq: it’s the metric symmetry of D with center a, and thus a is clearly also the midpoint of pq. Apart from this, to our knowledge, the butterfly theorem doesn’t yield any movement on Jacob’s ladder. The fact (Fig. II.3.2) that there exist four circles tangent to the three sides of any triangle in the Euclidean plane is not profound; we use the interior and exterior bisectors of the triangle (which gives a new way of obtaining a complete quadrilateral, cf. Sect. I.7). Nonetheless, we want to mention this figure, for its generalization to dimension 3 (and higher) sets a trap, pointed out to the author by his school mathematics teacher, the late Jean Itard, when he was 16; a trap into which practically all the mathematicians to whom the author has posed the problem have fallen, at least



Fig. II.2.4. An infinite chain of theorems: three lines: a circ*mscribed circle; four lines: four coincident circles; five lines: (what is the result?); next, go to infinity

Fig. II.2.5. Feuerbach’s result

when they were not given much time to reflect. The solution in every dimension does not require that we climb very high; it suffices to use barycentric coordinates (introduced in Sect. I.XYZ) and considerations of volume and area, all of which



Fig. II.2.6. The seven circles theorem and that of five circles. Above and below, the conditions for the tangency of the circles are satisfied, but the lines are not coincident; or the four cocyclic points, and “one time in two”

can be found in detail in Sect. 10.6.8 of [B]; but we are going to give the essentials, for the result is fascinating and the proof trivial, once barycentric coordinates have been introduced. One of the appeals of this very elementary geometry problem is that, once the dimension exceeds 2, the number of spheres tangent to the faces of a simplex depends on the initial figure; for example, in dimension 3 this number depends on the areas of the faces of the tetrahedron considered; it is equal to 8 typically, but decreases to 5 for the regular tetrahedron. It is equal to 2d in general. In dimension three, as far as pure geometry is concerned, it’s a matter of looking at the


Fig. II.2.7. Steiner’s alternative. Things either close up again no matter what the initial circle, or else they never close. I [B] Géométrie. Nathan (1977, 1990) réimp. Cassini c Nathan Édition (2009)

Fig. II.2.8. Eight circles tangent to three given circles. I [B] Géométrie. Nathan (1977, c Nathan Édition 1990) réimp. Cassini (2009)

Fig. II.2.9. Morley’s theorem. I [B] Géométrie. Nathan (1977, 1990) réimp. Cassini c Nathan Édition (2009)




Fig. II.2.10. Six circle theorem: whatever the initial circle, the operation closes up at the end of the six circles, or two rounds

Fig. II.2.11. Poncelet’s great theorem for circles. I [B] Géométrie. Nathan (1977, 1990) c Nathan Édition réimp. Cassini (2009)

combles (for the meaning of the French word “comble” (literally “attic” in English, see Fig. II.2.12 below). It’s also a nice example for showing that regularity isn’t always the gauge of optimality; we encounter this symmetry breaking in other places: Sects. III.3 and X.6. Here now are the details for ordinary space: we evidently use barycentric coordinates fi g (i D 1; 2; 3; 4) which yield the tetrahedron, with what was said in Sect. I.XYZ.

Fig. II.2.12. The interior, a truncation, a trihedron, two opposed combles

Now, if the ai denote the areas of the faces associated with the tetrahedron, V its volume and r the radius of a sphere tangent to the four faces, we must have



j1 j j2 j j3 j j4 j r D D D D : a1 a2 a3 a4 3V The sphere on the interior is that of radius r D 3V , a2 Ca3 Ca4 a1

3V , a1 Ca2 Ca3 Ca4

those of the

truncations have for radii r D etc. and thus always exist. In contrast, for two negative signs (which correspond to a pair of opposed combles, only one of the two numbers a1 C a2 a3 a4 and a3 C a4 a1 a2 is positive; moreover it may be zero for certain tetrahedra. Whence eight spheres in the general (“generic” is now the more common term) case. In particular, only five for a tetrahedron all of whose faces have equal areas (we leave to readers the study of these tetrahedra), and otherwise 6 or 7 according to the case. For a study, repulsive because of its quixotic refusal to use barycentric coordinates, see Rouché and de Comberousse (1912, v. II, p.653). Wallace’s theorem (Fig. II.2.3), long attributed to Simson, states that the three projections of a point onto the three sides of a triangle are collinear if and only if the point in question belongs to the circ*mscribed circle of the triangle. It prepared the way for numerous developments, for example finding when two points of the circle have orthogonal Wallace-Simson lines; or to find the envelope curve of these lines of Simson when the point traverses the circle; for all of this see the references given in 10.9.7 of [B]. However, the solution isn’t so very simple; for example, a brutal calculation using coordinates will discourage the majority of readers, even professionals. The solution “in depth” requires only a very small climb up the ladder, specifically the angular property of plane circles (called, in old French books, “l’arc capable”):

Fig. II.2.13.

However, it should be carefully noted that, for ordinary angles (between 0 and 180 degrees), there are two cases for the figures. We escape this difficulty with the notion of oriented angles for lines, but it is ultimately rather hard to explain well, see 8.7 of [B]. But we won’t linger here; for, to our knowledge, no path has been cleared for any very interesting developments.



However, regarding the theory of oriented angles, we mention an application of a rather rare type, specifically the chain of theorems (Fig. II.2.4) associated with finite sets of lines in the Euclidean plane. Three lines determine a triangle, and we have its well defined circ*mscribed circle. Four lines determine four triangles, and we can ask what happens regarding their four corresponding circ*mscribed circles: the answer, far beyond what we might hope, is that they have a point in common, are coincident. Next, let there be five lines: with four at a time we associate the point of the preceding result. Then the five points belong to a single circle. It is left to readers to formulate the sequel and to prove all that, which is not hard with the angular property of circles; if there is difficulty, they can read the classic works or Section 10.9.8 of [B], but above all (Coolidge, 1916). It is fascinating that, for the case of five lines, a weak version of this very old result was presented at the ICM (International Congress of Mathematicians) in Beijing in 2002 as a difficult and little known problem, with the consequence that Alain Connes chose to give his own proof! In contrast the three figures (Figs. II.2.5, II.2.6, II.2.7) that follow, associated with the names of Feuerbach, Steiner and (to the best of our knowledge) with an unknown person, require a new tool: inversion. This not only yields proofs of the three results that are rather obvious from the figures, but proofs that are very simple, that would be inextricable with direct calculations using coordinates. Furthermore, as we will see amply in the sequel, the inversions of the Euclidean plane are fundamental elements, the generators of the conformal group, and ultimately of hyperbolic geometry. We encounter the importance of hyperbolic geometry in numerous domains of mathematics: algebra, number theory, complex analysis, dynamical systems and theoretical physics. We now give the assertions underlying the three figures in question. Feuerbach’s theorem (Fig. II.2.5) asserts that the Euler’s circle of a triangle (also called the nine point circle), i.e. the one that passes through the midpoints of the three sides of the triangle, but also through the bases of the three altitudes and through three other points (for readers to discover!). This Euler’s circle is tangent to each of the four circles of Fig. II.2.2. We leave it to readers to show this with the aid of an appropriate inversion. The seven circles theorem (Fig. II.2.6) can be proved using only inversion, even though the proof is more sophisticated than for Feuerbach’s. However, we should beware that one time out of two holds. What does this mean? That one more condition on tangency of the circles is needed, that can only be defined by orienting each of the circles considered (by putting an arrow on them) and observing, at the point of contact, whether or not they agree. A conceptual proof thus requires placing oneself in the geometry of oriented circles. This geometry exists, but we don’t



Fig. II.2.14. The nine point circle. I [B] Géométrie. Nathan (1977, 1990) réimp. c Nathan Édition Cassini (2009)

know any good reference; what is said in Klein (1926–1949, §27), is too skeletal to permit finding the proofs that we need here. We know of no more modern exposition of them, all the elements are those given right at the end of Sect. II.XYZ. The circles are defined there (in the space of all circles) as points of a real projective space, thus via a quotient of an appropriate space by the real line R. To apportion these circles as oriented circles, it suffices to take the quotient by only the real positive half-line RC . See also the article on “conformal geometry” in the Encyclopedic Dictionary (1985). For an elementary proof (but of course via inversion) of this theorem and of the six circles theorem (Fig. II.2.10), see Evelyn, Money-Coutts, and Tyrelle (1975), but this “by hand”. The author doesn’t know how to attribute the seven circles theorem. The six circles theorem is astonishingly difficult to prove and is attributed to MoneyCoutts; see in Tabachnikov (2000) several avatars, for example the figure below:

c Springer and S. Tabachnikov Fig. II.2.15. Tabachnikov (2000)

The five circles theorem states see Fig. II.2.6 that when four circles are tangent two at a time in succession, then the four points of contact are concyclic.



As with seven circles, it is in fact also true only one time in two, as readers will verify. To prove it (in the favorable cases) we perform an inversion about one of the points of contact, which transforms two of the circles into parallel lines; and then the result follows from completely elementary angular properties of triangles and parallel lines. An amusing application (but related to the result of Rivin on the possible inscribability of polyhedra in spheres, which will be seen in Sect. VIII.8) is that, if a quadrilateral (skew) has sides tangent to a sphere, then the four points of contact lie on the same (“small”, as opposed to “great”) circle of this sphere (proof left to readers).

Fig. II.2.16.

The Steiner Alternative (Fig. II.2.7) concerns a pair of non intersecting circles; it asserts that the chain of circles (a priori infinite), starting out with an initial circle as in the figure, either closes on itself (but then this remains true for every other initial circle), or else never closes (and again, its then the same whatever the initial circle). Stated otherwise: the problem either has an infinite number of solutions (in fact a continuous infinity) or none. This depends only on the initial pair of circles: it’s an alternative of the all or nothing sort. We will see the proof in detail, using inversion, in the following section. Moreover, this proof permits us to answer practically every question suggested by the result, e.g. what is the necessary and sufficient relationship between the radii and the distance between the centers of the circles considered so that there will be closure for a given integer n? Figure II.2.8 doesn’t give rise, as best we know, to conceptual developments. In contrast, it is quite different for the search for conics tangent to five given conics, a problem that was a source of inspiration for more than a century, until at least 1952. We will treat this question, the most difficult in the realm of conics, amply in Sect. IV.9.



Morley’s theorem (Fig. II.2.9, strictly speaking without circles, but nonetheless of a Euclidean nature because of its angles) states that the trisectors of the angles of an arbitrary triangle always intersect in such a way a to form an equilateral triangle. It is celebrated as being, without any doubt, the simplest result of plane Euclidean geometry to state (not even circles appear in its statement) for which the proof is invisible; a good calculator in trigonometry will however get off without too much grief, but a non artificial geometric proof is still lacking. There wasn’t any conceptual proof, although that of Chap. IV from the second part of Lebesgue (1950) climbs the ladder a little, but remains very computational. It is not until Connes (1999) that we find a very conceptual and even very natural proof, since it is based on the group of isometries of the Euclidean plane and extends to the case of geometries over more general base fields. The six circles theorem (Fig. II.2.10) asserts the closure at the end of six steps in a chain of circles in a triangle, starting with an arbitrary tangent circle to two of the sides; the next circle must be tangent to two following sides and to the initial circle, etc. We don’t know of any conceptual proof; see the proof in Evelyn et al. (1975) or a dynamic proof can be constructed by studying the effect on the other bisectors of a movement of the center of the initial circle along the first bisector. It doesn’t seem that there is any connection with the theorem of closure of six spheres; see below in Sect. II.6. The great Poncelet theorem (for circles) is an alternative, like that of Steiner. It states that for a pair of circles C , C 0 and a given integer n, either there exists no polygon of n sides whose vertices are situated on C and whose sides are tangent to C 0 , or else there exists an infinity of such polygons. More precisely, we can choose as initial vertex an arbitrary point of C (with obvious restrictions on arcs, if the circles intersect), but there are always as many such polygons as real numbers (the power of the continuum). In fact, Poncelet proved this theorem on two circles in order to generalize it by projective transformation to an arbitrary pair of conics. His proof wasn’t completely conceptual, but in Sect. IV.8 there is a conceptual proof for circles, and a still more profound proof for the conics in general even over an arbitrary field with some references. This problem, as we will see, has haunted very many mathematicians, who have sought to comprehend it increasingly better. If there is a need, readers can gain respect for this result by attempting to prove it for themselves just for triangles (n D 3); in this case it’s an old result of Chapple from 1746, “Ponceletified” by Mackay only in 1887. By hand the case n D 4 will be found easier by an expert in “elementary” conics; the case n D 6 can also be treated by hand. The general case is of a whole other order of difficulty. See Sect. IV.8 for the relation between the integer n and the radii of the two circles, where we will also see that Poncelet’s theorem is still very much alive.



For configurations of regular hexagons, and their iteration (in the spirit of Sect. I.4), see the quite recent (Smith, 2000), typical of what new and profound things can always be done in very elementary geometry, such as the use of computers for divining new results, a trait that is common to the articles published by Experimental Mathematics. But see also Gutkin (2003), which studies the problem of finding triangles whose three vertices are situated on three concentric circles and which satisfy an extremal condition. See Sect. II.XYZ for some complementary objects relating to polarity with respect to a circle: pencils of circles. We describe them in detail, for it seems to us that this has been made necessary by the de-geometrization of mathematics instruction. For the configurations of spheres, and of circles in space, look to what accompanies the next topic. II.3. A solitary inversion and what can be done with it As an introduction we ask ourselves, in the spirit of the fundamental theorem of affine geometry seen in Sect. I.2: what are the bijective transformations of the Euclidean plane that preserve circles? For sure the isometries of the Euclidean plane, which preserve the metric, are included. These are: translations, rotations with arbitrary center and angle and the orientation reversing isometries, specifically the symmetry translations (glide-reflections); this is the place to remark, if needed, that many people believe that only the symmetries with respect to a line preserve distance while changing orientation. Then there are the hom*otheties, which of course change the radii of circles while preserving them as a whole. We define similitudes to be the transformations which are compositions of an arbitrary isometry and an arbitrary hom*othety; they form a group. It isn’t difficult to deduce from the fundamental theorem of affine geometry that the only bijections of the Euclidean plane that preserve circles are the similitudes; see Sect. II.XYZ and, if needed, Chap. 9 of [B], where other characterizations of these similitudes can be found.

Fig. II.3.1. Inversion: the product Om Om0 must remain constant



Now comes the miracle, discovered by Descartes but little clarified and exploited by him, a miracle that has flourished in exuberant fashion only since nineteenth century: by removing just one point from the plane, we obtain the existence of transformations, other than the similitudes, that preserve the family formed of the line and circles of the Euclidean plane; these are the inversions. The inversion IO;k with center O and power k (an arbitrary nonzero real) is, by definition, the bijection defined on the plane with the point O removed, given by m 7! m0 where m0 is the point situated on the line Om such that the product of the distances Om and Om0 is equal to k; if k is positive, we take m and m0 on the same side of O; if it is negative they are taken on opposite sides. Two inversions with the same center differ only by a hom*othety. It is practically trivial that IO;k permutes the circles not containing O among themselves and replaces lines (not containing O) with circles containing O, and finally conserves the lines passing through O (from which of course O has been removed). Moreover, just like the similitudes, inversion (we say inversion for short when it isn’t necessary to specify center and power) conserves angles, and this in a larger sense, specifically not only angles between circles, between lines, between circles and lines, but also for differentiable curves; (see Chap. V). The angle between differentiable curves that meet in a point is, by definition, the angle formed by their tangent lines. However, there can be a change of sign if we use oriented angles. Inversion also preserves osculating circles, see Sect. V.6. According to whether k is negative or positive, an inversion p has respectively either no fixed point or a whole circle of fixed points, of radius k, called the circle of inversion. In the k > 0 case, the interior of the associated disk is transformed to the exterior and this permits, according to a classical jest, catching a lion in the desert without difficulty or risk. Note finally that an inversion changes orientation,

Fig. II.3.2. Inversion preserves angles and osculating circles. I [B] Géométrie. Nathan c Nathan Édition (1977, 1990) réimp. Cassini (2009)



and that it is involutive, i.e. its square is the identity transformation so that iteration leads nowhere.

Fig. II.3.3. How to catch a lion in the desert

Inversions can be realized mechanically, with systems of articulated rods. In the two figures below the O points (centers of inversion) are fixed, and the inversion points for the two are m and m0 . To invert a line, we could demonstrate its property with Ptolemy’s theorem (see in Sect. II.XYZ the “scandal to be repaired”). Historically, since the appearance of steam engines, these inverters have had some importance, having to do with describing straight lines by articulated systems; such a system eluded Cayley in spite of his desire for one. It seems that these inverters have only exceptionally been applied in practice, such as for industrial systems or for tracing perfect lines in metrology; see the beginning of Sect. II.2 and above all Sect. II.XYZ of this chapter. Incidentally, Peaucellier’s inverter was used to ventilate the Crystal Palace in London. We will encounter this inverter at the end of Sect. IV.8. For the rest, in order to trace lines in practice, rulers (rather good, or sliders, according to what is required) are used, and not inverters: see Sect. II.XYZ for these considerations, which are rather little known.

Fig. II.3.4. Peaucellier’s inverter and Hart’s inverter



Fig. II.3.5.

We now prove Steiner’s alternative for a pair C , C 0 of circles. It is left to readers to create the figure below, i.e. to see that there exist two points p and p 0 such that the circles passing through them are orthogonal (intersect at right angles) to C and C 0 . It then follows that an inversion with center p or p 0 transforms the pair C , C 0 into a pair of concentric circles, the common center being the image under this inversion of the second point, since all lines passing through this point cut the new pair at a right angle, which is characteristic of the center of a circle. Now Steiner’s alternative is trivial for a pair of concentric circles; the profound reason is that these circles are conserved by the group of rotations for which the center is their common center. Interested readers can make some calculations for finding the condition linking the integer n with the number of circles in the chain and the triple formed by the two radii and the distance between their centers; it suffices to know how inversion transforms the radii of the circles. Finally, they can prove Feuerbach’s theorem, then the seven circles theorem (figures in Sect. II.2). In applying inversion, the problem is to properly choose an appropriate center (on the other hand the power, in general, matters little) in obtaining the simplest figure. If the center of inversion is the point of contact between two tangent circles, these latter are transformed to two parallel lines. In Sect. II.6 a typical and spectacular example of this technique in dimension three establishes the theorem known as that of six spheres. A naive question: why not use the same technique to show Poncelet’s great theorem, i.e. find a transformation (perhaps defined only in a portion of the plane) preserving lines and transforming a pair of arbitrary circles into a pair of concentric circles? If we want to preserve lines, we know (according to Sect. I.XYZ) that it is necessary to choose a projective transformation. But it is easy to see that there exists no projective transformation that takes nonconcentric circles into concentric circles; in fact, two concentric circles, when they have been complexified and projectified, become two bitangent conics at their cyclic points; see II.XYZ and Sects. IV.8. This is why the great theorem of Poncelet is of a whole other order of difficulty; see Sects. IV.8 and see there also, at the end, a result of Emch that encompasses the Steiner and Poncelet theorems.



II.4. How do we compose inversions? First solution: the conformal group on the disk and the geometry of the hyperbolic plane And here we have the great desire the great and good temptation! to compose inversions with different centers (the product of two inversions with the same center is a hom*othety with the same center), and this for various reasons. For example, we could hope to obtain results that are stronger than in the three examples above; but also to form a group, whether just because we like groups or, following Klein, ca. 1900 we know that the study of a geometry is very strongly tied to the structure of its groups of isomorphisms. So the idea is to compose inversions, of arbitrary number, hoping all the while that things will terminate at the end of finitely many steps; for the inversions would then just generate a group of finite dimension, as do plane isometries having symmetries about lines as generators. We would then have, in the case of inversions, generators that are easy to treat. Note that an inversion is a type of symmetry, not with respect to a line, but with respect to a circle. Finally, the great hope: the group obtained will be the one that preserves circles and also the one consisting of all transformations that preserve angles all this in a broader sense to be defined, because only the similitudes accomplish it in the strict sense of bijections of the Euclidean plane, as seen at the beginning of this section.

Fig. II.4.1.

Suppose we compose two inversions with respect to circles as in the above figure; with a bit of sketching we see that we obtain a type of rotation about the two points of intersection ! and ! 0 of the two circles, which is rather nice. On the other hand it will now be necessary to remove not only two points from the plane, specifically the two centers, but also their images and inverse images sufficient for this transformation being well defined on the set considered. Surely our composition preserves the set of lines and circles. It also preserves angles and orientation, but as we continue, we must remove more and more points, and this is not satisfying. The



definitive and global solution will come momentarily in Sect. II.5, but we start with an intermediate solution, which will be essential for the sequel. The idea is to find a domain of the plane and lots of inversions that preserve it, without having to remove any points. We will then be able to compose as many times as we wish. The right domain is simple; it’s the interior of a circle, any circle whatsoever.

Fig. II.4.2. Inversions that preserve the area of a circle

Each point on the exterior of the circle is the center of a well defined inversion that preserves the circle entirely; we can thus compose any number of such inversions without any problem. However, when do we stop? We can see it geometrically, but it’s really hopelessly complicated. It suffices to write the formula giving the inversion about the pole outside the disk to see that once our Euclidean plane is identified with C, taking the center of the disk as origin it can be written in the zz0 form of the hom*ography z 7! e i 1z composed with complex conjugation z 7! zN 0z (with the sole condition jz0 j < 1, being any real number). We only need remark zz0 that the transformations z 7! e i 1z of this type form a group, and a larger group 0z if we add their compositions with conjugation z 7! z. N This group is called the conformal group of the disk, but we will now give it, with some explanation, the name Möbius group of the circle S1 or for dimension 1 and denote it by Möb.S1 /. Without the conjugation, we obtain the connected component of the identity element. In both cases we have exactly three real parameters. Be aware that, to generate it geometrically, we need to add to the inversions that give rise to it the inversions with center infinity, i.e. the symmetries with respect to the origin of the disk (take a limit in order to convince yourself). The group Möb.S1 / contains all rotations about the origin, the symmetries about lines through the center, the compositions of two inversions whose circles intersect at a point interior to the disk; these are generalized rotations, oblique in some way. This group plays an essential role in the study of holomorphic functions. We show, as a particular case of more general results to come, that this group coincides with that of transformations of the disk that preserve circles (more precisely



the part intersected by the disk), or alternatively (differentiable) transformations of the disk which preserve angles. Finally, we now justify now the appearance of the circle S1 in the description. In fact, the hom*ographies above or the inversions evidently preserve the circle S1 , for it is the boundary of the unit disk. Moreover the action on the circle determines, and is determined by, what happens on the interior. These mappings of S1 into itself cannot be characterized as conformal, i.e. (true or infinitesimal) angle preserving, because we are in dimension 1; but it will be different in Sect. II.5, which follows. The geometry invariant under this group is that of the hyperbolic plane; see below in this section and in Sect. II.XYZ, where we will have a metric and angles.

Fig. II.4.3. Poincaré’s half-plane: the lines are the semi-circles orthogonal to the boundary; Euclidean circles are also hyperbolic circles

A very useful model of the geometry of the hyperbolic plane see for example Sect. IX.3 is the Poincaré half-plane H . To obtain this model it suffices simply to perform an inversion whose center is on the boundary circle of the preceding disk. The interior of the disk thus becomes an open half-plane, whose boundary is taken as the real axis in R2 D C and the hyperbolic group becomes the group of mappings azCb H ! H written z 7! czCd where a ; b, c, d are real and ad bc D ˙1. The connected component containing the identity element is isomorphic to SL.2I R/. In this model the lines are semi-circles that are orthogonal to the real axis and the vertical half-lines. In this model it is the easiest to prove the fundamental formula for the area of a triangle T in the hyperbolic plane as a function of its angles ˛, ˇ, (compare with formula (III.1.2) for the spherical case): .II:4:1/

Area.T/ D .˛ C ˇ C /:

Fig. II.4.4. Models of the hyperbolic plane: Poincaré half-plane and Poincaré’s disk correspond under inversion



Fig. II.4.5. Circles, triangle, ideal triangle in Poincaré’s disk

A remarkable consequence, essential for certain applications, is that the area of a triangle is always less than and equals only for the ideal triangles, i.e. those whose three vertices are at infinity. We now have all that is necessary in order to introduce the hyperbolic plane; see the historical remark at the end of this section. We will not dwell on hyperbolic geometry (also not in higher dimensions), despite its essential importance in many branches of mathematics and the fact the it was “the solution” to Euclid’s fifth postulate. Expositions can be found in many works on geometry and other subjects. We note only that hyperbolic geometry is represented by five very different models, which, although we won’t show it, of course yield completely isomorphic geometries. Many authors develop just one or two models, explaining that these are the best or simplest. This is an illusion: on the contrary, it is the case that one or another model is best adapted to the particular problem being studied. In Chap. 19 of [B] we gave the four models that emerge from “elementary” geometry. They will be presented briefly in Sect. II.XYZ . The fifth is not given in [B], for it uses Riemannian geometry: hyperbolic geometry is the Riemannian structure induced on a 2 semi-hyperboloid in RnC1 by the quadratic form q D x12 xn2 C xnC1 . In this hyperbolic geometry that we want to define, as in Euclidean geometry, there are angles but above all a metric. The angles are those induced locally by the Euclidean structure on the disk. They are indeed conserved by the group Möb.S1 / since its elements are holomorphic functions of a complex variable (or antiholomorphic if zN appears). This property is in fact nothing other than the definition of holomorphic functions. These are functions U ! C (where U is an open subset of



C, identified with E2 ; see Sect. II.XYZ) for which the (complex) derivative f 0 .z/ exists. When f 0 .z/ ¤ 0 they are then automatically local similitudes, since multiplication by a nonzero complex number f 0 .z/ is a similitude with ratio equal to the modulus of f 0 .z/ and angle equal to its argument. We need to find a metric invariant under this group.

Fig. II.4.6.

Now is the moment to recall that the hom*ographies have an invariant, i.e. the cross ratio encountered in Sect. I.6. Given two points p, q in the interior of the disk, consider the unique circle through p, q that is orthogonal to the unit circle bounding the disk. It intersects the unit circle at two points r, s, whence we have four points p; q; r; s and we define the (hyperbolic) distance ı of p to q by ı.p; q/ D j log.Œp; q; r; s/j, where Œp; q; r; s is the cross ratio of the four complex numbers p; q; r; s on the affine line C. We can then prove the basic formula from which all others can be derived, i.e. that if a triangle has sides a, b, c and angle ˛ at a, then .II:4:2/

ch a D ch b ch c sh a sh b cos ˛;

where ch and sh denote hyperbolic cosine and sine. And we can verify without difficulty that in this hyperbolic geometry Hyp2 , the lines are the arcs of circles (or segments of lines passing through the origin) orthogonal to the unit circle, i.e. that these are the shortest paths from one point to another. Finally, the group with which we began is the group of isometries of this metric. For more on plane hyperbolic geometry (and in all dimensions: Hypn ), see Sect. II.XYZ and the references mentioned there. Finally, in the spirit of Sect. I.7 in particular, the geometry at infinity of the hyperbolic plane is that of the circle; each of its points is a point at infinity, as opposed to the complex affine line, which has but a single point at infinity (which will play an essential role in the following section), or the real affine plane whose points at infinity are associated with directions of lines, and not as here with directions of vectors (or oriented lines). Historical remark: hyperbolic geometry had to wait a long time before being solidly defined by Riemann in 1854. Apart from philosophical reasons, there was the mathematical fact shown by Hilbert (see Sect. VI.7) that this geometry cannot be completely realized by a surface in three-dimensional space. The surface that Beltrami



proposed had a double disadvantage: it wasn’t complete and it wasn’t simply connected it had to be developed by unrolling it; see Sect. VI.7 for all this. II.5. Second solution: the conformal group of the sphere, first seen algebraically, then geometrically, with inversions in dimension 3 (and three-dimensional hyperbolic geometry). Historical appearance of the first fractals We can find the unconditional algebraic solution for the composition of inversions if we remember the preceding section. We continue to identify E2 with the complex line C. Then the inversion through the origin and the power 1 is nothing zCb N other than z 7! z1N ; thus each inversion of the plane has the form z 7! cazCd . But we N know how to complete C with a single point at infinity: C [ 1, which we write as azCb the sphere S2 . As for z 7! czCd , we have seen in Sect. I.6 that the right extension a d N is given by 1 7! c and c 7! 1, and likewise with z. Written in hom*ogeneous coordinates (see Sect. I.5), the inversions become quasilinear: .z; t / 7! .azN C bt; c zN C dt /. Finally, the composition of an arbitrary number of inversions belongs to the group GP.1I C/ [ GP.1I C/ , which comes in two pieces. The first of these is the projective group of the complex line, denoted GP.1I C/; the second piece can be denoted GP.1I C/ , having the form of elements of GP.1I C/ composed with complex conjugation. We denote it by Möb.S2 / and call it the Möbius group in dimension 2 or else the conformal group of S2 . It has six real parameters (we can divide numerator and denominator of the fraction by an arbitrary complex number) and two connected components. The component containing the identity element consists of orientation preserving transformations. It can be shown (see 18.10 of [B]) that this group coincides with the group of those differentiable transformations of S2 that preserve infinitesimal (local) angles, i.e. those for which the derivative is a similitude. But the algebraic construction above, albeit sound, doesn’t tell us much about the geometric nature of the elements of Möb.S2 /. The geometric trick here is to introduce stereographic projection, which is the geometric realization of the completion S2 D C [ 1 of C. This doesn’t take us much up the ladder, but is a new use of three dimensions for resolving questions posed in dimension 2.

Fig. II.5.1. Stereographic projection



Stereographic projection see the figure puts the points of the Euclidean plane in bijection (one-to one-correspondence) with the points of the sphere minus the north pole n. Thus, if we identify the Euclidean plane with the complex line C, there is a bijection between the points of the sphere in its entirety and the points of the complex projective line CP1 D C [ 1, the north pole now corresponding to the point 1 at infinity of CP1 . First of all, the definition of inversion extends to Euclidean spaces of arbitrary dimension without any change. These inversions preserve the set of hyperplanes and spheres (we don’t say hypersphere) and hence circles in the space. They also preserve angles between them as well as the angles between tangents to curves at their points of intersection. Stereographic projection is nothing other than the restriction to the sphere of inversion through the north pole. It thus preserves circles (the “small” circles of the sphere), except those that pass through the same pole, which yield lines of the plane (Fig. II.5.2); and it preserves angles. The fact that it preserves circles explains why it has been used in astronomy for so long, for if our sphere is thought of as the celestial vault, the stars there all trace out circles; we can thus represent their trajectories by circles on a plane map. For us, its interest is to furnish a geometric compactification that preserves the angles of the Euclidean plane.

Fig. II.5.2.

We now proceed as in Sect. 11.4 and consider the inversions of three dimensional space, where we effect stereographic projection and restrict ourselves to inversions of the space whose center is exterior to the solid ball defined by the sphere and which preserve this ball (Fig. II.5.3): they preserve simultaneously the interior of the ball and also the sphere. We can thus compose them as many times as we wish.

Fig. II.5.3.



Now it an entertaining but nontrivial exercise to see that, transported by stereographic projection in the plane, the effect of these inversions on the sphere becomes an inversion of C [ 1, always with the convention that the north pole corresponds to the point at infinity 1 of the plane. Proceeding in the reverse sense, we can compose plane inversions as many times as we want. It can be shown, as usual, that this group, taken on S2 , coincides with the group of differentiable transformations of S2 that are conformal, i.e. preserve angles. This group has already been denoted Möb.S2 /. We will see that it is canonically isomorphic to that of hyperbolic geometry of dimension 3. Here now is an historical example proceeding from the composition ad infinitum of planar inversions; it’s the first historical appearance of fractals. We consider a family of mutually tangent circles and the inversions for which the invariant circles are the circles of the family. Next we consider the effect of these inversions on the set of these circles, and we repeat ad infinitum. What is the “curve” formed by the points of contact of all these circles? This doesn’t have to do with a curve in the

c Mandelbrot Fig. II.5.4. Mandelbrot (1982)



strict sense (like those of Chap. V), but in a broader sense, since we have an infinite, but denumerable, number of points. The answer is a fractal see Fig. II.5.4 is a “curve” that has a tangent nowhere (we except the case where the initial circles are all orthogonal to the same circle, in which case the curve is that circle itself). This fractal, studied by Klein at the end of the nineteenth century, is doubly interesting: on the one hand, it is a natural example of a fractal object appearing where a smooth object was expected. But above all it arises naturally in studying certain discrete subgroups of Möb.S2 /, called Kleinian, which occur in the analysis of functions of a complex variable. A recent book is entirely devoted to them: Maskit (1988). In fact, the group here is that generated by the (finite) set of inversions for which the circle of inversion is one of the given circles. These groups, defined by circles, are called Schottky groups. They play a natural and essential role in various problems; they are used in Sects. II.8 and II.9 for the study of certain pencils of circles. We now define hyperbolic geometry of dimension 3, denoted by Hyp3 , and its group of transformations, as the space constituted by the open ball above, acted on by Möb.S2 /. Distance is defined as for the hyperbolic plane in the preceding section. The trouble is that we can’t read the group algebraically in the real coordinates of dimension 3. The solution requires climbing the ladder into dimension 4, utilizing the linear projective model of hyperbolic geometry. Furthermore, it’s no harder in any higher dimension. Let us say here only that the geometry studied, viewed on Sn , is in fact the conformal geometry of the Euclidean space of dimension n, but compactified with a single point at infinity (compactification without regard to real projective geometry of dimension n, that of RPn ). For the study of transformations in Möb.S2 / viewed on the sphere, see Sect. III.2. To conclude with the transformation of angles, we are right to ask whether there might exist still more marvelous transformations than inversions (and their compositions) that preserve angles. We have just seen that the answer is no, but we utilized the fact that the transformations considered are bijective. Defined on the whole Euclidean space, these are the similitudes; on the compactified space in Sn , these are the inversions and their compositions. But let us suppose that we are concerned only with local geometry; then, in the case of the Euclidean plane viewed as C, each holomorphic function defined on an open set is conformal except possibly at points where the derivative is zero: it preserves angles. We thus have an enormous class (not depending at all on a finite number of parameters) of local transformations that preserve angles (but not circles). For bijections of the whole plane, the theorems cited earlier on characterization of similitudes and inversions are the generalizations of Liouville’s theorem, which states that the only holomorphic and bijective transformations of C are the similitudes, and those of C [ 1 are the functions of the azCb form z 7! czCd . What happens in dimension 3? The question was settled by the very same Liouville:



Each transformation of an open subset (no matter how small) of the Euclidean space of dimension n > 3 that preserves angles is necessarily the restriction to this open set of an element of Möb.Sn /. This is a good example of passage from local to global; we also speak of rigidity. However beautiful Liouville’s proof may be, it remains in part artificial and complicated as it comes down in the end to showing that the transformation in question transforms spheres into spheres. Moreover, as was typical of the time, it was formulated in only three dimensions. Liouville’s result has attracted numerous mathematicians, who have given varied proofs, all rather complicated. The first direct proof is due to Nevanlinna and is relatively simple: it consists solely in applying three times the condition that the derivative of the mapping considered is a similitude at each point. Such a calculation is almost impossible when expressed in coordinates, i.e. with partial derivatives. Before Nevanlinna (1960) all the differential calculus on the surface of the planet was expressed in coordinates. In 1960 Nevanlinna wrote a whole little text to show that the modern language of linear algebra permitted intrinsic expression of derivatives, a language introduced for the first time in great profusion in Dieudonné (1960) and which then spread like lightning. Nevanlinna, as a great mathematician, apologized for the simplicity of his article, but added that for a new notation to be really interesting it must have applications. He then gave the application to Liouville’s theorem, see of [B]. In conclusion, beginning with dimension 3 conformal geometry is reduced to that controlled by Möb.Sn /. For the proofs and more, see Sect. 9.4 of [B]. A recent proof is presented in Frances (2003). Calculation of volumes of tetrahedra in hyperbolic geometry of dimension 3 is both useful and very difficult. For these reasons it is much studied. There is no magic formula, but only calculations in particular cases, which are closely and fascinatingly connected with number theory, topology and analytic function theory. See, for example, Vinberg (1993) and Cho and Kim (1999) and the references mentioned therein. In Langevin and Walczak (2008) there is an interesting geometric characterization of holomorphic functions with the aid of sphere pencils. II.6. Inversion in space: the sextuple and its generalization thanks to the sphere of dimension 3 Inversion in space has applications just a spectacular as those of Steiner’s alternative. We deal first with that of the sextuple: consider three spheres in three dimensional Euclidean space (our space), which are mutually tangent and exterior. We easily convince ourselves that there exists a continuous infinity (in one real parameter) of spheres that are tangent simultaneously to these three spheres. Now, as in Steiner’s alternative, start with one such sphere tangent to three given spheres and construct the next sphere of the chain, requiring that it be tangent to four initial spheres. Continue progressively in this manner, choosing a sphere tangent to the three initial spheres and the preceding sphere of the chain. Drawings are difficult



to produce because the spheres pass alternatively above and below the three initial spheres, which makes it hard to see what is happening. Do we have an alternative a la Steiner? The solution is immediate if we effect an inversion whose center is one of the points of contact between two of the three spheres considered. Then the figure image is made up of two parallel planes and a tangent sphere “sandwiched” between the two. But any sphere tangent to these three is itself sandwiched, thus of the same radius as the third, and is tangent furthermore, so the center is on a circle: and things then close up always at the end of six times exactly. There is no alternative here but only automatic closure as in the theorem of the six circles in Sect. II.2.

Fig. II.6.1.

The generalized sextuple is scarcely more difficult to define and besides isn’t a sextuple in general. We start with an arbitrary torus of revolution and associate with it two continuous families of spheres, the first that of spheres tangent to the torus on the interior along its meridians, the second that of exterior spheres tangent along its parallels, a family to which we need to adjoin for continuity the two planes tangent to the torus along the extreme parallels. We proceed as before, starting with a given interior sphere and constructing a chain of interior spheres such that each is tangent to the preceding and ask whether the resulting chain is finite. This is a generalization of the problem of the sextuple inasmuch as we begin with three mutually exterior spheres (not necessarily tangent) instead of starting with a torus. With an inversion we get reduction to the case where the spheres are of equal radius, and the family of spheres tangent to the three spheres coincides with the second family associated with the torus that they define. Since it’s a torus of revolution, things are as in Sect. II.3: either the chain never closes or it closes no matter how the initial sphere is chosen. We let p denote the associated rational number, since things are in general starred. More precisely, we say that the chain is of type p D ab , where a, b are relatively prime integers, if it includes a spheres and closes up after making b circuits about the center. Now an inversion shows that we have exactly the same dichotomy for the exterior spheres; if there is closure there will be an associated rational number q.



Fig. II.6.2. The generalized sextuple: five spheres in a torus, how many are there on the outside?

The elegant theorem of the generalized sextuple, discovered by Steiner, is as follows: Either the torus admits no chain of type p (whatever the initial spheres), in which case the same is true for the exterior chains: they never close; or the torus admits an interior chain of type p (and thus infinitely many, with any initial sphere), but then it admits exterior chains (with any initial sphere) of 2p the same type q, where q is given by the relation p1 C q1 D 12 , i.e. q D p2 . The first pairs are .3; 6/, .4; 4/, .6; 3/, .5; 10 /, etc. 3 This theorem was considered by Steiner to be one of the most beautiful in all of geometry. The case .3; 6/ is that of the sextuple above; we leave it to readers to prove the case .4; 4/ by hand with the inversion utilized for the sextuple. As for the symmetry of the relation between p and q, it can’t be a surprise: we can, with an inversion, interchange the interior and the exterior of a torus. Now for the integers > 3, beginning with 5, we can’t get away with an inversion in dimension 3, we must “climb” into dimension 4 and study the geometry of the sphere S3 (threedimensional, but situated in E4 ); then everything becomes transparent, as we will see right away. We can also say that we lift the situation studied in dimension 3, by a



stereographic projection onto S3 , into the space of dimension 4. Readers who wish to stay in dimension 3 can try a proof of their own and thus appreciate the force of Jacob’s vision. The sphere S3 is a principal object of modern geometry; it is central to topology and, like S1 , admits a group structure the group of quaternions of norm equal to 1, see 8.9 of [B]. Our main interest here is that S3 is a covering of just two sheets of the group SO.3/ of rotations of E3 . For the intrinsic geometry of a sphere we refer to the next chapter. We will see in Sect. III.5 that we are far from knowing everything about S3 . To prove the generalized sextuple, we begin by identifying the Euclidean space E4 of dimension 4 with the complex plane C 2 of pairs .z; z 0 / of complex numbers. Then S3 consists of the pairs such that z zN C z 0 z 0 D 1 (or x 2 C y 2 C s 2 C t 2 D 1 if z D x C iy and z 0 D s C i t ). We have two distinguished circles C and C0 on S3 that are associated with the pairs of type .z; 0/ and .0; z 0 /, respectively. These belong to two orthogonal planes of E4 . Since these planes generate E4 , we see that S3 is the union of circular arcs that terminate on either C or C0 and which have length equal to =2; these circular arcs are automatically orthogonal to C and C0 at the points that define them; we will denote the set of all of them by A.

Fig. II.6.3.

We now have everything we need to solve the problem of the generalized sextuple posed earlier. We easily see that stereographic projection transforms a torus of revolution, if the axis of revolution is the axis of stereographic projection, into the set T of points of S3 consisting of points situated on all the arcs of A at a given distance d from their initial point in C. Then this torus T clearly contains a chain of inscribed spheres of type the rational number p if this distance d is equal to =p. Now the spheres exterior to T correspond to the torus T , but seen as a torus for which the



interior is the complement in S3 of the interior of T complementary to T a torus now regarded as formed by the points situated on the arcs of A which are located this time at a distance =2 =p from the initial point, but on the circle C0 . But by the same reasoning as above this complementary torus contains a chain of type the rational number q if and only if its radius equals =q. Now by construction its radius equals =2 =p, Q.E.D. We can’t talk about S3 without mentioning the Clifford parallels and the Hopf fibration. The latter is essential in algebraic topology, where it is the archetype of certain geometric fibrations. It will also be utilized at the end of the following section. A coordinate-free, purely geometrical, construction can be given; see Chap. 18 of [B]. Here we will proceed as in most of the texts which deal with this matter: we parachute using a description in terms of complex numbers. It suffices to remark that the one-parameter group formed by multiplication by complex numbers of modulus 1 acts on C 2 by the mapping .z; z 0 / 7! .z; z 0 /, operations which are isometries and thus preserve distances on S3 . The orbits, i.e. trajectories, are circles;



Fig. II.6.4. Two models of fibration by Clifford parallels. In the model at the top we obtain the sphere S3 by compactifying the space R3 by a single point at infinity (and then the vertical axis becomes a circle). The linear model is more complicated to explain, and the figure is imperfect at the level of the horizontal plane. It has in fact to do with a Hopf fibration of the projective space RP3 . The space R3 of the figure must now be compactified in RP3 by adjoining the plane at infinity (the lines however all become topological circles); we remark that the completion of a hyperboloid of one sheet is a (topological) torus. We have of course to remove from the hyperboloids all c Springer; (b) Hilbert, generators of one of the two families. (a) Penrose (1978) c Springer Cohn-Vossen (1996)



and these circles have the property that, taken two at a time, each point of one is at a constant distance from the other (which explains the term parallel). These properties of S3 are often presented with the aid of quaternions, even though this is not at all essential; again see Chap. 18 of [B]. The lower figure, based on quadric surfaces, is a linear model of the one above (see Sect. IV.10). It is also very important to find the structure of this family of circles in their entirety, insofar as there is one. The answer is that they naturally form the sphere S2 . We have all that is needed to explain this, since the quotient obtained by the equivalence provided by .z; z 0 / .z; z 0 / is nothing other than the complex projective line CP1 (see Sect. I.XYZ); we have simply to remark that S3 C 2 doesn’t omit any complex line of C 2 . The mapping of S3 ! S2 thus obtained, whose fibers are all circles, is called the Hopf fibration. It plays a considerable role in algebraic topology, as do its generalizations for the spheres S2nC1 and S4nC1 obtained from the projective spaces CPn and HPn in all dimensions (in the latter case, the fibers are the quaternions of modulus 1, namely the spheres S3 ). The Clifford tori are tori formed by points at a given distance from a great circle of S3 . These are the inverse images in the Hopf fibration of a “small” circle of S2 . Certain authors reserve the name Clifford torus for the torus equidistant to two orthogonal circles of S3 : it is conjectured, see Sect. III.5, that these are the only surfaces in S3 that are minimal and of toroidal type. II.7. Higher up the ladder: the global geometry of circles and spheres As things stand, the associated geometry Möb.Sn / remains insufficient for giving account of the immensity of theorems on circles and spheres, results that have accumulated over centuries. We need to climb just one more rung up the ladder for a concept that is quite natural: we seek to endow the space of all spheres in the Euclidean space of dimension n with an appropriate structure. We present this structure very briefly, then give a few examples of configurations that it allows us to treat conceptually. We stop there; for more “depth” see Chap. 20 of [B]. However, we must mention Coolidge (1916, 1999) above all for all the classic results on the subject. The reason that we stop is that, in the spirit of Jacob’s ladder, the tool that follows allows us to treat all possible problems in this branch of the subject; to our knowledge it isn’t necessary to climb higher. We use the language and notations set forth at the end of Sect. II.XYZ. The first configuration (yet another six circles theorem) for which we owe the interpretation to Adrien Douady is classic and concerns circles in the plane. It states that, in the figure of four circles below (Fig. II.7.2): if the four points a; b; c; d are on a circle, then so are the four other points e; f; g; h on a circle. This can be proved “by hand” with the classic notions of power with respect to a circle and of radical axis, etc.; see Sect. II.XYZ. However, the conceptual proof is meteoric: in the space C of circles constructed in Sect. II.XYZ, which is a projective space of dimension 3, the set of circles passing


Fig. II.7.1. The space of circles in the plane, Im.s/ denotes the set of point circles (radius zero) that constitute the points of a circle s ; the point circles form a sphere. [B] c Nathan Édition Géométrie. Nathan (1977, 1990) réimp. Cassini (2009)

Fig. II.7.2. Yet another six circles theorem. [B] Géométrie. Nathan (1977, 1990) réimp. c Nathan Édition Cassini (2009)

Fig. II.7.3.




through two given points form a projective line of C . Then, if in C our four circles are denoted p; q; r; s, the hypothesis of the theorem is that the lines D and D0 defined by the pairs .p; q/ and .r; s/ respectively have a common point x. But these two lines are in the same plane and thus the lines defined by .p; s/ and .q; r/ intersect in a point y of C , which is the desired circle! Interested readers will subsequently see a third circle, but then will be traumatized by the fact that the desired points of intersection don’t exist. This reflects the fact, see for example Chap. 20 of [B], that the lines of C don’t correspond just to circles having two points in common, but also to a pencil of circles of type “having no point in common”:

Fig. II.7.4. A pencil of circles without a common point is a line in the space of circles

The second configuration is in the space of spheres S of the Euclidean space E3 . It figures in many French books on “classical” geometry from the first half of the twentieth century, in the sections called “anallagmatic geometry”. We don’t cite them, but they use 50 pages at a time for proving results that we will develop much more briefly, this length being due either to an outdated desire for geometric purity (they even speak occasionally of “higher geometry”, with a fragrance of romanticism) turned toward the past or to conceal composition of inversions in the group Möb.S2 /. Yet we will see from a particular case that all problems of this sort are resolved very rapidly, and above all “in depth”, as soon as the geometry of S is utilized. Moreover the concepts work well, without additional effort, in arbitrary dimension. We have allowed ourselves this little polemic because the geometry of S was widely known outside France by the end of the nineteenth century, especially in Germany by Felix Klein. Here is the configuration we need, the paratactic ring. We begin with the case of the ring of orthogonal circles, easily treated by inversion, that. A circle C of E3 and its axis, the line D, have the property that each sphere passing through C intersects D at a right angle; but also each plane passing through D intersects C at a right angle. Now a general inversion transforms the union of D and C into a pair of circles such



that each sphere passing through the one cuts the other at a right angle (this of course applying to both).

Fig. II.7.5. An orthogonal ring and a paratactic ring

But the thing is true for any angle: for each angle ˛ there exist pairs of circles such that each sphere passing through the one cuts the other by the angle ˛; the same thing for the other circle and the same angle ˛. These pairs of circles are called the paratactic ring for the angle ˛. By hand, and even with inversion, it is rather difficult and long to establish their properties, still much longer to show that, within an element of the conformal group, there is but one paratactic ring of angle ˛; this takes a good fifty pages, for example in Hadamard (1911). If one of the circles is a line, it is called a focal line of the other circle; this designation resulting because, in projecting onto a plane perpendicular to the line, the line must yield a point which is a focus of the elliptical projection of the circle (see Section IV). Things are much simpler in the present case and are left as an exercise. But this is not the good conceptual approach. To prove the existence and uniqueness of paratactic rings with the language of the space of spheres S, we must first know that in S the circles are represented by lines (made up of points corresponding to the spheres that contain the circle). Therefore we look for pairs of lines such that each point of the one is at a constant distance to the other (the distance being the same for both) given by the angle ˛. But we have found such lines in Sect. II.6, the Clifford parallels. We only need still find the sphere, “Euclidean of dimension 3”, in S. In fact, what we find are the projective spaces endowed with their elliptic structure (quotient geometry that of the sphere). To do this, it suffices to take any imaginary sphere whatsoever; considered as a point in S it has a dual hyperplane with respect to the basic quadric of S. This dual hyperplane is thus endowed with the restriction of the basic quadratic form, which here is well defined and positive if an imaginary sphere has been chosen from the start. This is then certainly an elliptic projective space; see more details as needed in Sect. II.XYZ and 20.5.4 of [B]. To establish that such rings are essentially unique, it suffices to utilize the strong transitivity of the group Möb.S3 /. The case where a circle is a line is of course covered, but it is not so easy to prove by hand.



Fig. II.7.6. The focal lines of a circle and the paratactic rings

Now I can’t resist speaking about the result in the geometry of ordinary space which made the greatest impression on my life (along with, in truth, Poncelet’s theorem on conics: see Sect. IV.8). We can see it by regarding what is given by Clifford parallelism in the sphere S3 under stereographic projection to ordinary space and concerns the so-called Villarceau circles; see 18.9 of [B]. We consider a torus of revolution and we cut it with the tangent planes on the interior: the astonishing thing is that the section always decomposes into two circles. So we find, on the tori of revolution, four continuous families of circles, the parallels, the meridians and the two Villarceau families. Moreover, these “exotic” circles are the loxodromes of the torus: they cut the meridians by a constant angle, the double of the angle at their points of intersection. I will soon cease being very personal, but at 16 was so astonished that I actually wanted to saw out a ring in wood to verify the theorem. It was in my father’s terminal instructional material (Rouché and de Comberousse, 1912) for the baccalaureate degree where I found the theorem. The proof is completely elementary, but good spatial perception is needed. Then, in the preparatory class for the university, I learned the clear and profound proof not unlike the one here. The torus contains the umbilical (see Sect. II.XYZ) as double line, but a tangent plane cuts it in a curve that has two double points (at the points of contact). Whence, finally, for this curve section, there are four double points; but since it is of degree four, it is decomposed into two conics (see Sect. V.14), which are thus circles. This proof is more than quick, and nonetheless it’s just this way that it was taught us. In fact you will understand all that follows: first projectify the space where our torus is considered, then complexify it. Then the complexified torus is that to which the above reasoning is applied. The two circles obtained are in fact complex circles, they have the topology of a sphere S2 . We subsequently keep only the real part of the figure. This going and coming from the real to the complex is short-circuited in entire books, or entire parts of (now old) textbooks, of so-called modern geometry. Things go well because a torus is defined by an algebraic equation (a polynomial of degree four), and complexifying and projectifying is automatic, we keep the same



c Pour la Sciénce, Fig. II.7.7. See other nice photos in Berger (2002). Berger (2002) Éditions Belin

equation, but throw in hom*ogeneous quadruples; then we allow complex hom*ogeneous quadruples, and there we have the torus with which we can reason as we have just done. To my knowledge, some historical points remain open: who really first discovered these circles, known in at least one particular case to the stone masons of the middle ages (see the Fig. II.7.7)? Who discovered the loxodromic property? Moreover, the view in S shows that an arbitrary pair of Villarceau circles of the same family form a paratactic ring for the angle ˛ of loxodromy. The topologist will not have failed to note that the Villarceau circles, thus all the paratactic rings, are formed from enlaced circles. If it is surprising that four distinct families of circles



Fig. II.7.8. Readers will convince themselves in their own ways of the reality of the Villarceau circles

can exist on a single surface as simple as the torus (its coordinate equation is only of degree 4); it can be seen in 20.7 of [B] that there even exist surfaces, always of degree four, called cyclides, that contain six distinct families of circles. A special class is formed by the Dupin cyclides, which are nothing other than the inverses of tori of revolution; besides [B], see also 11.21 of [BG]. Yvon Villarceau, who has a street in Paris named after him, was an astronomer. The oldest reference that we know is Yvon Villarceau (1948), where the circles are noted in about ten lines and without any explanation! Readers will ask if there exist other surfaces that contain as many or more oneparameter families of circles. This question was completely settled in Darboux (1880). The answer is that ten families of circles is the maximum, and that ten is only attained by the circles discussed earlier. However, Darboux’s text is ambiguous, for as described in Darboux (1917) (see also [B]), the real cyclides can’t actually have ten families of circles, for that they must be complexified, as it seems to us. This sort of ambiguity is frequent in the texts of the nineteenth century and has often fed quarrels, the one between Poncelet and Cauchy being most celebrated. In Darboux (1880) the problem studied is that of surfaces that contain lots of conics. The answer is that the maximum number of parameters for continuous families of conics is ten (only six in the real domain), except for the ellipsoids and Steiner’s Roman surface, already encountered in Sect. I.7, for which we have an infinity of conics dependent on two parameters.



II.8. Hexagonal packings of circles and conformal representation What has preceded is beautiful geometry, with some ascent of the ladder, but was essentially known and explained in depth prior to the twentieth century. The results of this section and the one that follows are different both with respect to topicality and depth. The matter begins with the drawing exercise that follows, which we advise readers to carry out for themselves with paper and pencil, especially if they are good drawers, for it is very instructive. Start with a system of seven circles, as in Fig. II.8.1, consisting of a central circle surrounded by six circles tangent to it and the corresponding tangent lines, and continue the figure by first adjoining a ring of twelve circles, with the condition that each of the first six circles around the central circle will be surrounded by six circles, each tangent to it and to two of the other six. Such a packing is called hexagonal. For the moment we ask nothing of boundary circles regarding the number of circles they touch. The numbers of the generations is indicated in the figure. It’s a matter of continuing indefinitely, if possible, so as to fill the entire plane.

Fig. II.8.1.

Fig. II.8.2.



However well we draw, it will be confirmed that this is impossible. Very quickly a catastrophe occurs: we can no longer continue. If this happens with the figure on the left of Fig. II.8.3, it’s because the large initial circle has many more than six neighbors! But nonetheless we know of an hexagonal packing, of the whole plane, called the regular hexagonal packing. Where is the contradiction? There isn’t any; it simply happens that in the other drawings the six imposed circles of the first generation didn’t have equal radii. Precisely, we introduce the default def.C/ of a circle surrounded by six circles with radii ri (i D 1; 2; 3; 4; 5; 6): ˇ ri ˇˇ ˇ def.C/ D sup ˇ1 ˇ: rj i6Dj We speak later of packings having n generations: see Fig. II.8.1 above.

c John Wiley & Sons, Inc Fig. II.8.3. Pach, Agarwal (1995)

In 1987 Rodin and Sullivan showed that: (II.8.1) There exists a sequence fun g indexed by the positive integers n, such that un tends to zero as n tends toward infinity, and which has the property that if a circle C is the starting place (zero-th generation) of an hexagonal packing having n generations, then def.C/ < un . Therefore if we fill the plane the number of generations is infinite and thus def.C/ D 0; in particular the circles about C have the same radius, necessarily equal to the radius of C itself, and thus step by step all the radii are equal, Q.E.D. Thus: (II.8.2) The only hexagonal packings of circles that fill the whole plane are those with equal radii. We can thus say that such a packing is unique within similitude. We are dealing here with an extremely difficult result. The amateur can try proving it “by hand”,



but this author has wracked his brains over it during very many hours of waiting in airports. Before speaking of motivation, applications and proof, here is a bit of history, with some additional remarks for interested readers. We mention first that the result of Rodin and Sullivan immediately unleashed a veritable avalanche of work: an up-to-date bibliography on the subject would contain well over a hundred titles. We also study generalizations of the Rodin-Sullivan result: packings of circles in the plane, hexagonal and also more general, under the condition that they be of finite valence (the valence is the number of circles that touch a given circle); and indeed plane packings of still more general objects. We will see another motivation, apart from the geometric beauty, below. Never had a result as strong as that of Rodin-Sullivan been anticipated. Fejes Tóth had merely conjectured in 1977 that an hexagonal packing of the whole plane would have one of the radii either tending toward zero or toward infinity as the number of generations increases. In 1984 Barany, Füredi and Pach proved the conjecture of Fejes Tóth, and even a bit more. But that is far from Rodin and Sullivan; and it furthermore tells us nothing about the sequence fun g, apart from the fact that it tends toward zero. The first thing to do is to define the optimal un , the only interesting values. That is to say, we define, for each integer n, the number sn as equal to the largest default def.C/ possible so that an initial circle C can admit a packing having n generations. Readers will easily calculate the default of the initial circle in the packing of n generations obtained by an inversion through a center as close as possible to the packing of equal radii; they will thus prove that sn > n4 . But what is clearly interesting is finding an upper bound for sn . We have almost complete answers at present, due to Doyle, He and Rodin: (II.8.3) On the one hand there exists a constant K such that sn 6 Kn for each n; p on the other hand, limn!1 nsn D 2 3 2 2 . 13 /= . 32 /, where denotes the gamma function. This number is not as mysterious as it might seem; it is in fact equal to 4=R, where R is the conformal radius of the regular hexagon; see just below for the definition. But we still don’t yet know the best K possible in complete generality. For the proofs, the essential difficulty is that the algebraic relation (it certainly exists) that connects the six radii ri of the circles about a given circle of radius r is unworkable. The seven circles theorem (see Sect. II.2) is in any case useless. Barany, Füredi and Pach proved the two inequalities X 1 6 > ri r i



rin > 6r n

.for each n > 1/:


They concluded in a relatively elementary way with the left equality by reasoning on the graph of the centers, where we can it is now classic define a Laplacian (of discrete type) and speak of subharmonic functions. But there isn’t any



subharmonic function except for the constants a la Liouville’s. Interested readers can consult Sect. 8 of Pach and Agarwal (1995). The proof of Rodin and Sullivan has at first the appearance of a vicious circle; to prove (II.8.1), they used (II.8.2). In fact, to deduce (II.8.1) from (II.8.2) is elementary, proceeding by contradiction with the help of a pretty lemma illustrated by the figure below and passing to the limit:

c D. Sullivan Fig. II.8.4. Rodin, Sullivan (1987)

(II.8.4) (Ring lemma.) There exists a constant r.n/ (i.e. depending only on n) such that, if we are given a system of n circles tangent two at a time in succession and all tangent to the unit disk, then all these circles have radius greater than r.n/. The proof of (II.8.2) (uniqueness of hexagonal packings in the plane) uses the theory of the geometry at infinity including some quite recent results of the discrete groups studied by Klein, so-called Kleinian groups, which are discrete subgroups of GL.1I C/, encountered in Sect. II.5. They enter analysis in an essential way, since they are connected by definition to functions of a complex variable. The effect of these groups on S2 is examined by studying their effect on the circles of S2 . We must see what happens when we iterate this or that transformation and so encounter a geometrically dynamic composition that we have already seen with Pappus’s theorem in Sect. I.1. The Rodin-Sullivan proof thus utilizes recent and profound results on Kleinian groups. But this still does not explain the depth of matters in (II.8.2). The starting idea is common to much rather recent work in analysis and geometry and uses the notion of a quasiconformal mapping. Such a mapping, say from a portion of S2 into itself (although this notion is in fact very general, applicable to every metric space), preserves spheres within a deformation factor that is universally bounded on the open set considered; in addition it is required that we have a homeomorphism. Precisely, there must exist a k > 0 such that the image under f of each sphere S.p I r/ D fq W d.p; q/ D rg with center p and radius r satisfies the two conditions: f .S.p I r// is contained in the ball which is the interior (a ball) of the sphere S.f .p/ I kr/ and f .S.p I r// contains in its interior the sphere S.f .p/ I k 1 r/. Such a definition evidently has interest only if we go to infinity, i.e. if its domain of definition isn’t compact. Now we consider an hexagonal packing H



of the plane and the regular packing H 0 (with equal radii). It may seem natural to define a quasi-conformal mapping of the plane by conformally mapping the disks of H onto the corresponding disks of H 0 , worrying subsequently about the interstices between the disks. But this approach is too naive: it is impossible to attain agreement at the points of contact, since a conformal mapping of one disk into another is determined once we know the images of three points of the boundary. In fact, it’s with the interstices that we must begin: there exists a unique conformal mapping that takes an interstice defined by three mutually tangent circles onto an interstice of the same type, taking the three vertices of the first interstice onto the three vertices of the second. To see this, it suffices to consider the circle passing through the three vertices of the interstice: c′

b a′




Fig. II.8.5.

Next, for the interiors of the circles of H we do the same thing for the image circles of the circles of H by the inversions for which the circle of inversion is in H . Likewise for H 0 . We thus create new interstices (shaded in Fig. II.8.6) which are made to correspond by new conformal mappings. By iterating this operation, we obtain on both sides a partial darkening of the plane by interstices, the images of the original interstices by the Schottky groups (see Sect. II.5) of H and H 0 ; the two darkenings correspond to each other by a mapping which can be shown to be quasiconformal. But what remains is easily seen to be of measure zero; see Sect. II.9. We can also interpret all this in S2 by stereographic projection. But in this situation a fundamental result of the theory states that a quasiconformal mapping of a “very large” part of S2 onto itself arises from a quasiconformal mapping on all S2 , thus an element of Möb.S2 /. Hence H and H 0 are “the same”, and thus the elements of H also have equal radii.

Fig. II.8.6. H1 3 and H2 3 are the images of H1 and H2 by inversion in the circle H3



It seems that as yet there is no very simple and conceptual proof of the uniqueness of hexagonal packings of the plane. That of Schramm from 1991 provides a rigidity for all packings of a rather general type. This result is topological, rather long and hard to follow in detail. The essential property that allows conclusion in the case of circles is that the difference of two circles, in contrast to convex sets generally, never has more than two connected components. This excludes for example rectangles with different side ratios, which relates to the fact that the side ratio of a rectangle is a conformal invariant.

Fig. II.8.7.

So much for (II.8.2). But how do we prove (II.8.3)? For the upper bound in we do exactly the same thing as in the proof of Rodin-Sullivan, except that we abandon the finite packings Hn and Hn0 : Hn has equal radii and n generations (think of a regular hexagon), whereas Hn0 is arbitrary hexagonal and at least n generations. We still have interstices, finite in number. We show that what remains once the interstices are removed (which evidently is not of measure zero) has a very small 0 measure, i.e. less than nK2 (where K0 is a universal constant). Let us say that if a point is thrown at random, the probability is very strong that it will fall in one of the interstices. It remains then to show by a (reasonable) analysis that the mapping constructed which is conformal on a subset of very large measure since it is here defined on a compact set, has the global property of quasiconformity and is explicitly evaluable as a function of n. This allows an estimation of the default of the initial circle in Hn0 . Finally, to get the asymptotic estimate of sn in (II.8.3), we apply the preceding technique with Hn and Hn0 to an arbitrary packing, but whose combinatoric is exactly the canonical one with n generations, and thus the interior of a regular hexagon; see Fig. II.8.1. We therefore have the mappings gn W Hn ! Hn0 . We show that they have a limit and that this limit is a conformal representation of the regular hexagon on the disk. K n

This provides us with a connection to the motivations: where could Rodin and Sullivan have gotten the idea for their result? It in fact did not appear as a result in the theory of Kleinian groups, but rather their motivation lay in a conjecture of Thurston from 1985 concerning the geometric construction an algorithm for the conformal representation of a domain in the plane. We have seen in Sect. II.5 that there exist plenty of local conformal mappings: every holomorphic function defined on an



arbitrary open subset U of C is conformal, i.e. conserves angles; computationally its complex derivative (real differential) is a similitude. The problem of conformal representation is of considerable mathematical importance. We are given an open subset U of C and seek a conformal transformation (a biunique conformal function) which maps it onto the open unit disk D. We would have thus “parametrized” the open set U with the disk, which is trivial, but not if conformality is required. We have what is called a conformal representation of U. We could thus reduce the study of this or that problem on U to a problem on the unit disk. Apart from its pure mathematical interest, conformal transformations are encountered in applications. There is the case of the study of heat dispersion and that of subsonic flow about an airfoil. We will now describe in detail an application to electricity. We fix a point O of U and consider a long cylinder in space having section U, representing an electric conductor or electrostatic, with a central wire represented by O. A voltage of 10 000 V is applied to the exterior, the boundary of the cylinder, and 0 V to the central conductor. We want to find how the voltage is distributed on the interior, between the central conductor and the exterior surface. In other words we seek the equipotentials of the problem.

Fig. II.8.8. Conformal mappings are well-known for their use in determining electrical equipotentials and flow about an airfoil. To calculate the equipotentials in the case of the cylindrical surface indicated above on the left, we deform the problem by a conformal mapping so as to obtain a circle about its center for which the equipotentials are known: these are concentric circles. In effecting the inverse mapping, we obtain the equipotentials for the original problem under study. Pour la Science, no 176, June c Pour la Sciénce, Éditions Belin (1992), Marcel Berger, Les placements de cercles



It’s easy to see that the answer is given by the level curves of the conformal mapping taking the pair .U; O/ onto the pair .D; 0/. We deduce from this a physical demonstration of the existence and uniqueness of the conformal representation of U. This result was announced by Riemann in a more general setting, by the way with a completely insufficient proof, in terms of convergence of mappings minimizing an appropriate functional. Still other mathematicians thought that the theorem did not require proof, given the physical interpretation. It was not until toward the beginning of the twentieth century that Koebe gave the first complete proof. As for uniqueness, it is easy if a single image point is fixed, for example the center. Then we have uniqueness, within transformations of the conformal group for the disk, i.e. the group Möb.S1 / encountered in Sect. II.4. This allows the imposition of three boundary points; see the figure below. We mustn’t forget that the necessary condition is evidently that the topology of U be that of the disk, which is easily seen to be equivalent to requiring that U be simply connected. Further on we will see what happens in the general case.

Fig. II.8.9. Conformal representation of a square

The problem of Thurston is to obtain a geometric realization for the conformal transformation. With a succession of improvements over a long period of time, we have extremely rapid algorithms at our disposal; see the references in the introduction of Pommerenke (1992). But these have nothing to do with geometry. Here is Thurston’s strategy: we fill U with regular hexagonal lattices with a rather small common radius r. We see (in terms of the hyperbolic polyhedra in Sect. VIII.8) that there always exists an “image” in the disk of such a packing by an hexagonal packing (except at the boundary) and having the same combinatoric as the initial packing of the lattice, this combinatoric being the one imposed by the behavior at the boundary.

Fig. II.8.10.



With each r we associate an fr W U ! D by requiring first that centers of circles in U be sent to centers of the corresponding circles in D. We then fill the triangles thus obtained by affine mappings (which exist and are unique, as seen in Sect. I.2). Thurston conjectured that, if r tends to zero, the mappings fr have a limit mapping and that this is conformal. The proof of this result was the goal of the article of Rodin-Sullivan, and (II.8.1) gives a preview: at the center of circle C, the default in conformity is carried over by the default def.C/ of this circle. But we know that this default tends to zero with r, Q.E.D. We can now also say that it is constant by (II.8.2), namely 4=R, where the so-called conformal radius R of the regular hexagon is the similitude ratio (ratio of conformity) at the origin of the conformal mapping (unique within a rotation) that sends the regular hexagon onto the unit disk while preserving the centers.

Fig. II.8.11. The conformal radius

What happens next is very interesting. The various authors mentioned earlier who attacked the generalization of (II.8.1) had begun to sense a connection between circle packings and problems of conformal representation. We point out this important fact: that although the numerous results obtained so far indisputably demonstrate such a connection, a deep and truly conceptual result remains to be found. We return to the case of conformal representation, this time for domains U that are not necessarily simply connected, i.e. U is punctured by (simply connected) compact sets on its interior, e.g. points, etc. The Kreisnormierungsproblem is to generalize the existence of classic conformal representation by showing that, for any U, there always exists a conformal representation onto the unit disk D that is punctured in holes that are either points or disks. So the boundary will be completely traversed. Thanks to the intuition forged by the study of packings, the specialists knew how to show this for a disk punctured in no more than a denumerable number of holes. However, a point of history: when Schramm recounted this result at a conference, Sachs also a specialist in conformal representation pointed out that, at least for a finite number of holes, the proof had already been obtained by Koebe by 1936, but only in the simplicial case, i.e. triangulations (in fact decompositions into polygons) where each vertex belongs to only three polygons, or the dual situation where the polygons are all triangles; see also Sect. VII.9 and Ziegler (1995) for references. Koebe utilized a passage to the limit, easy to see with hexagonal packings. It remains to be discovered today whether or not the Kreisnormierungsproblem has a solution in the case of a non denumerable number of holes.



Fig. II.8.12. In these geometric examples we can see the conformal representation

Fig. II.8.13.

For all the preceding, readers can consult the general exposition (Berger, 1993) and the references therein, to which we need to append some more recent ones, e.g. Doyle, He, and Rodin (1994) and the references mentioned there, as well as Mathéus (1999). A recent experimental reference is Bobenko and Hoffmann (2001). Recent expositions are Stephenson (2003) and the book entirely dedicated to it (Stephenson, 2005). This whole section might seem to readers like a an aesthetically pleasing game of pure geometry, but Thurston’s Thurston technique for realizing conformal representation with circle packings has just recently been applied to construct planar rep-



resentations of the brain which may give better account of all the connections within its complete internal structure; see the site of M. Hurdal. II.9. Circles of Apollonius

Fig. II.9.1. The circles of Apollonius and the Apollonian packing of a square

The problem is to know what happens when we fill the interstices formed by three mutually tangent circles ad infinitum. That is, we trace the circle tangent on the interior to the three given circle, then we do the same with the three new interstices obtained, and we iterate indefinitely. Here is a new intrusion of iteration into geometry, in other words a certain dynamic feature. The first problem lies in knowing the area (the measure) of the surface that remains at the end of these infinitely many operations. Readers can try to show that this final area is, might be suspected, zero, which can be done quite nicely by hand. It is interesting to note that the first reference to the proof seems to date from 1943; for this whole section readers can consult (Apéry, 1982). We remark that we are in the situation inverse to that of the preceding section: here it’s the interstices “that survive”, for which the set itself is of measure zero. For those who know the construction of Lebesgue measure with the aid of squares, the preceding discussion shows that we can just as well use circular disks to define this measure. Such disks are sufficient to fill a square “completely”, in the sense of the measure. Now there is a conceptual ascent into the theory of Kleinian groups that explains or puts in a good context the following result on Apollonian circles. Here briefly is the essence of things: the set of disks for removal by the discrete (Kleinian) group are defined thus: Let A, B, C, D be the four circles of the Fig. II.9.2. There exist two Möbius transformations f, g that are well defined by the conditions: first they are transformations called parabolic, i.e. conjugates of translations. We can obtain them all since translation is the product of two symmetries about parallel lines by composing two inversions whose circles are tangent to each



other (two parallel lines, in conformal geometry, are tangent at the point at infinity). Second, we require that: f .A/ D A; f .B/ D B; f .C/ D D; g.C/ D C; g.D/ D D; g.A/ D B: We can draw things on the sphere, thanks to stereographic projection:

Fig. II.9.2. We view things on the sphere by stereographic projection. Apéry (1982) c Societé Mathématiques de France and F. Apéry

We see for example that f slides the circles in the space included between A and B, beginning by sending C onto D. Finally, the group generated by f and g (which is a free group by the way) is exactly the one which provides, beginning with the three initial circles, all the disks of the Apollonian Circles. Now we need to interpret as a subgroup of the group of hyperbolic space H 3 of three dimensions. Its geometry is tied to that of its fundamental domain (also called a Dirichlet or Voronoi domain); we encounter this type of domain in Sects. III.3 and IX.3. This domain .p/ (associated with a point p, but they are all isomorphic by the definition itself) is the set of points that are closer to p than all the other points .p/, where .p/ runs through all the elements of other than the identity. In hyperbolic geometry of dimension 3, the points at an equal distance from two distinct points always constitute, as in Euclidean geometry, a plane. The boundary of .p/ is thus made up of pieces of such planes. We see that this domain is a tetrahedron of H 3 . But, as a discrete subgroup † more general than Möb.S2 / D Isom.H 3 /, this boundary may be made up of an infinite number of planes. We say that † is geometrically finite in the case where the boundary of the fundamental domain is made up of a finite number of planes. Then the theorem on the Apollonian Circles is a very, very special case of a general theorem of Ahlfors from 1966: for each geometrically finite Kleinian group, its limit set on S2 is of measure zero. The limit set of † is associated with it just as the Apollonian circles were associated with the group . That is what puts the Apollonian Circles in the right conceptual context.



A question: The Apollonian circles give a packing, within measure zero, of the sphere S2 by disks: is this the optimal way? This question seems open, but we can nonetheless think that a fractal dimension provides part of the answer. For fractals see the classic references: among others, after the historic (Mandelbrot, 1977– 1882), the three books Feder (1988), Falconer (1990) and Cromwell (1997). We consider here, along with the Apollonian Circles, the subset of C made up of circles (not disks) an infinite (but still denumerable) set of curves. The “finite” curves have a positive 1-dimensional measure but are of measure zero with respect to 2-dimensional measure. Now, what is it for the infinite set of curves C ? It’s a set of fractal type obtained by an iteration process repeated infinitely many times, as in a snow crystal, which is the best known. For each part of S2 , and of a metric space more generally, we define its Hausdorff dimension. For a nonempty open subset of S2 , it is always equal to 2; for each (differentiable) curve of S2 , it is equal to 1. This dimension measures in some way the density, the complexity, the packing efficiency, etc. To evaluate this efficiency, we evaluate the total “˛-dimensional” measure of the balls required to cover the object considered as the maximum radius tends P to zero. Precisely, we seek the limit inferior, for r tending toward zero, of the sums .radius.Bi //˛ , where the balls Bi cover our set and are of radius 6 r. It is shown that there is at most one value of ˛ for which this limit is neither infinite nor zero. This real number ˛ is, by definition, the Hausdorff dimension of the set. In considering the value of this limit for our set and its subsets, we have furthermore a corresponding measure for this dimension: the Hausdorff measure. Note that in simple cases we can be content with finite coverings by balls of equal radius and replace the above sum by N.r/ r ˛ , where N.r/ designates the number of balls of radius r necessary for covering the set; but there are situations where denumerable coverings are required, and then the balls can not all have the same radius. For the Apollonian Circles C , we are obviously between 1 and 2 and as for the snow crystal we therefore expect, in one dimension, a Hausdorff dimension between 0 and 1. For the snow crystal, the answer is classic: its Hausdorff dimension 4 equals log . log 3 The matter is infinitely more difficult for the Apollonian circles; in fact the exact value of its Hausdorff dimension is unknown. The profound reason why we know it for snow crystals is that these are self-similar fractals, i.e. defined by an iterative procedure involving similitudes, for which we know the ratios for the areas and lengths when we iterate to infinity. In contrast, for the Apollonian circles, it is inversions that are used and that must be iterated, and they transform neither areas nor lengths uniformly. For computer enthusiasts, calculations can aid in formulating a result and they may believe that they need only program the calculations on the right computer. But it is essential to realize that Hausdorff dimension, which requires an infinite process in order to obtain an exact limit, is by its very nature inaccessible to computer calculation to the extent that such calculation is “explicit” and nonmathematical: the computer knows only rational numbers for which the number of significant digits



is bounded (certainly the bound depends on the computer, but it is always finite). Nonetheless we owe to Sullivan, in a very subtle study, a method for calculating the Hausdorff dimension of C theoretically, which furnishes the crude estimate of 1:3. The title of the text (Sullivan, 1984) shows how fully up the ladder of our subject it is situated. II. XYZ The practical measurement of lengths (distances). Before presenting any mathematical theory, we have the pleasure of giving for inquisitive minds some information on the way in which lengths and angles are measured from practical and theoretical points of view. On the one hand because there is mathematics behind the practice, but on the other because, as it seems to us, the practice is most often unknown to professional mathematicians, teachers and researchers. There are roughly three levels of precision to be distinguished: to start that of rulers, the protractors of schoolchildren or graduated straightedges for “ordinary” drawing. Then that of industry and scientific laboratories: in this case the degrees of precision demanded can vary enormously according to the objects to be made. For angles, the scholarly word is goniometer. Today the machining of car engines and reactor rotors is performed with increasing precision, but such precision is also needed in scientific laboratories. The exemplary case is that of telescope mirrors. The final level is that of metrology, for which the laboratories furnish on the one hand theoretical results so that measures on the planet might be, as much as possible, the same in different places, and on the other hand instruments utilized by industry. There are objects called calipers, both for distances or for angles. In order to produce straightedges (see also Sect. V.7): defective, approximate rulers are rubbed (with an appropriate abrasive) one against the other, dealing with three of them (with two only we obtain circles); imperfections are further repaired by interferometry or, recently, by laser imperfections that are then corrected “by hand”, with or without a polishing tool, and more and more today with CAD (computer aided design).

Fig. II.XYZ.1. The curve of contact necessarily being of constant curvature, it’s a piece of a circle. With three pieces we therefore end up, in theory, with portions of perfect lines. Bouasse (1917). With kind permission from Delagrave Éditions, public domain



Fig. II.XYZ.2. A “marble” Bouasse (1917). With kind permission from Delagrave Éditions, public domain

As for planes, we will see in Sect. VI.6 why, when three “defective” planes are rubbed each against the others, three perfect planes are obtained in the limit. In practice two things are done. First, proceeding as just indicated in the case of straightedges, finishing with corrections by hand (since time is limited and reality isn’t theory). In industry, such planes are called marbles. To make other planes (or, as is said, to prepare a surface), a “bad plane to be prepared” is rubbed with a dye against the reference marble and corrected by abrading the colored portions by hand, repeating until everything finally is just about evenly colored. In metrology, the “perfect” marble is of a transparent material, for example of the ceramic glass ZERODUR; it is applied to the future plane and then imperfections are repaired by interferometry (classic or laser), always with subsequent correction “by hand”. Of course in all these cases the operations are iterated until a satisfactory result is obtained. Ceramics are employed, in that they are materials for which the coefficient of thermal expansion is extremely low. For all these fabrications, the inquisitive mind will pose the question about the abrasion paradox: “how can we wear down steel with a felt grinder?” The answer is given on p. 46 of (Bouasse, 1917 # 1728): when two different bodies rub against each other with interposition of an abrasive, it isn’t the harder material that wears less quickly, but the one that better retains the abrasive. The grains of abrasive, of emery, find lodging in the felt, the paper, or the cloth. Everything happens as if the steel were rubbing against a block of emery. This provides lines, we dare say sufficient for the purposes of affine geometry. But now we need to measure lengths. The first thing to do is define a unit of length common to all nations. Progress in this definition has been historically almost simultaneous with that in graduation, the measurement of lengths as a function of a unit length. Finally we have witnessed a spectacular turn of events. The world community of metrologists, having made better and better measurements of the speed of light with rather sophisticated meter standards and using optical means of measurement based on classical interferometry, then based on fringes for well determined wave lengths of radiation corresponding to transitions between energy levels of given atoms this community has reversed everything in a double fashion and ended up defining the unit of length (always called meter) as follows. In 1963 the meter was equal to 1; 650; 763:73 wave lengths, in a vacuum, of radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton 86 atom. Since 1983: the meter is the length of the path traversed by light in a vacuum during



1=299; 792; 458 seconds. Technically what has made these successively new definitions possible is the appearance of lasers in 1958 and their stabilization in frequency at molecular transitions. The second (unit of time) is defined in atomic fashion (the actual precision in the measure of time is known). There remains today the mass, which still hasn’t been defined in an atomic fashion and for which the kilo remains the mass of an object that is an iridium alloy of platinum and kept at Sèvres (which still keeps the meter standard, but now only as an historic souvenir). But it is clear that the atomic definition of unit mass is much studied, but nevertheless still poses difficulties for physicists. As possible references for the history of the definition of length, readers can consult (Bouasse, 1917) (whose humor and virulent attacks are to be taken with caution and forewarning), but above all Bouchareine (1996) for interferometry, and Giacomo (1995). Let’s return to the practice of the measurement of lengths. The crudest procedure is with a graduated ruler, which always remains difficult, more difficult still as we seek not only equal graduation (that would be trivial enough) but we need first to have an accurate basic length, say a meter, and then to divide it. The problem of the good length encompasses first of all that of expansion, see above. Metals have the advantage of not rotting, but in return they expand with heat considerably. This is why the old standard was kept precisely at 4° centigrade. Today, just as for lengths, we have the same for angles (angular calibration): the ceramic glass called ZERODUR mentioned above is used. It is also used now for the mirrors of large telescopes. Primitive graduations are made by division and by repetition for verification. For refining measurement beyond that, a practical device is the vernier, whose theory is closely related to the continued fractions, which are discussed in Sect. IX.1. This trick permits measurement of a tenth of a millimeter, and even of a twentieth, by interpolation on a ruler graduated in millimeters. The micrometer, which was (and is) used in average precision industry, permits measurement to better than a tenth of a millimeter, it uses a screw thread (and the theory of continued fractions, at the lowest level to be sure). We will come back to the screw thread in a moment. For the theory of the vernier and that of the micrometer (which is always an entry point of continued fractions), see again Bouasse (1917). In industry, distances are now easily measured to the order of a micron. To go to a higher degree of precision in the measurement of length, optics and interferometry are used, which had fundamentally been done for the first precise measurement of the speed of light by Michelson in 1881, then still more precisely by Michelson and Morley in 1887. Now we have come so far that the laser is an everyday object, to the extent that the tracks of high speed trains are corrected to the ideal curve with the aid of lasers; they have come to be used likewise for correcting errors in linearity and planarity. The best performances in the domain of dimensional metrology are obtained by a laser system with fringe counting. The wave length of a laser in a vacuum can be stabilized to within a few multiples of 109 , and the measurement of air pressure and temperature permit determination of wave length in air to several 107 . These devices can display a tenth of a micron by interpolation on the interior of a fringe. The wave length exploited is that of the transition of neon (633 nm) and the scrolling of a fringe cor-



Fig. II.XYZ.3. Bouasse (1917). With kind permission from Delagrave Éditions, public domain

responding to displacement of a half wave length, say 0:316 m. Screw threads are used, their role however being limited to that of moving the displacements, which are read either by a scale engraved on rulers with the aid of a microscope or by numerical rulers on which an optical reader counts lines that scroll, with an inter-

Fig. II.XYZ.4. A caliper and a micrometer. Bouasse (1917). With kind permission from Delagrave Éditions, public domain



polation system between lines. Such a reader can handle about a tenth of a micron, thus a precision of 107 . It is important to remark that here we are concerned with the measurement of absolute lengths. In contrast, in measuring variations of lengths, precisions of a whole other of magnitude are now attained, totally astonishing; specifically, of order 1021 , it is hoped that gravitational waves will soon be detected in this way. Their existence is one of the great problems of contemporary physics, for these waves are predicted by the general theory of relativity. This is astonishing for the following reason: the diameter of the atomic nucleus is of the order of 109 μm, the diameter of the atom (with its electron cloud) of order 103 μm only. Well, presently, measurements of 30 m are made with this relative error of 1021 , or thus an absolute error of order 3:1014 μm. It seems that there is a contradiction here, since the atomic dimensions are much larger. The explanation is that, if the surface mirror that is used in the experiments is almost perfectly plane or spherical, it will produce some compensations, thus an averaging effect. As for detecting gravitational waves, it’s not precision that is lacking presently, but the time for constructing tubes sufficiently accurate and three kilometers long. All these experiments require excellent parabolic mirrors, whose construction is being refined constantly. The screw thread is a much prized device in dimensional metrology. Making perfect screw threads is just as simple as making perfect spheres (cf. Sect. VII.13), for here again there aren’t three but only two objects to adjust one against the other. It suffices to take a long bolt, a long screw thread and a nut approximations and to rub them carefully; see e.g. Fig. II.XYZ.3. When all goes well, when they have been rubbed well and when the red powder used leaves a uniform trace on the screw thread, that’s what matters for a “perfect” helix. This may be seen in two ways, either that we know that the helices are the only curves of E3 of constant curvature and torsion (see 8.6 of [BG] for the curves of E3 , about which we have practically no time here to speak explicitly and readily acknowledge that they will need to be broached in Chap. V); or again that the only one-parameter subgroups of the group of motions of E3 are the helical displacements. Having a perfect helix, we have a correspondence an exact proportionality between lengths and angles that had been sought. Nowadays interferometry is used. We point out that, historically, the measurement of the speed of light by Michelson succeeded with a very high precision because, being a very, very, good experimenter, he had made very good sliders and an extremely precise helix. For all this history and more, see the reference Bouasse (1917) which still remains up-to-date for the majority of fabrications of objects with strong geometric constraints. For surface fabrication, and for that of spheres and planes, see VI.6. For the fabrication of balls see Sects. VI.8 and VII.13.C. The problem of angle measurement is simultaneously easier theoretically but more difficult practically. For what is easy, first there is no problem in tracing circles with a compass. Then, about its center, we are certain, even if the circle itself is



badly drawn, to turn 360 degrees. It’s not like having to define a unit of length such as the meter. Of course we need to take care to turn steadily while remaining in a plane, which is one of the major difficulties in practical astronomy. Besides, dilation has practically no effect on angles. Nevertheless more and more the ceramic glass ZERODUR is being used for angular calibration. Now the problem is to divide our circle into equal parts to obtain the best possible goniometers. The method is incredibly primitive, but there isn’t any other. We divide approximately and correct until reaching the point where, by repeating such and such an angle, the division is visually good. We obtain in this way goniometers with engravings to the order of several seconds. To measure some angle with precision, to be verified to some standard, it has been discovered rather recently that the best thing to do is to make the dozen measurements obtained in turning the object twelve times through 30 degrees and to take the mean. The precision of this mean is in theory several tenths of a second (a degree is divided into 60 min and the minute into 60s). Better results are obtained in this way than by the classical least squares method applied to repeated single measurements. If things go well, it’s because we have turned through exactly 360 degrees (see however the caveat above). There is also the fact that an error in the center of the goniometer is compensated by the symmetry about the center. For the rest, the measure of angles today in the best metrological laboratories on the planet don’t achieve maximum precision of an order less than that of the second, and again it is necessary to make comparisons between such laboratories. As for measurements made by astronomers and every other profession requiring precision angles, things are much more difficult because of the variation in axes of rotation, etc. We end up at best with an accuracy of order 106 . Let us return to “pure” mathematics with an eye only for the formalization of geometry. Recall at first, if necessary, that a metric space is a set E endowed with a function d with nonnegative real values that associates with each pair of points x; y of E their distance d.x; y/, satisfying only three axioms: the symmetry axiom d.x; y/ D d.y; x/ for all x; y of E, the triangle inequality d.x; z/ 6 d.x; y/ C d.y; z/ for all x; y; z in E and finally the separation axiom: d.x; y/ D 0 if and only if x D y. It’s often this last that is most difficult to show when defining a metric space. Thanks to linear (and multilinear) algebra we nowadays define a Euclidean vector space of dimension n simply as a vector space of dimension n endowed with, among other things, a positive definite quadratic form q. We then define the norm of a vector v as the square root of the value p we get upon applying the quadratic form of the definition, denoted by kvk D q.v; v/, the scalar product being denoted by hv; wi or v w. The angle ˛ (between 0 and ) between two nonzero vectors v and w is defined by cos ˛ D v w=kvk kwk. The Euclidean space itself is the associated affine space whose subspaces (lines, planes, hyperplanes) are defined. We have in addition a metric d defined by d.p; q/ D kq pk. We note the Euclidean heredity:



all the subspaces (of whatever dimension) of a Euclidean space are automatically endowed with a canonical Euclidean structure, inherited trivially. All the Euclidean spaces of the same dimension n being isomorphic to one another, we often speak of the Euclidean space of dimension n and use the notation En (or rigorously Rn ). It is often of interest to identify the Euclidean plane E2 with the field C of complex numbers; to do this we choose an orthonormal basis f0; v; wg (a point and two orthogonal unitary vectors). Then the complex number associated with the point with coordinates .x; y/ in this basis is the complex number z D x C iy. The (direct, i.e. orientation preserving) similitudes with center O are represented by the operation z 7! kz (k an arbitrary nonzero complex number), and if we want to change the orientation (to find the inverse similitudes) it is necessary to consider the z 7! k z. N In particular, multiplication by i is nothing other than rotation about the origin (or vectorial) through an angle 2 . Poles and polars with respect to a circle. Let C be the circle with center O and radius R. We say that two points of the plane m and m0 are conjugate with respect to C if ! !0 Om Om D R2 : The points of the circle C are those that are conjugate to themselves. If m isn’t the center of C, the set of points conjugate to m is a line perpendicular to Om, situated a distance R2 =d.O; m/ from O. We call it the line polar to m, or simply the polar of m, denoted m . If m is interior to C, then m doesn’t intersect C, if m is exterior, then m intersects C, and if m is a point of C, then m is the tangent to C at the point m. These instructions suffice to give a construction of the polar of a point m exterior to C: it’s the line that joins the points of contact of the tangents leading to m. The mapping m 7! m possesses all the properties of a duality if we agree to eliminate the point O from the plane P and the lines passing through O from the set of lines of P . First of all, it’s a bijection. In particular, for a line d not passing through O, the unique point d for which d is the polar is called the pole of the

Fig. II.XYZ.5.



Fig. II.XYZ.6.

line. Next, it transforms collinear points into intersecting lines. Moreover, when the point m describes a line d (see the figure below), its polar m describes, with a single exception, the pencil of lines passing through d , denoted again d . This correspondence between d and d is hom*ographic, i.e. it preserves the cross ratio. A second way of defining the conjugate points will lead to another definition of the polar: two points m and m0 of the plane are conjugate with respect to C if the points of intersection p and q of the line form with m and m0 an harmonic division: Œm; m0 ; p; q D 1. The points p and q can be complex or identical (the case where mm0 is tangent to C and m D m0 is the point of contact). A rather simple calculation allows us to show the equivalence with the first definition, but the deep reason is as follows: if we are restricted to a line D on which an ! ! abscissa x is chosen, the relation Om Om0 D R2 takes on the (symmetric) form axx 0 C b.x C x 0 / C c D 0: We solve for x 0 as a hom*ographic function of x. This hom*ography is an involution that admits the abscissas of the points p and q as fixed points (since they are solutions of the equation ax 2 C 2bx D c D 0). By expressing that they preserve the cross ration we obtain the relation Œm; m0 ; p; q D Œm0 ; m; p; q, from which we deduce Œm; m0 ; p; q D 1; see Sect. I.6. To conclude, we indicate two other ways of defining conjugation with respect to a circle, valid in all circ*mstances, and which avoid, for those who care, recourse to the scalar product and to complex numbers. Two points m and m0 are conjugate with respect to the circle C if the circle with diameter mm0 is orthogonal to C. In the figure below, the points m and m0 are conjugate with respect to C. To see this, we complete the figure (see Fig. II.XYZ.8): the point ˛ is conjugate to the point m with respect to the pair of lines D and D0 (figure of the complete quadrilateral). It is thus the conjugate harmonic of m with respect to the points of intersection of the line m˛ with D and D0 , which are also the points of intersection



m′ m

Fig. II.XYZ.7. Construction of a conjugate point (we can begin either with m, or with m0 )

α m′ m β

Fig. II.XYZ.8. Justification of the construction of Fig. II.XYZ.7

of this line with C. In other words, ˛ is the conjugate of m with respect to the circle C, and it similarly for ˇ, which shows that the polar of m with respect to the circle C is the line ˛ˇ, and in particular that this polar passes through m0 . It remains to dispose of the exceptions in the pole-polar correspondence. This is done by passing to the projective plane and by giving the definition of conjugation in hom*ogeneous coordinates. The equation of the circle is given in hom*ogeneous coordinates by setting a quadratic form equal to zero: Q.x; y; z/ D x 2 C y 2 R2 z 2 D 0 if the coordinates are well chosen, but the left hand side of the equation remains a quadratic form in any coordinate system. In this case, the points .x W y W z/ and .x 0 W y 0 W z 0 / are conjugate with respect to C if we have xx 0 C yy 0 R2 zz 0 D 0: For points at a finite distance, we verify immediately that this relation is equivalent ! ! to the relation Om Om0 D R2 . In general two points m and m0 of the projective plane are conjugate with respect to C if the corresponding triples .x W y W z/ and .x 0 W y 0 W z 0 / are orthogonal (we also say conjugate) with respect to the bilinear form B associated with Q. Frequently B is called the polar form of Q, and in coordinates it is obtained by polarizing the quadratic form: a square term z 2 gives zz 0 and a



rectangular term yz gives 12 .yz 0 Cy 0 z/. The coincidence of vocabularies is evidently not a coincidence. We can now associate a polar with the point O: it’s the line at infinity, and the poles of the lines passing through O: these are the points at infinity situated in the perpendicular direction. We thus have a perfect duality between P and the set P of lines of P . It is clear that, in the form that we have just given it, the theory applies to an arbitrary conic; see Chap. IV. For the mapping m 7! m to be bijective, it is clearly necessary that the quadratic form Q be nondegenerate, but the mapping is interesting even when Q is degenerate. In the case where Q is only of rank 2, i.e. where Q is the product of two equations of distinct lines D and D0 , the polar of a point m with respect to the conic is well defined if m is distinct from the point a of intersection of D and D0 , and it is a line passing through a: i.e. the locus of the points m0 of the plane such that Œm; m0 ; p; q D 1, where p and q denote the points of intersection of the line mm0 with D and D0 . We thus recover the classic notion of polar with respect to two lines.



Fig. II.XYZ.9. Conjugate points with respect to two lines

Harmonic hom*ology. hom*ology comes up in the proof of the butterfly theorem; see Fig. II.2.1. Let D be a line of the projective plane P , and a a point not belonging to D. The harmonic hom*ology with center a and axis D is the mapping h that associates with a point m of the plane distinct from a and not belonging to D the harmonic conjugate of m with respect to the point of intersection of the line am with D, in other words the unique point m0 collinear with a and m such that Œa; p; m; m0 D 1, where p is the point of intersection of the line am with D. According to the usual conventions regarding the cross ration, we have Œa; m; m; m D 1, Œa; a; a; m D 1 on the line am, provided that m is a distinct point of a, which allows extending h to D and to the point a: h leaves the point a and all the points of D invariant. An extension by continuity will have the same effect. If P denotes the projectified affine plane, and if a is a point at infinity, the harmonic hom*ology with center a and axis D is the (oblique) symmetry with respect to D, parallel to the direction of a. If a is at finite distance and if D is the line at infinity, the harmonic hom*ology is the symmetry with center a. These examples show,



without need for an additional calculation, that the general harmonic hom*ology is a hom*ography. Pencils of circles. Let C be a circle in the plane with center O and radius R; moreover let m be a point of the plane and D be a line passing through m and intersecting C at the points p and q. Then the product mp mq is independent of the line D chosen. The value of this product is called the power of the point m with respect to the circle C, and is denoted by pC .m/. If m is exterior to the circle, then pC .m/ is the square of the length of the tangents leading from m to C (which is called the tangential distance of m to the circle). In general we have pC .m/ D Om2 R2 , and pC .m/ D 0 if and only if m is a point of the circle. In coordinates, pC .m/ is nothing other than the first term of the equation of the circle, provided that this term is normalized so as to begin with x 2 C y 2 . t q p O m



Fig. II.XYZ.10. Power of a point with respect to a circle: mp mq D mp 0 mq 0 D mt 2 D Om2 R2

We call the line defined by the equation pC .m/ D pC0 .m/ the radical axis of the two circles C and C0 . If the circles intersect in two distinct points a and b, their radical axis is the line ab; if the circles don’t intersect, the radical axis is the line that passes through the centers of their common tangents; if the circles are tangent, the radical axis is the common tangent.

Fig. II.XYZ.11.

The pencil of circles defined by two circles C and C0 is the set of circles given by the equation pC .m/ C 0 pC0 .m/ D 0: For D 0 , we obtain not a circle but a line: the radical axis. We can normalize the above equation by writing pC .m/ C ˛pC0 .m/ D 0, but we then lack the circle C0 , for which we must stipulate that it is obtained for ˛ D 1. We can also normalize it by writing ˇpC .m/ C .1 ˇ/pC0 .m/ D 0, but we must stipulate in this case that the



radical axis is obtained for ˇ D 1. For ˇ finite, the first member of the equation has for higher degree terms x 2 C y 2 and thus represents the power of m with respect to the circle C.ˇ/ with parameter ˇ. Two properties are thus evident: the points of the radical axis have the same power (or the same tangential distance) with respect to all the circles of the pencil; the ratio of powers of a point of a given circle of the pencil of circles with basis C and C0 is independent of the point chosen. When the circles C and C0 intersect, the pencil defined by C and C0 is the set of circles passing through the points of intersection a and b of C and C0 (see the figures taken from [B]). When C and C0 don’t intersect, the pencil corresponds to two circles of radius zero situated on the line of the centers. When C and C0 are tangent at a point a, the pencil is the set of circles tangent at a to the common tangent of C and C0 . We note that in all these cases, the radical axis of two circles of the pencil is the same as that of C and C0 . We can now forget the two base circles C and C0 and give the classification of the pencils: pencil with two base points: the set of circles passing through two given distinct points a and b; pencil with two limit points: the set of circles defined by an equation of the form mp=mq D const., where p and q are two given points; tangent pencil: the set of circles tangent to a given line D at a given point a. Completing the plane by a line at infinity and introducing the cyclic point may clarify the situation to a certain extent: a pencil of generic conics is defined as the set of conics passing through four given points a, b, c, d ; it entails three degenerate conics: the pairs of lines ab and cd , ac and bd , ad and bc. The pencil with base points a and b is simply the pencil of conics passing through the four points a, b, I, J. Of the three degenerate conics that it contains, two are complex, but the third is the union of the radical axis and the line at infinity, which is a degenerate circle, as it contains the line at infinity. In the case of a pencil with limit points p and q, the points common to all the circles of the pencil aren’t interesting, but the three degenerate circles are clearly visible: first that formed by the radical axis and the line at infinity, then the circles p and q of radius zero, which can be considered as being the union of lines of slope i and i (isotropic lines). Here is another way of (partially) clarifying the situation: the space C of circles is a real projective space of dimension 3 (see toward the end of XYZ), and a circle pencil F is a line in the space of circles, whereas the set of circles of radius zero is a quadratic in this space. For the circle with equation x 2 C y 2 2ax 2by C c D 0, we have R2 D a2 C b 2 c; it’s a quadratic form if we pass to hom*ogeneous coordinates. There are thus three possible situations: F doesn’t intersect this quadric, F intersects this quadric in two points, F is tangent to this quadric. A necessary and sufficient condition for two circles C and C0 with centers O and O0 and radii R and R0 to be orthogonal is that R2 C R02 OO02 D 0. What is



remarkable is that the quantity OO02 R2 R02 , within a factor of 2, is the polar form associated with the quadratic R2 , and which is expressed in bilinear fashion with respect to the equations of two circles. In fact, if these are written respectively x 2 C y 2 2ax 2by C c D 0 x 2 C y 2 2a0 x 2b 0 y C c 0 D 0 we have

OO02 R2 R02 D 2aa0 C 2bb 0 c c 0 :

We can call the quantity 2aa0 C 2bb 0 c c 0 the scalar product of the circles C and C0 ; and two circles are orthogonal if they are, in the space of circles, conjugate with respect to the quadric of circles of radius zero. Further, the set of circles orthogonal to two given circles is a line in the space of circles, i.e. a pencil of circles, and each pencil can be defined in this way. In particular, the set of circles orthogonal to the circles of a pencil is a pencil the pencil with limit points a and b when the given pencil is the pencil with base points a and b, and conversely. This is particularly interesting when we complete the plane with a single point at infinity, in the context of conformal geometry. First, the radical axis becomes a circle like the others, provided that we adjoin the point at infinity. But above all, the fact that a pencil can be defined as the set of circles orthogonal to two given circles

Fig. II.XYZ.12. Pencil of limit points (alternation of the white and the gray) and its orthogonal pencil. Pencil of base points and its orthogonal pencil



Fig. II.XYZ.13. Tangent pencil

provides an immediate proof of the fact that the image by an inversion of the pencil of circles is again a circle pencil which isn’t at all evident from the calculation in the case where this is necessary, i.e. in the case of limit point pencil. A scandal to repair. In all this we have neither defined a circle nor, a fortiori, a sphere. Recall that a sphere of radius r and center O in a Euclidean space is the set S.0I r/ formed of points situated a distance r from the point O, and which we call a circle in the case of a plane. Let us speak primarily of the plane and of circles and the problem of having tools for studying different figures, like those of Sect. II.2. We first remark that three noncollinear points, forming a true triangle, have a well defined, and unique, circ*mscribed circle. Therefore: four points being given, when are they on a circle? We saw in Sect. II.2 a condition that is necessary and sufficient, based on angles. But we are right to expect a purely metric condition that allows intervention of the six mutual distances that separate the four points. In fact, the notion of angle is not primary, and we have seen that it is necessary to be very sophisticated if we want to include at the same time the case of the convex quadrilaterals and ones that cross. Now such a condition is purely metric, non angular, and almost never taught in any school or university curriculum. That is a scandal, to be repaired, even though there is a dazzling excuse: it’s practically impossible to prove the overwhelming majority of theorems on circles using this relation. Here it is, however; it’s called Ptolemy’s theorem and states that, for the four points a, b, c, d to be on a circle, it is necessary and sufficient that one of the four relations ab cd ˙ ac db ˙ ad bc D 0 between the six distances in question be satisfied. We utilize this expression so as not to have to specify whether the quadrilateral is convex, in which case the relation is very pretty: “the product of the diagonals is equal to the sum of the products of the opposite sides”. A perverse but miraculous proof uses angles; for a purely metric proof and its extension to arbitrary dimensions (five point on a sphere in ordinary



space, etc.), see 9.7 of [B] or the great classic (Blumenthal, 1970). All the relations are written in the language of determinants. If we apply Ptolemy’s theorem when two points are the ends of a diameter, we find the relation that gives the sine of the sum of two angles; in fact, Ptolemy used his theorem for constructing tables of the sine function (or the equivalent at his time: tables of “chords”.: crd ˛ D 2 sin ˛=2)

Fig. II.XYZ.14. Ptolemy’s theorem: ab cd R C ad bc D ac bd . In particular, in the case of a diameter, we find the formula sin.˛ C ˇ / D sin ˛ cos ˇ C sin ˇ cos ˛ by using the basic fact: in a triangle the ratio of a side to the sine of the opposite angle is always equal to the diameter of the circ*mscribed circle

It’s also nice to know that the six distances of four (arbitrary) points aren’t independent, as can be seen in the references and as one might discover from a plane figure, for when five distances are given there are just two possibilities for the sixth, since three sides determine a triangle.

Fig. II.XYZ.15.

It can be shown that there exists a universal relation between the six distances of each quadruple of points in the Euclidean plane but this is, contrary to that of



Ptolemy, of second degree in each distance; see once again 9.7 of [B]. This relation is almost unwritable by hand, without the language of determinants. Interested readers might ask if there exist other geometries (of given dimension n) where the .n C 2/.n C 1/=2 mutual distances of each .n C 2/-tuple of points are always connected by a universal relation. In fact, there aren’t any such geometries other than Euclidean, spherical and hyperbolic; see Berger (1981). These Euclidean relations don’t only serve the interests of geometers, but are used in particular in celestial mechanics; see Albouy and Chenciner (1998). The isometries of a Euclidean space are the mappings that preserve distances. In the case of the plane we have seen that these are the rotations about a point, the translations and the symmetry-translations, specifically compositions of symmetry about a line and a translation in the direction of this line. Every isometry is a product of at most three symmetries with respect to lines, symmetries that thus form a pleasing (and useful) set of generators. The classification of isometries of Euclidean spaces of arbitrary dimension is given in [B]; Theorem 9.3.1 is taken from Frenkel (1973), where it seems to appear for the first time in general and precise form. The symmetries with respect to hyperplanes remain generators, a much more general result; see Chap. 13 of [B] and the references mentioned there. Note the separation of isometries into two classes, according to whether or not they preserve orientation; see Sect. 2.7.2 of [B] for this notion. The similarity transformations (similitudes or similarities for short) of Euclidean spaces are the transformations that multiply distances by a fixed ratio, called the similitude ratio or aspect ratio of the similarity. They differ from isometries, in the main, only by the simplest similarities, specifically the hom*otheties (see Sect. I.XYZ). A hom*othety of center O and ratio the real number is the mapping that, when the affine space is vectorialized about the point O, is written v 7! v. Among the bijections, the similarities are characterized, in arbitrary dimension, by the following equivalent conditions: they preserve spheres, preserve angles, preserve right angles, see 9.5 and 9.6 of [B]. The initial idea is to arrive at using the fundamental theorem of affine geometry (of Sect. I.3). In Sect. II.2 we promised readers that we would “retrieve” Euclidean geometry with the help of projective geometry; here, briefly, is how we can achieve a good bit of this, which we explain in the planar case. We have seen in Sect. I.7 how to retrieve the affine invariant ac of three points arranged on a line F: its value is nothing ab other than the cross ration Œc; b; a; 1F , where 1F of course denotes the point at infinity of the line F, i.e. 1F D D \ F when the affine plane considered has been projectified. We can calculate ratios of distances, so it thus now suffices to know how to calculate angles in order to control all of Euclidean geometry. For that, we need to do two things: first, complexify our (Euclidean) real affine R2 in the complex affine plane C 2 ; we thus embed R2 trivially in C 2 . We could do the complexification more intrinsically, but then much more abstractly; but this is too complicated for the goal



pursued; see Chap. 7 of [B]. Then we projectify C 2 in CP2 . The essential idea is that the Euclidean structure (to within a change of scale) of R2 is coded in the pair fI; Jg of points of CP2 , called cyclic points of R2 , and which are, in complex projective coordinates, the two points at infinity f.0; 1; i/; .0; 1; i /g (here i is the square root of 1, that which in real coordinates is written .0; 1/) in C. Be aware that the real dimension of C 2 , like that of CP2 , is equal to 4. An important remark: we cannot choose any one of the cyclic points in a canonical fashion; it necessary to consider the pair, to the extent that we only know the Euclidean structure of the plane considered, identified in fact with R2 by an “unknown” isometry; thus we can’t say that we know the point .0; 1; i /, for we know only the quadratic form and not its expression in coordinates. However, interested readers can show that we can specify the cyclic points in a pair fI; Jg if the Euclidean plane is, in addition, oriented; see 8.8.7 and 8.5.1 of [B] for this and what follows. Briefly, the notion of cross ratio for 4 points of C (or of S2 D CP1 ) furnishes a nice characterization of the cocyclicity of points: they are cocyclic (or collinear) if and only their cross ratio is real. Now we have hope of retrieving Euclidean geometry, i.e. the quadratic form that defines it, for this form is x 2 C y 2 . Now, in C 2 , the equation x 2 C y 2 D 0 defines a conic (see Chap. IV), here degenerated into the pair of lines x ˙ iy D 0. We can in an equivalent fashion replace consideration of cyclic points by that of these two lines, called isotropic lines; we denote the pair of them by fI ; J g. Their points at infinity are exactly the cyclic points. This accomplished, all that remains is to give the formula, due to Laguerre: angle.D; D0 / D

ˇ ˇ 1 ˇˇ 1ˇ log Œ1D ; 1D ; I; J ˇ D ˇlog ŒD; D0 ; I ; J ˇ; 2 2

where the angle.D; D0 / of two lines D, D0 passing through the origin of R2 is expressed with the help of the cross ratio, with a choice either of four points 1D , 1D , I, J of the line at infinity of CP2 , or of four lines D, D0 , I , J passing through the origin. The young Laguerre was still a student in the preparatory class when he concocted this formula, not being really satisfied with his professor’s course. The proof today is banal, it utilizes only the fact that the complex exponential e it equals cos t C i sin t . We are thus finished with the problem. We will see in Sect. IV.7 how the whole “elementary” theory of the Euclidean conics can be treated with the considerations above. In the geometry of space, the role of the cyclic points is played by the umbilical. In Euclidean projective space, it’s the curve with hom*ogeneous equations x 2 Cy 2 C z 2 D 0, t D 0. It collects the cyclic points of all the planes of space (two parallel planes have the same cyclic points) and, for a non degenerate quadric to be a sphere, it is necessary and sufficient that it contain the umbilical. A torus of revolution contains the umbilical as a double line (curve of self intersection); we have used this observation apropos Vilarceau circles. The umbilical plays a large role in the modern theory of stereoscopic vision and its applications to computer vision (see the work of Olivier Faugeras on epipolar geometry).



To now make clear the nature of hyperbolic geometry of dimension n, denoted Hypn , we follow Chap. 19 of [B], i.e. the four models of hyperbolic geometry, to which we adjoin a fifth, the Riemannian model, that was not presented in [B] since it requires the notion of Riemannian manifold. Other references are mentioned in [B], but after the appearance of numerous works on geometry, we remain very fond of Benedetti and Petronio (1992) and Ratcliffe (1994), since both go very high on the ladder. The most efficient model for exhibiting hyperbolic geometry in any dimension n is the projective model (also called linear or Klein’s model), where the price for rapidity that must be paid is knowledge of the notion of projective space, always difficult to visualize, since it requires an increase by one in dimension. We are placed in RnC1 , where the starting point is no longer a Euclidean structure but that defined 2 by the quadratic form q D x12 xn2 C xnC1 . For what follows, carefully examine the figures below. The basic geometric object is the cone Q D q 1 .0/; its interior consists of points where q is negative. But we now need to consider it in the projective space RPn associated with RnC1 . We find two things: the projections of the points where q is negative, the set of which forms an open ball Bn of dimension n, and the projection of the cone Q. We can, for example, obtain this ball (a disk when n D 3) as the interior of the cone cut by the hyperplane xnC1 D 1. For what follows the space of spheres it is well to remark that in RPn the projection of the cone is a projective quadric (see Chap. IV) for which the topology is that of a sphere Sn1 , since it is the boundary of the ball Bn .

Fig. II.XYZ.16. II [B] Géométrie. Nathan (1977, 1990) réimp. c Nathan Édition Cassini (2009)

now remains to define the metric d.p; q/ on Hypn : it is equal to the quantity where Œp; q; u; v denotes the cross ratio of the four points considered in the projective space RPn . Here p and q are seen in the projective and fu; vg is by definition the pair of points of the boundary of Bn where it is cut by the projective line joining p with q. We might prefer to see the cross ratio as that of four lines D, D0 , U, V passing through the origin associated with the four points considered. With the quadratic form q and its associated polar form P (i.e. the ana1 j log.Œp; q; u; v/j, 2



log of the scalar product in the Euclidean case), we can calculate d.p; q/ by the formula utilizing the hyperbolic cosine: P.; / ch.d.p; q// D p ; q./q./ where and are any two representatives of p and q in RnC1 . In Hypn , by definition, the subspaces (lines, planes, hyperplanes) are those of RPn restricted to Hypn . The lines are effectively those that attain minimum length among curves joining two points. By construction, all these linear mappings of RnC1 that preserve q, lowered into n RP , are isometries of Hypn , and it is easy to see that there aren’t any others. We denote this group Isom.Hypn /. Here it appears in linear form in each dimension, more precisely in linear projective form. But if interpreted on the cone, then the projective quadric Sn1 is the Möbius group Möb.Sn1 /. This is the projective version, let us say linear, of hyperbolic geometry. We can say it is linear because the lines, the subspaces, are the same as in the affine geometry of the unit ball embedded in Rn . But of course the angles are no longer the same as in the canonical Euclidean geometry of Rn . The presentation we saw in Sects. II.4 and II.5, in contrast to the preceding, does not conserve lines but does conserve angles. The lines are pieces of circles orthogonal to the boundary. It is rightly called the conformal model or the Poincaré model. We can pass from the linear model to the conformal model by the inverse of a stereographic projection followed by an orthogonal Euclidean projection, as can be seen in the figure below:

Fig. II.XYZ.17. Correspondence between the Poincaré’s disk model and the linear c Springer model Hilbert, Cohn-Vossen (1996)

The Poincaré’s half-space model consists simply in transforming this conformal model by an inversion through a center situated on the spherical boundary of the conformal model. The interior of the ball is thus transformed into an open half-space. This model allows very simple calculations for certain problems, in particular in the dimension 2 case; see Sect. II.4. The lines are the semicircles of the plane that are orthogonal to the boundary line of the half-space. The quickest and simplest definition is that of Hypn as a Riemannian model. Its drawback is being nonelementary, i.e. it requires knowledge of the basics of



Riemannian geometry dealt with in Chap. III; also see Berger (2003) or any book on Riemannian geometry. It consists of saying that the underlying set is the upper sheet q of the hyperboloid (see Sect. IV.10) defined by the equation q D 1, i.e.

xnC1 D 1 C x12 C C xn2 . It’s a hypersurface of RnC1 for which the tangent space, translated to the origin, never intersects the cone Q and thus on which the restriction of the quadratic form q is positive definite. By definition, this makes the sheet of the hyperboloid into a Riemannian manifold. A Riemannian geometry in fact consists of putting a Euclidean geometry (infinitesimal thus) on each tangent space. Its Riemannian geometry is nothing other than that of Hypn , which can be seen, for example, from the group of isometries being the same. For more on hyperbolic geometry, see the references mentioned.

Fig. II.XYZ.18. Three versions of hyperbolic geometry. From left to right: Klein’s linear model, Poincaré’s disk model, Poincaré’s half-plane model. From above to below: a line, a pencil of lines passing through a point, a family of circles passing through a point and tangent at that point to a given fixed direction. We note the horicycles, circles for which the center has been moved to infinity: they are represented by circles tangent to the horizon in the two Poincaré models and by ellipses that are surosculating to the horizon in the Klein model

In Sect. II.7, we encountered the space C of circles of the Euclidean plane and the space S of spheres of the Euclidean space of dimension 3. As a reference for what follows, in modern language, we know only Chap. 19 of [B]. To define them correctly (things are the same in each dimension) we will thus construct the space S n of spheres of the space En (we might say hypersphere beginning with n D 4, as we say hyperplane, but the term sphere is presently most common). The idea is to describe a sphere in RnC1 by its equation, which may always be written in the



Fig. II.XYZ.19. The quadratic form x 2 C y 2 z 2 restricted to the plane is clearly positive definite

form x12 C C xn2 C b1 x1 C C bn xn C c D 0. But if we wish to calculate, to add, we can’t do it, the form changes. The idea is to write the form slightly more generally, specifically a.x12 C C xn2 / C b1 x1 C C bn xn C c D 0; but then two proportional expressions give the same sphere. So that this doesn’t happen, we pass to projective space! Here it is thus a RPnC1 resulting from RnC2 , defined by the n C 2 coordinates .a; b1 ; ::::; bn ; c/. In taking all RnC2 we have appended to the “true” spheres the object of the equation 1 D 0, which is therefore empty but that we interpret as the sphere of radius zero at infinity, which we like, for we have seen that it interposed too the objects where a D 0, which are the planes, and we know that they must be introduced at the same time as spheres in conformal geometry. To have a geometry, and hopefully a distance, on S n D RPnC1 , the idea is to introduce a quadratic form, which is in the disguise of the square of the radius. We define on RnC2 above by ..a; b1 ; ::::; bn ; c// D 14 .b12 C C bn2 4ac/. But, within a trivial change of coordinates, this quadratic form r is the same as the quadratic form q of hyperbolic geometry (we say that they are of signature .1; n C 1/, by having in diagonal form a single positive square and nC1 negative squares). We let S n denote the projective space RPnC1 endowed with the quadratic form . The group of this geometry, called generalized spheres, denoted here Möb.Sn /, is thus the same as Isom.Hypn / seen above. We now see why this answers practically all the questions about spheres, see [B] for the details. We can say that the geometry of the sphere is an extension of hyperbolic geometry. First, there is the hyperbolic geometry in S n ; these are spheres for which the square of the radius is négative. We can say that the radius is a pure imaginary, i.e. that is negative. The spheres of radius zero are the points of the space to which one adds the point at infinity; the totality makes up Sn . The true spheres are those for which is positive, but there are also the hyperplanes on which is positive. We can see geometrically where the hyperplanes in S n are: these are the points of the hyperplane tangent to S n at its point at infinity. The space of the true spheres is thus



difficult to “see”, for its topology is that of RPnC1 from which we have removed an open ball; it’s what remains when we have removed the hyperbolic geometry, a sort of complement of this geometry. As a quadric, not yet projectified in RnC2 , it’s that of the hyperboloid of one sheet with equation D 1 (see Sect. IV.10). It is immediate, for example, that the (generalized) circles are represented, in S 3 , by the projective lines (made up of the set of spheres that contain that circle). We do the same with all the k-spheres and the projective subspaces of all dimensions. Finally, the angle of two spheres plays a role analogous to that of the hyperbolic metric; if R is the polar form of , then we define the angle ˛ of two spheres s and s 0 as the real 0 /j number in Œ0; =2 such that cos ˛ D pjR.s;s . This angle doesn’t alway exist, but .s/.s 0 / when it does, i.e. if in fact the spheres intersect, it is of course the angle they make in the ordinary sense. The geometry of oriented spheres consists simply instead of considering S n D nC1 RP , where RPnC1 is the set of vectorial lines of RnC2 of taking the space n .S / of half-lines, i.e. the quotient of RnC2 by the equivalence relation v w if w D kv, with k a positive real number. We thus obtain a space for which the topology is that of the sphere SnC1 . See an alternative presentation in the article on conformal geometry in the Encyclopedic Dictionary (1985). Bibliography [B] Berger, M. (1987, 2009). Geometry I,II. Berlin/Heidelberg/New York: Springer [BG] Berger, M., & Gostiaux, B. (1987). Differential geometry: Manifolds, curves and surfaces. Berlin/Heidelberg/New York: Springer Aigner, M., & Ziegler, G. (1998). Proofs from THE BOOK. Berlin/Heidelberg/New York: Springer Albouy, A., & Chenciner, A. (1998). Le problème des n corps et les distances mutuelles. Inventiones Mathematicae, 131, 151–184 Apéry, F. (1982). La baderne d’Apollonius. Gazette de la Société math. France, nı 19 (juin), 57–86 Benedetti, R., & Petronio, C. (1992). Lectures on hyperbolic geometry. Berlin/Heidelberg/New York: Springer Berger, M. (1981). Une caractérisation purement métrique des variétés riemanniennes à courbure constante. In P. Butzer & F. Fehér (Eds.), E. B. Christoffel (pp. 480–492). Basel: Birkhäuser Berger, M. (1993). Les paquets de cercles. In C. E. Tricerri (Ed.), Differential geometry and topology. Singapore: World Scientific Berger, M. (2002). Les cercles de Villarceau. Pour la Science, nı 292 (février 2002), 90–91 Berger, M. (2003). A panoramic view of Riemannian geometry. Berlin/Heidelberg/New York: Springer Blumenthal, L. (1970). Theory and applications of distance geometry. New York: Chelsea Bobenko, A. & Hoffmann, T. (2001). Conformally symmetric circle packings : a generalization of Doyle’s spirals, Experimental Mathematics, 10, 141–150 Boltyanski, V., Martini, H., & Soltan, P. (1997). Excursions into combinatorial geometry. Berlin/Heidelberg/New York: Springer Borsuk, K. (1933). Drei Sätze über die dreidimensionale euklidische Sphäre. Fundamenta Mathematicae, 20, 177–190 Bouasse, H. (1917). Appareils de mesure. Delagrave, reprinted by Blanchard en 1986 Bouchareine, P. (1996). Charles Fabry métrologue. Annales de physique, 21, 589–600 Cho, Y., & Kim, H. (1999). On the volume formula for hyperbolic polyhedra. Discrete & Computational Geometry, 22, 347–366



Connes, A. (1999). Le théorème de Morley. Publ. Math. Inst. Hautes Études Sci., numéro special des 40 ans Coolidge, J. (1916, 1999). A treatise on the circle and the sphere. Oxford/New York: Clarendon Press/AMS Chelsea Cromwell, P. (1997). Polyhedra. Cambridge: Cambridge University Press Darboux, G. (1880). Sur le contact des coniques et des surfaces. Comptes Rendus, AcadÈmie des sciences de Paris, 91, 969–971 Darboux, G. (1917). Principes de géométrie analytique. Paris: Gauthier-Villars Davis, P. (1995). The rise, fall and possible transfiguration of triangle Geometry: a mini-history. The American Mathematical Monthly, 102, 204–214 Dieudonné, J. (1960). Foundations of modern analysis. New York: Academic Press Doyle, P., He, Z.-X., & Rodin, B. (1994). The asymptotic value of the circle packing constant. Discrete & Computational Geometry, 12, 105–116 Eggleston, H. (1957). Covering a three-dimensional set with sets of smaller diameter. Journal of the London Mathematical Society, Second Series, 30, 11–24 Encyclopedic Dictionary. (1985). Encyclopedic dictionary of mathematics (3rd ed.). Cambridge: M.I.T Evelyn, C., Money-Coutts, G., & Tyrelle, J. (1975). Le théorème des sept cercles. Paris: CEDIC Falconer, K. (1990). Fractal geometry. New York: John Wiley Faugeras, O. (1993). Three-dimensional computer vision: A geometric viewpoint. Cambridge: MIT Press Feder, J. (1988). Fractals. New York: Plenum Press Frances, C. (2003). Une preuve du théorème de Liouville en géométrie conforme dans le cas analytique. Líenseignement mathématique, 49, 95–100 Frenkel, J. (1973). Géométrie pour l’élève professeur. Paris: Hermann Giacomo, P. (1995). Du platine à la lumière. Bulletin BNM, 102, 5–14 Grünbaum, B. (1993). Convex Polytopes. Berlin/Heidelberg/New York: Springer Gutkin, E. (2003). Extremal triangles for a triple of concentric circles. Preprint IHES, M03-46 Hadamard, J. (1911). Leçons de géométrie élémentaire. Paris: Armand Colin, reprint Jacques Gabay, 2004 Hilbert, D., & Cohn-Vossen, S. (1996), Anschauliche Geometrie. Berlin/Heidelberg/New York: Springer Kahn, J., & Kalai, G. (1993). A counterexample to Borsuk’s conjecture. Bulletin of the American Mathematical Society, 29, 60–62 Klein, F. (1926–1949). Vorlesungen über höhere Geometrie, 3ème édition, bearbeitet und herausgegeben von W. Blaschke. Berlin/Heidelberg/New York: Springer, 1926, reprint Chelsea 1949 Lalesco, T. (1952). La géométrie du triangle. Vuibert, reprint Jacques Gabay, 1987 Langevin, R. & Walczak, P. (2008). Holomorphic maps and pencils of circles, American Mathematical Monthly, 115(8), 690–700 Lebesgue, H. (1950). Leçons sur les constructions géométriques. Paris: Gauthier-Villars, reprint Jacques Gabay, 1987 Mandelbrot, B. (1977). Fractals: Form, chance and dimension. New York: Freeman Mandelbrot, B. B. (1982). The fractal geometry of nature. New York: W.H. Freeman and Company Maskit, B. (1988). Kleinean groups. Berlin/Heidelberg/New York: Springer Mathéus, F. (1999). Empilements de cercles et discrétisation quasiconforme: comportement asymptotique des rayons. Discrete & Computational Geometry, 22, 41–61 Nevanlinna, R. (1960). On differentiable mappings. In Analytic functions (pp. 3–9). Princeton, NJ: Princeton University Press Pach, J., & Agarwal, P. (1995). Combinatorial geometry. New York: Wiley Penrose, R. (1978). Geometry of the universe. In L. A. Steen (Ed.), Mathematics today - twelve informal essays. Berlin/Heidelberg/New York: Springer Pommerenke, C. (1992). Boundary behavior of conformal maps. Berlin/Heidelberg/New York: Springer



Raigorodskii, A. (2004). The Borsuk partition problem: The seventieth anniversary. The Mathematical Intelligencer, 26, 4–12 Ratcliffe, J. (1994). Foundations of hyperbolic manifolds. Berlin/Heidelberg/New York: Springer Rodin, B., & Sullivan, D. (1987). The convergence of circle packings to the Riemann mapping. Journal of Differential Geometry, 26, 349–360 Rouché, E. & de Comberousse, C. (1912). Traité de Géométrie (two volumes). Paris: GauthierVillars, reprint Jacques Gabay, 2004 Smith, A. (2000). Infinite regular sequences of hexagons. Experimental Mathematics, 9, 397–406 Stephenson, K. (2003). Circle packing: A mathematical tale. Notices of the American Mathematical Society, 50, 1376–1388 Stephenson, K. (2005). Introduction to circle packing. Cambridge: Cambridge University Press Sullivan, D. (1984). Entropy, Hausdorff measures old and new, and limit sets of geometrically finite groups. Acta Mathematica, 153, 259–278 Tabachnikov, S. (2000). Going in circles: variations on the Money-Coutts theorem, Geometriae Dedicata, 80, 201–209 Vinberg, E. (1993). Volumes of non-Euclidean polyhedra. Russian Mathematical Surveys, 48, 15–48 Yvon Villarceau, A. (1948). Note concernant un troisième système de sections circulaires qu’admet le tore circulaire ordinaire, C. R. Acad. Sciences Paris, 27, 246 Ziegler, D. (1995). Lectures on Polytopes, Graduate Texts in Mathematics, New York: Springer Verlag Zong, C. (1996). Strange phenomena in convex and discrete geometry. Berlin/Heidelberg/New York: Springer

Chapter III

The sphere by itself: can we distribute points on it evenly? III.1. The metric of the sphere and spherical trigonometry As we shall see throughout this chapter, the geometry of the “ordinary sphere” S2 two dimensional in a space of three dimensions harbors many pitfalls. It’s much more subtle than we might think, given the nice roundness and all the symmetries of the object. Its geometry is indeed not made easier at least for certain questions by its being round, compact, and bounded, in contrast to the Euclidean plane. Sect. III.3 will be the most representative in this regard; but, much simpler and more fundamental, we encounter the “impossible” problem of maps of Earth, which we will scarcely mention, except in Sect. III.3; see also 18.1 of [B]. One of the reasons for the difficulties the sphere poses is that its group of isometries is not at all commutative, whereas the Euclidean plane admits a commutative group of translations. But this group of rotations about the origin of E3 the orthogonal group O.3/ is crucial for our lives. Here is the place to quote (Gromov, 1988b): “O.3/ pervades all the essential properties of the physical world. But we remain intellectually blind to this symmetry, even if we encounter it frequently and use it in everyday life, for instance when we experience or engender mechanical movements, such as walking. This is due in part to the non commutativity of O.3/, which is difficult to grasp.” Since spherical geometry is not much treated in the various curricula, we permit ourselves at the outset some very elementary recollections, which are treated in detail in Chap. 18 of [B]. Typically we will deal with the sphere S2 of radius 1 centered at the origin of the Euclidean space E3 , but all spherical geometries are the same within a change of scale. The interesting distance between two points p and q of S2 isn’t their distance in E3 but that on S2 , i.e. the length of a shortest path on S2 that joins p to q. We are dealing here with the metric that is called intrinsic or internal, as opposed to the induced metric that is the distance between these points in E3 and evidently without much interest for Earth’s inhabitants: digging tunnels is expensive. It’s classical that this path is unique and that is the arc of a great circle that joins p to q, which can be proved rigorously by a symmetry argument, or here with the fundamental formula (III.1.1) below. Note that in saying the arc of a great circle we necessarily exclude antipodes, for which all the great circles starting from the one arrive at the other after a distance . We let d.p; q/ denote this distance, which varies from 0 to . It is given in terms of the scalar product by the formula: M. Berger, Geometry Revealed, DOI 10.1007/978-3-540-70997-8_3, c Springer-Verlag Berlin Heidelberg 2010




cos.d.p; q// D p q. But it may also be regarded as the angle through which we see the points p and q from the origin O. We can say, in an entirely equivalent manner, that we are dealing here with seeing the sphere as a set of half-lines emanating from the origin, which is crucial in descriptive astronomy: the sphere is the celestial vault and there is an essential need of the formulas that will follow. As astronomy is very old, these formulas date from several centuries ago, but we will read them on S2 , not as would astronomers.

Fig. III.1.1.

We remain on S2 . It is clear that when we have three points p, q, r of S2 we may speak in analogy to Euclidean geometry of the spherical triangle fp; q; rg. Now if we know the distances of d.p; q/ and d.p; r/, as well as the angle between the sides that emanate from p, we sense in advance that the distance of d.q; r/ is determined, thus theoretically calculable. The formula that provides this distance is called the first fundamental formula of spherical trigonometry. Let us call a, b, c the lengths (distances) of the sides of fp; q; rg, and ˛, ˇ, its angles (between 0 and ). Then we always have: .III:1:1/

cos a D cos b cos c C sin b sin c cos ˛:

The proof is a direct application of the definitions and of the scalar product, whose neglect in school mathematics explains why this formula isn’t better known to students. It was prohibited in the French lycées until about 1950, which explains the incredible contortions in treating the inequality of days and nights, which was part of the required program. From it we can deduce cautioning that the value of the sine does not determine the angle unambiguously all the formulary given in detail in [B]. The practical importance is considerable; this or analogous formulas provide the solution to the following problem relating to three directions (or three half-lines) in space. Such a configuration is associated, as is the corresponding spherical triangle, with six numbers: the three angles between each pair of lines, and also the angles between the planes determined by pairs of these lines. This play with formulas allows us to calculate all these elements as a function of only three of them, but with some precaution. Think about the “dubious” case of equality of triangles. Readers



will be able to investigate some practical applications: to astronomy, computation of the length of days in the course of the year, positions of artificial satellites, geodesy, petroleum research, GPS, etc.

Fig. III.1.2. Construction of the spherical triangle dual to fp; q; rg. In the order: construction of p 0 , construction of q 0 and r 0 in the plane orthogonal to the ray Op ( p , q , r represent the projections onto the plane of points p , q , r ), the triangle fp; q; rg and the dual triangle. II [B] Géométrie. Nathan (1977, c Nathan Édition 1990) réimp. Cassini (2009)

It is important to mention the duality that exists for spherical triangles, that simplifies the derivation of the formulas. This duality associates with the triangle fp; q; rg, by taking appropriate perpendiculars, a (the) dual triangle fp 0 ; q 0 ; r 0 g whose elements (denoted using apostrophes) are simply the angles complementary to , but exchanging the roles of sides and angles: a C ˛ 0 D b C ˇ 0 D c C 0 D a0 C ˛ D b 0 C ˇ D c 0 C D : The triangle dual to fp 0 ; q 0 ; r 0 g is the original triangle fp; q; rg. This duality is the spherical version of that given in Sect. I.7 for the real projective plane P . In the language of half-lines, i.e. of trihedra, it is expressed thus: with the trihedron fD; E; Fg we associate the trihedron fD0 ; E0 ; F0 g, called supplementary, defined by the conditions: F0 is perpendicular to D and E and is located on the same side of the plane defined by D and E as F, and likewise for the two other half-lines. An historical note on “school” mathematics, interesting but doubly depressing: in French instruction the duality of supplementary trihedra was included in the program for the terminal classes at the lycées. Until 1950 the proof of duality (involutivity) was done geometrically, since use of the scalar product was strictly forbidden. Now with the notion of “same side” involutivity was rather obscure to prove, even in the very good work (Hadamard, 1911). It took several pages, with complicated reasoning about the poles associated with a great circle, etc. Such reasoning is always subtle; we will see this clearly in Sect. III.4 for the problem of “the thirteenth sphere”. Now this involutivity is trivial with the use of the scalar product: if fx; y; zg and fx 0 ; y 0 ; z 0 g are nonzero vectors representing our trihedra of our spherical triangles, the six conditions defining fx 0 ; y 0 ; z 0 g are:



x 0 y D x 0 z D 0; y 0 x D y 0 z D 0; z 0 x D z 0 y D 0 x 0 x > 0; y 0 y > 0; z 0 z > 0: The set in its totality is clearly symmetric. The author learned this algebraic formulation from his classmate André Aragnol when he was preparing for his “agrégation” (French competition for specially privileged teaching positions at the secondary level). We ask ourselves why so many schoolchildren and their teachers need suffer because of a perverse sort of love for pure geometry. We have seen this phenomenon above regarding the fundamental formula, we will encounter it again regarding the duality with respect to a circle in Sect. IV.2. Avoiding use of coordinates and still worse the scalar product, is an obsession that has long haunted “pure” geometers. Now the scalar product figured in Grassmann’s work from 1840, but in writing that was far too prophetic; it wasn’t really popularized until Gibbs in 1860. But resistance to Gibbs raged from the disciples of Hamilton, who wanted to keep the scalar product exclusively for quaternions (thus in dimension 4) and prepared the way for a long and bitter correspondence, which is today hilarious: see Gibbs (1961). The second point, sadder perhaps, is that this discussion is hardly relevant today, since geometry, especially that of space, has all but completely disappeared from the mathematics curriculum of schools and universities this while space visualization is becoming more and more necessary for practical applications such as three dimensional animation and robotics. So, with the fundamental formula and the duality, plus some calculations, astronomers have all the necessary formulas; see the formulary 18.6.13 of [B]. They were much more developed until the introduction of computers; the calculations were done by hand using logarithmic tables. So that they could be done as quickly as possible, it was necessary to transform as much as possible all those operations bearing on trigonometric properties, sums in particular, into products or quotients. The sphere possesses a canonical measure that is unique within a scalar factor if we require invariance under O.3/; see Sect. III.6. This is what physicists call a solid angle, in visualizing half-lines emanating from the origin. The total area of the sphere equals 4. The spectacular theorem that goes back to Thomas Harriott in 1603 but that was published only later by Albert Girard (1595–1632) states: (III.1.2) The area of any spherical triangle with angles ˛, ˇ, is equal to ˛ C ˇ C . We can compare this with formula (II.4.1) from hyperbolic geometry. If the three angles are commensurable with (are rational multiples of ), then the surface area of such a triangle will also be commensurable with . In Sect. III.7 we will see that the analogous question in higher dimensions remains open. The unpublished proof of Harriott amounts to looking at the classic figure below. By calculation alone things are much more difficult and do not give a hint of the final simplicity of the formula. To make use of the figure, it suffices to know that the area of a lune on a



sphere formed by planes through a diameter of angle ˛ equals 2˛, which is evident from the proportion to the total area 4 of the sphere (calculated in Sect. VII.6.B).

Fig. III.1.3. Harriott (1560–1627), Girard (1595–1632)

Fig. III.1.4. A spherical lune with angle ˛

Here we briefly mention the isoperimetric problem on the sphere (we will encounter isoperimetric problems extensively in this book, in particular those on convex sets in Chap. VII). First, the isoperimetric inequality is true for triangles in S2 : among all the triangles of S2 the maximal area (measure of the surface) is attained for the equilateral triangles and for them only a direct consequence of the formula that gives this area as a function of the lengths a, b, c of its three sides: p tan D sin p sin.p a/ sin.p b/ sin.p c/ 4 . This formula for the area may be compared to Heron’s, given where p D aCbCc 2 in Sect. VIII.3. We will see a comparable formula in Sect. III.5 for the sphere of dimension 3. At the other extreme of the subsets of S2 , we have the general domains for which an area and a perimeter can be defined. As in the Euclidean plane (see Sect. V.11) the general isoperimetric inequality is true on the sphere: among the



domains having a given perimeter, it is the disks, and they alone, that have maximum surface area. The disks of S2 are the closed balls for the canonical metric, the B.p I r/ D fq W d.p; q/ 6 rg of center p and radius r. In the literature the term spherical cap is frequently used. We encounter the canonical measure of the sphere in Sect. III.6.

Fig. III.1.5. The spherical caps attain, for a given area, the minimum perimeter

Although a little outside of our general theme because of the setting in more abstract geometries, we mention the problem of finding geometries for which we know how to calculate the distance between two points, starting with a universal formula in which nothing appears except for the distances of the two points to another point (fixed, so to speak) and the angle subtended by the two points from the fixed point. These geometries are all known, and in fact they are nothing other than Euclidean geometry (where the formula is a2 D b 2 C c 2 2bc cos ˛), spherical geometry and hyperbolic geometry (see Sect. II.XYZ). In fact, if we want notions of distance and angle in the context of Riemannian geometry, then the spaces sought are those for which the local curvature is constant. We can then show that there are only the three geometries mentioned; see any work on Riemannian geometry, for example Berger (2003). For more general spaces we don’t know much about the problem; one of the difficulties is knowing which are the most general metric spaces for which we can define a reasonable notion of angle. Another problem, concerning geometric optics, was posed in 1927 by Blaschke: do there exist other metric geometries on the topological sphere for which each point has a well determined antipodal point and on which each point can be joined to its antipode by a shortest path, starting in any direction from the initial point? It’s the problem of constructing a perfect optical instrument that is completely stigmatic. In practice the corresponding instrument is realized in certain animals: it’s Maxwell’s fish eye. In fact only the canonical metric of the sphere is everywhere stigmatic, but this has only been known since 1963 and is due to Leon Green; see



Deschamps (1982) for the systematic construction of partially stigmatic objects, as well as for references.

Fig. III.1.6. Only the sphere is a completely perfect lens. But there exist other objects that are stigmatic on just a portion of their surface. Deschamps (1982) Gauthier-Villars. c Elsevier

III.2. The Möbius group: applications In Sect. II.5 we introduced a group on the ordinary sphere S2 that is larger than the group of isometries (which has three parameters), i.e. the Möbius group Möb.S2 /, which has six parameters. It’s the group of its conformal transformations, generated by restricting to the sphere those inversions of space that preserve the sphere. We can study the geometric nature of all these transformations, interesting for being more general than isometries, but all the while preserving angles. The conformal type mappings most removed from isometries are those that lift the hom*otheties of R2 centered at the origin onto S2 by stereographic projection. We see what they do: they preserve the north and south poles; they preserve meridians while moving individual points (to a variable extent that depends on the latitude) toward the north pole if the hom*othety ratio is greater than 1. For a very large hom*othety ratio, points are moved radically toward the north pole. If from the outset the sphere is considered as a distribution of masses, there will be more and more mass shifted toward the north pole as the hom*othety ratio increases. Matters are still more graphic if we (think dynamically! ) iterate such a transformation f . In the limit all the mass is going to be concentrated at the north pole; all other points, the south pole in particular, are left without mass. By composing such a transformation with a rotation about the north-south axis, we obtain what are called loxodromic transformations. For example, we can simultaneously compose the group of rotations with the oneparameter group of hom*otheties of variable ratio and look at the trajectory of a point: this will be a curve, one that cuts all the meridians at a constant angle. These are then, for navigators, the curves that describe the movement of a boat for which the



heading is constant; they are called loxodromies; and on marine maps they appear as lines. These curves are liftings of logarithmic spirals by stereographic projection, written in polar coordinates .; / : . / D e k . Indeed, stereograhic projection preserves angles and the logarithmic spirals are exactly the curves that make a constant angle with the radial lines.

Fig. III.2.1. Action of a hom*othety of the plane lifted stereographically onto the sphere

For interested readers, we mention two applications of such transformations to geometry. The first concerns rather general metrics g, called Riemannian, on the projective plane P . Here two constants are associated with g: the first is the total measure, the area Area.g/; the second is its systole Sys.g/, defined as the shortest length, relative to g, of the curves of P that are topologically equivalent to projective lines (in fact they are curves that are not contractable to a point, for P is not simply connected). Thanks to the Möbius group, we can show that we have an inequality of global isoperimetric type: Area.g/ > 2 Sys2 .g/ with equality only for the elliptic metric, thus canonical. The proof uses the fact that the Möbius group is very large. For such isosystolic inequalities, and for references, see e.g. Berger (1998) or Berger (2003). Let us return to the sphere S2 and endow it too with a Riemannian metric g. The theorem on conformal representation encountered at the end of Sect. II.8 implies here that we can always assume that g D f d , where f is a numerical function on the sphere (we say that g is consistent with the canonical metric d ). We can thus regard the pair .S2 ; g D f d / as a variable mass distribution on the sphere, the density being f . If we consider this object as vibrating, it will have frequencies: it will be musical. The lowest frequency is the one of reference, call it ƒ.g/. Then we have an isomusical inequality (characteristic of the standard sphere) between this frequency ƒ.g/ and the total area Area.g/, namely ƒ.g/ Area.g/ 6 8 (equality holding



only for the standard metric). The proof consists of showing that the Möbius group allows conformal transformation of the density f such that the center of gravity in the space of this massive sphere is the origin O of the space. Intuitively, this is how we proceed: we have seen above that we can move all the mass toward the north pole, but likewise toward any point of the sphere. This shows, by a covering type argument, that the initial center of gravity, subjected to all these transformations, must surely “travel through the origin” at least one time. The proof is completed with a classical minimal property for the function that gives the vibration of least frequency. For references and more, see Berger (1998) or Berger (2003). III.3. Mission impossible: to uniformly distribute points on the sphere S2 : ozone, electrons, enemy dictators, golf balls, virology, physics of condensed matter The problem that arises in very many practical applications is to equally place points, more or less large in number, on a sphere. We can always assume we are dealing with S2 . It is first a matter of esthetics. A bit more serious is the case of golf balls, to which we will return; but in fact the problem is much more important. We can think about the heart of a spherical nuclear reactor: how should we distribute the combustible bars?. In architecture, it has to do with constructing a sphere (or almost a whole sphere) as a geodesic dome, i.e. with finding an appropriate triangulation; an example is the Géode in the Villette Park in Paris. But it also has to do, in physics, with studying the distribution of electrons in atoms and with studying various molecular structures. It arises in molecular chemistry, in solid state physics, in virology and the physics of condensed matter. Finally there is the document scanner and medical imaging. Readers may reproach us for spending too much time on a single problem. Other than the motivations above, we mention that the problem relates to the seventh in the list of 18 mathematical problems that Smale has proposed for the twenty-first Century (Smale, 1998). Smale in fact encountered this problem in a totally different context, different at least in appearance: that of discovering good algorithms for finding the roots of a polynomial. The connection between the sphere S2 and such algorithms comes from this: a good quantity for considering the complexity of these algorithms is a pair of complex numbers, or rather their ratio, or still more precisely such pairs modulo their product with an arbitrary complex number. But we know that this is nothing other than the complex projective line; in Sect. I.XYZ we correctly identified it with the sphere S2 . Uniform distribution of points on the sphere is not just a study of configurations to be studied or discovered. More important, we will encounter this problem in each discipline where distribution of points on a sphere is involved: information receptors at the given points, numerous enough for the information to be good, but not too numerous for economic reasons. A typical case is that of ozone: if we want to know the total quantity of ozone in the atmosphere, it is necessary to measure its density at different points and take the mean of these values. It is clear that these



points must be well distributed, so as not to bias this or that portion and so as “not to neglect any”. The points considered may be various receptors, or various analyzers: sounding balloons, but also satellites.

Fig. III.3.1. The carbon molecule C60 : a fullerene discovered rather recently and named in honor of Buckminster Fuller, architect and inventor of the geodesic dome. Spherical fullerenes are also called “buckyballs”

Fig. III.3.2. Hexagonal radiolarian: Aulonia hexagona. The skeleton of this animal can’t consist only of hexagons (we observe pentagons)

For the mathematician, the problem presents itself when it is required to approximate the integral of a function on the sphere. We know that, for ordinary integrals, the interval of R on which the function is to be integrated gets partitioned, ideally into N equal intervals; then N is made to tend toward infinity. These matters are also rather easy on the circle, which can easily be partitioned into equal arcs of whatever number, but also for integrals over the plane and over



Fig. III.3.3. The Géode in the Villette Park

Fig. III.3.4.

space: we take points that form a lattice (see Chap. IX), etc. For the plane, the best functioning lattice is “obviously” regular hexagonal, an old friend encountered in Sect. II.8. On the sphere we would like if the points are denoted by xi (i D P 1; : : : ; N) and for any reasonable function f (say continuous) that the sum N1 i f .xi / R approximate well the total mass S2 f .x/ dx of the function, or rather its mean



R 1 value 4 S2 f .x/ dx, where dx denotes the canonical measure on the sphere. What is asked of the approximation evidently varies according to the problem. We might suspect that there could be several sorts of criteria. We are going to review several of them, but the conclusion will be rather negative: it is very difficult to distribute points well on the sphere. It is ultimately impossible, except for some small values of N, that don’t suffice for computing integrals; or on the contrary as has just recently been discovered it seems that such partitions are possible, but require more than 10 000 points. Before studying the different approaches to this problem and concluding that it is finally and in a certain sense impossible, some heuristic words for getting a sense of its difficulty. A first observation is that the sphere, even a tiny piece of it, can never be isometric to a piece of the Euclidean plane, i.e. that a perfect map does not exist. It is possible for a map to preserve angles, which is almost always done for reasons of navigational bearings, but not distances; there will always be some distortion. Otherwise we would only need to “place” a plane (or a part of the plane) on the sphere and we could mark out regular hexagonal lattices, with increasingly smaller scale. We might say, as has already been noted, that an essential reason for this impossibility regarding distances is the non commutativity of the group of isometries of the sphere, whereas the affine spaces possess the commutative translation group. Another observation: we might think of using points that are the vertices of regular polyhedra. Now, contrary to the case of the circle where there are regular polygons with arbitrarily many sides, there do not exist in space regular polyhedra with a large number of vertices, not even semi-regular polyhedra (which would suffice); see Sect. VIII.4. Still worse, we will see that the cube and the dodecahedron do not yield optimal lattices. We barely touch on the related problem of scanning a solid. Theoretically it has to do with reconstructing the density of a ball starting with all plane projections, in other words with classic x-rays taken in all directions. In practice, only a finite (but perhaps rather large) number of projections are used and there is a need for effective algorithms for distributing these “evenly”; see Helgason (1980). Historically, we can’t but be surprised by the fact that mathematical work on the subject is so incredibly recent. But above all, if we except the “foundations” by Habicht and van der Waerden (1951), the first edition of Fejes Tóth (1972) in 1953, and several others, e.g. Stolarski (1973) and Delsarte, Goethals and Seidel (1977), the subject didn’t really begin to be studied until 1990. We can convince ourselves of this with the bibliography of basic references in Conway and Sloane (1993), to which it is necessary to add the brief and informal synthesis (Saff and Kuijlaars, 1997) that we largely follow, but above all the recent (Hardin and Saff, 2004) which is, with its technical references, the new global reference text. This is corroborated by the nonexistence of books treating numerical integration on the sphere, whereas the works treating numerical integration on the line or the plane are legion. We also cite the book (Melissen, 1997), very detailed for the “elementary” problems, for example for configurations with few points, and which has a lot of pictures. We are now going to study the different approaches to the problem. See also how to use



the theorem of Dvoretsky (cf. Sect. VII.12) for evenly distributing points on spheres of arbitrary dimension in Wagner (1993), or the end of 7.3 of Giannopoulos and Milman (2001). The Tammes problem. The formulation that is historically first and the simplest for geometry consists of being given an integer N and deciding that “the best configuration” for these N points is the one furnished by “the” solution of the problem of the enemy dictators, also known as the Tammes problem. It is studied in the synthesis (Saff and Kuijlaars, 1997), but the first treatment historically is in a set of works of van der Waerden and his collaborators; all the references are in van der Waerden (1952). The case of small values of N is treated in great detail in Melissen (1997), whereas Croft, Falconer and Guy (1991) is no longer sufficiently up to date. The “best partition”, by whatever criterion, is of course understood modulo an isometry of the sphere. Tammes was a Dutch botanist who in 1930 published an article on the partition of tiny holes on the surface of pollen grains. The criterion for the desired configuration fxi g (i D 1; : : : ; N) is to require that each of the points be as far as possible from the others, i.e. to maximize the minimum of the distances d.xi ; xj / (i 6D j ). It amounts to the same thing as seeking the largest possible radius r for N disjoint disks of radius r placed on the sphere. The points sought should be as far as possible from one another in order to maximize their territory, whence the language of the dictators. The territory of the dictator p will of course be a Dirichlet (or Voronoi) domain about p, a notion we will encounter in Chaps. IX and X, i.e. the set of points of the sphere that are closer to p than to any of the other dictators (this defines the realm of the dictator). For reasons of compactness, there always exist such optimal configurations: consider the product of the sphere with itself N times and the (continuous) function equal to the smallest of these mutual distances. The solution is furnished by the point (or points) of the product space that gives the function its maximum value. We will say at first what happens for small values of N, this to give a feeling for the difficulty of the problem, also because it is very geometric and a source of pitfalls! Two points get placed at antipodes, three are placed on an equator, the vertices of an equilateral triangle; four points are vertices of a regular tetrahedron, the proof being left to readers. We thus confirm that there is essentially one optimal solution for the values 2, 3, 4, and that these are configurations that have the greatest symmetry possible, which conforms well to the intuition mentioned above. It can be shown next that, for 6 points, there is only one configuration possible, the vertices of a regular octahedron. Up until now our intuition has worked well. Having gotten started, it is natural to think that we can do better for 5 points than for 6. We remove a point from the regular octahedron and thus have room to move the 5 remaining points, distancing them from each other. The answer is no, you can displace them in such a way as to find a different configuration that gives the same



Fig. III.3.5. The optimal configurations for 2, 3, 4, 5 and 6 points. For 5 points there is no uniqueness, and the configurations are not very aesthetic!

value of the optimal distance as do 6 points, which equals 90ı D =2 incidentally (not less!). Readers can search for this second configuration. We pass to the case of 8 and distribute the 8 points on the sphere at the vertices of a cube. Can we do better? Yes, we turn one of the faces, which preserves the distances between vertices in two faces, the one that we turn and the one opposite, but strictly increases all the other distances. Whence the idea, if the turned face and its opposite are horizontal and we lower the top face and raise the other, while keeping it a square, then we increase the sides of these squares; and if we move them just enough the distances between points of the resulting (non-regular) octahedron will be equal, and greater than they were initially (equaling 70ı 320 , by way). Finally, Schütte and van der Waerden showed that for 8 points the optimal configuration can be obtained precisely by the above method; it consists of what is called a square antiprism i.e. two equal squares with the same axis, which we take to be vertical, but one of the squares is turned by 45ı in relation to the other. Subsequently we adjust their distance apart so that the sides joining the vertices of these opposing squares have the same length as the sides of the squares themselves. Such a figure admits symmetries (find all of them!), but fewer than the cube. Thus our intuition wasn’t well founded; the solution of the Tammes problem for the values 4, 6, 8, 12, 20, isn’t always given by regular polyhedra. It is for 4 and 6, but not for 8. Readers familiar with the notion of symmetry breaking, very important in physics, know that it isn’t always configurations having maximum symmetry that provide optimal solutions. Nature loves symmetry only tepidly! The simplest known symmetry breaking concerns the square: we seek to connect its four vertices, e.g. four villages, by a set of paths for which the total length is as small as possible. The solution is given in the figure below. It does not have all the symmetries of the square and so there are two solutions.



c Pour la Sciénce, Éditions Belin Fig. III.3.6. Square antiprism. Berger (1992)

Fig. III.3.7. A very classical symmetry breaking

Alain Connes likes to give an example of symmetry breaking where at a circular restaurant table the server puts the bread, or anything else, symmetrically between the guests, for whom there are no firm left-right conventions. The first person who takes his bread breaks the symmetry! For 7 points of the sphere the configuration isn’t very pretty, which isn’t surprising. We will encounter symmetry breaking again in Sect. VII.11.C. In physics, for the states of a system admitting symmetries, there was first Curie’s principle that affirmed that each solution admitted all these symmetries. A false principle! What is true is that it’s the set of solutions that admits these symmetries. A physics text on symmetry breaking is Brézin (2001). Where are we at present? To the knowledge of the author, the only numbers for which the optimal value are known include all N through 12 but only one additional value, i.e. N D 24. For N D 12 we get the regular icosahedron, the unique solution, but we also see the phenomenon already encountered for 5 and 6. I.e. for N D 11 we can’t do better than the regular icosahedron minus a point, and this is not the only configuration! For 20, it is known that we can do much better than with the regular dodecahedron, i.e. 47ı 260 as against only 41ı 490 for the regular dodecahedron. Readers will easily see that an improvement is possible; proceed as with the cube. For N D 16 some alloys are found in nature where Friauf’s configuration is encountered, and we might think that it seeks to solve the Tammes problem, but it seems that no one knows if it is the best possible. For this subject,



Fig. III.3.8. Drawings by Geoffroy Wagon

see Nature, vol. 352, July 1991. In Nature, vol. 351, May 1991, there is still better: for the analogous (dual) problem of covering the sphere most economically with a number of identical spherical caps of the same radius (that of the sphere), for the cases of 16 and 20 caps, objects are found in nature (cages of clathrine clathrine is a protein) which have done better than the conjectures of mathematicians. In the case 16, the dual polyhedron, with 16 vertices, is exactly Friauf’s configuration. We should also say that we know that the semi-regular polyhedra, like the soccer ball which furnishes a configuration with 60 points, don’t in general yield the best value. The case N D 24 is thus special. Moreover, the only configuration giving the optimal angle value (43ı 410 ) is the one that is called the snub cube. This semiregular polyhedron is a most interesting object, astonishingly little known. Together with the snub dodecahedron it is the only semi-regular polyhedron that doesn’t admit an orientation-changing symmetry, thus in particular a mirror symmetry, not even simply a center of symmetry! Returning to the general problem, the proof for 24



Fig. III.3.9. (a) Cardboard models for the covering of a sphere by disks of the same radius: a and c are the optimal solutions conjectured by mathematicians for the cases of 16 and 20 discs; b and d are the actual solutions observed by Tarnai in the study of cages formed by the molecule of clathrine. (b) Wire polyhedra with equal sides corresponding to modules from A above: a, the “heptagonal drum”, formed from two special heptagons surrounded by two rings of seven pentagons; b, polyhedral cage formed from 12 pentagons and 4 hexagons placed tetrahedrally; c, the “hexagonal barrel”, polyhedral cage formed from two rings of 6 pentagons, a ring of 6 hexagons and 2 special hexagons; d, the “tennis ball”, polyhedral cage formed from two concave bands of 4 hexagons, separated by a belt of 12 pentagons placed in space like the seam of c Nature Publishing Group, a division of Macmillan a tennis ball. Stewart (1991). Publishers Limited

c Columbia Fig. III.3.10. The snub cube and the soccer ball. Holden (1971) University Press

is very long and is not conceptual, but of course makes extensive and subtle use of spherical trigonometry; see Robinson (1961). It’s the moment for reassuring or perhaps of disquieting computer enthusiasts. It’s necessary to realize that devising a computer program for solving the Tammes problem (obtaining the best bound and configuration) is extremely difficult. We will return to this later and say here only that it’s an optimization problem having to do with a number of variables equal to 3N. Many have devised programs, by different methods, but all those programs require enormous computational resources for which the value of N is important. In all these numerical studies, generally treating other criteria in addition to the distribution of points on the sphere, there is a major difficulty due to the existence of a number of local minima that is exponential in N.



Fig. III.3.11. How to obtain a bound for the d.N/ of the Tammes problem

As in numerous other problems that we encounter, where exact solutions either don’t exist or are only theoretical, we must be content with asymptotic estimates. It’s definitely not easy either to obtain these estimates or to find configurations that satisfy them. Here in the Tammes problem the task is to study the optimal valued.N/, i.e. the upper bound of the minima inffd.xi ; xj / W .i 6D j /g, ranging here over all possible configurations fxi g (i D 1; : : : ; N) of N points on the sphere. For this is exactly the requirement that each of the points be as far as possible from the others. Even not knowing d.N/ exactly, we desire a best possible approximation. This was accomplished in Habicht and van der Waerden (1951), where we find the double estimate: 8 1=2 1 8 1=2 1 1 6 d.N/ 6 p p p Cp p N N 3 3 N3 which is thus very satisfying, apart from the value of the constant C, which is not well known. In fact, the circ*mstance that the inequality is double with the same factor p1N shows that things necessarily have this precise asymptotic order. The inequality on the right is not difficult to obtain; at best the spherical caps of an optimal configuration touch each other three at a time (the boundary circles are tangent to each other). Three such centers form the vertices of a spherical triangle with sides of length equal to the optimal distance d.N/ that is sought. Thus the triangle is known on the sphere and we can calculate its surface area as an explicit function of d.N/ by the formulas of spherical trigonometry. At best all these triangles cover the sphere without holes; they are N in number and the total surface area of the sphere equals 4. All calculations done, we find that d is bounded above by the value given by the equation cos d.N/ D cos e=.1 cos e/ where e D N=3.N 2/: We have derived the bound on the right. To get the inequality on the left, it is necessary to explicitly construct configurations with a rather large d.N/. This geometry problem has been encountered by



makers of golf balls, which generally contain hundreds of dimples. For those who don’t know, it is interesting to recall that originally these balls were smooth and that one day a player, lacking new balls, recognized that the very worn balls went farther than the newer smooth balls. We know that the dimples (originally just scratches) have the effect of reducing the drag, i.e. the force caused by turbulence behind the ball; the dimples allow diminution of separation from the boundary layer.

Fig. III.3.12. How air flows around a smooth golf ball and around a rough ball. Walker c J. Walker (1979)

The patent filed by the Slazenger, for which the principle coincides with the construction of Schütte and van der Waerden, was natural enough. We can’t impose a regular hexagonal lattice on the sphere, as we have seen; but we can do something similar on the interior of a spherical equilateral triangle on the sphere: we take its center which gives us four triangles and we iterate the operation as many times as we wish. Note that we start with an equilateral triangle, but the triangles that make up the successive subdivisions are not equilateral on the sphere. We can also project pieces of a planar hexagonal lattice onto the sphere. It remains to find a nice partition of the sphere into equilateral triangles, as numerous as possible: we see below that 12 is the maximum, but as we know the regular icosahedron, we proceed thus: we place twelve points on the ball at the vertices of a regular icosahedron, which thus partitions the sphere into twelve equilateral spherical triangles. Now the faces of the icosahedron are planar equilateral triangles, so we can fill them with as many triangles as we wish in hexagonal packing as explained above; these are the integers of the form n.n C 1/=2, there being in total N D 10n.n C 1/=2 points. Proceeding as we have seems like the most natural way to distribute a significant number of points on the sphere. But now a brief commercial digression. If you obtain a certain model of Slazenger ball and mark the points with a felt-tip pen where a dimple touches only 5 others, you will see that they are located at the vertices of a regular icosahedron. Now we don’t advise you to make and market such balls Slazenger has patented that; but if this game interests you, obtain some other models



Fig. III.3.13. How to obtain the left hand bound for the d.N/ of the Tammes problem, c Pour la Sciénce, Éditions also how to make golf balls à la Slazenger. Berger (1992) Belin

Fig. III.3.14.

of balls. Some of them will be of a completely different type, about which we won’t speak. But you will find balls where the 12 exceptional points are drawn so that at least one whose position has been shifted, and thus the Slazenger patent has been “gotten ’round”! Let us see why 12 is an inevitable minimum. The reason for it is Euler’s formula for convex polyhedra (see Sect. VIII.4), which says that saCf D 2, where s, a, f are the respective numbers of vertices, edges, faces of the polyhedron considered. But if we regard them closely and accept that some vertices belong to but 5 triangles (each spherical cap touches only 5 others with some exceptions, hopefully small in number, and everywhere else there will be six), then Euler’s formula shows that this works with exactly 12 exceptions. In fact, since each face is a triangle and each edge determines two faces, we have f D 2a=3. Let s5 and s6 be the respective number of vertices where 5 or 6 triangles meet. Then, since each edge produces two vertices, we must have 2a D 5s5 C 6s6 . All together we have (the s6 disappears, thus we can place as many as we wish, as we have seen!) s65 D 2, where s5 D 12. Readers should investigate the exceptional pentagons in Fig. III.3.2 showing radiolaria and Fig. III.3.3 showing the Villette Géode. Typically in Fig. III.3.15 are shown both the points (the triangulation is apparent) and the tiling of the sphere; the dual obtained from the Dirichlet–Voronoi domains, composed of hexagons and 12 pentagons. We can apply Euler’s formula to it with, of course, the same conclusion.



Let us return to one of our motivations for regular distribution of points on the sphere, i.e. for knowing how to approximate the integral or the mean value of a function defined on the sphere, while knowing its value only at certain points, more and more numerous if we want to approach better and better. The method of van der Waerden –Slazenger isn’t good for integrating arbitrary functions on the sphere, not only because the twelve points distort things close to them, but also because the triangles that partition the faces of the icosahedron have more or less unequal areas. It seems thus that we are at an impasse. We are going to see where we are at the present moment, which we will do a bit in caricature for it is difficult to get our bearings in the sea of studies that regularly appear on the subject, given its practical importance returning once more to the references already given. The energy viewpoint of physicists. The physicist who reads this, or someone whose thinking has been partly influenced by physics, will likely be exasperated. For such a person it is so simple, so trivial, to find a good distribution for N points by a physical principle: take N points fxi g and put them anywhere on the sphere, thinking of them as equally and positively charged particles; they repel each other mutually according to Coulomb’s law (i.e. in r12 , where r is the distance between them) and thus they are certainly going to uniformly distribute themselves ultimately, i.e. P 1 they will minimize the total energy i6Dj jxi x . We could also consider small jj magnets, likewise signed with the same magnetic force.We consider here the Euclidean distances jxi xj j, but since they are intrinsically connected to distances on the sphere, it amounts practically to the same thing. Finally, your physicist will say, you can make your calculations on a computer, for you have only to minimize an explicit function of a set of points, which is much simpler than maximizing inffd.xi ; xj / W .i 6D j /g as above in the Tammes problem. But other energies P 1 can also be introduced, e.g. all those of the form i6Dj jxi x s , where s is a posijj tive real number. We remark that, as s tends toward infinity, with N fixed, we tend toward the Tammes problem, since the energy is dominatedQ by the terms of small distances. Finally, we might think of maximizing the product i6Dj jxi xj j of mutual P 1 . This distances, which amounts to minimizing the energy given by i6Dj log jxi x jj should be mentioned since this question was introduced in Shub and Smale (1993), where the problem is to study the complexity of the algorithmic solution of polynomial equations!; see the end of the present section. What will we then obtain as a consequence of these energies? We recall first the double algorithmic difficulty of the problem on computers. In the first place, the number of variables is large, i.e. 3N if N is the number of implanted points. More seriously, if we are to have any prospects, is the fact that the energy functionals considered have all the messiness of local minima. These minima are moreover very near each other; we must find the true minimum from among them. A simple example: if you take 12 points at random on the sphere and apply a dynamic program



that follows a descending energy path, the limiting configuration obtained won’t always be a regular icosahedron. This having been said, recentlyP we have seen emerge results of computer calcu1 lations for the minima of energies i6Dj jxi x s for various s and for numbers of jj points of the order of several hundreds at most. There are several crucial remarks to make in view of the configurations obtained, like that below in Fig. III.3.15 for 122 electrons. First, astonishingly, they practically all resemble each other, whatever the exponent s in the definition of the energy: they are of Schütte-van der Waerden Slazenger type, i.e. partitioning the sphere into hexagons, with twelve exceptional pentagons. Secondly, none of these configurations is good for integrating arbitrary functions on the sphere. This is not surprising; see the remarks already made previously on the configurations of Schütte-van der Waerden-Slazenger type. We will subsequently see that it is possible to be very effective, but at the price of only integrating polynomials whose degree is bounded by a given constant or of working with much larger numbers of points.

Fig. III.3.15. 122 electrons in equilibrium and their Dirichlet–Voronoi domains. Saff, c Springer and E. Saff Kuijlaars (1997)

There is also the problem of estimating the minimum energy obtained, at least in asymptotic fashion, as a function of the number of points considered. We will return to this later, at the very end of the present section. Integrating only polynomials. The impossible mission is realizable in the case of polynomials of degree with a fixed bound, presently of low degree. Since (Delsarte et al., 1977) we know how to construct sets of points fxi g that give the exact value



R 1 f .xi / D 4 S2 f .x/ dx for no matter which polynomial f (polynomial in three variables of R3 ) of degree less than a given integer. We find the present state of affairs in Saff and Kuijlaars (1997). We know, for example, how to construct 94 points which work for degrees up to 13 this requires large-scale computer calculations. For higher degrees, other than explicit construction, we ask for the minimum number of points necessary for a degree bounded by k; it is conjectured that it is of order k 2 . 1 N



Other attempts. We find in (Saff and Kuijlaars, 1997) a couple more ideas, one by partitioning the sphere into domains of the same surface area, the other with spirals; see Fig. III.3.16 below. The idea with spirals is very natural for some: place the points, from the north pole to the south pole, deftly arranged on a slicing of the sphere into pieces obtained by cutting across equally spaced planes orthogonal to the north–south axis. This yields an explicit formula and some sketches like those below.

Fig. III.3.16. 400 parts of equal area; 700 points on a generalized spiral. Saff, Kuijlaars c Springer and E. Saff (1997)



Uniformly distributed and optimal (?) infinite subdivisions. For those who don’t fear the infinite, a good subdivision can be defined for an infinite sequence of points fxi giD1;2;::: by means of an integral. We say that the sequence is asymptotically uniformly distributed if for each function f on the sphere we have: Z ˇ ˇ1 X 1 ˇ ˇ lim ˇ f .xi / f .x/ dx ˇ D 0: N!1 N 4 S2 iD1;:::;N

On the other hand, the geometer will prefer to count points in the spherical caps A, that being more visual. We have a good subdivision if for any cap A, the average number of points it contains tends to a limit, i.e. towards the mean area of the cap. In a formula: #f1 6 i 6 N W xi 2 Ag Area.A/ lim D : N!1 N 4 The analyst won’t have any difficulty seeing the equivalence of these two definitions, for we can recover each continuous function with the help of characteristic functions of caps (the characteristic function of a set has value 1 on the set and value 0 on the complement). On the other hand, we repeat here that what matters in practice for having a manageable number of points isn’t that there is a limit, but that we have an estimate of the speed of convergence, i.e. that we have integrals of the form Z ˇ1 X ˇ 1 ˇ ˇ f .x / f .x/ dx ˇ < “something in N” ˇ i N 4 S2 iD1;:::;N

and the same for caps. In Lubotsky, Phillips, and Sarnak (1987) we find a rather revolutionary method, but it answers the question in an optimal manner only with two disadvantages that we will examine after first having seen how the sequences of points are constructed. We give the simplest case, but the other constructions of the authors are completely analogous, for the method is very general. For more information, beyond the original paper, see Colin de Verdière (1989) and the book Sarnak (1990); see also the very similar Lubotzky (1994). Readers will perceive a rather lofty ascent of Jacob’s ladder.

Fig. III.3.17.



We fix three rectangular coordinate axes D, E, F in space and perform rotations about each of these axes through the angle 126ı 520 , more precisely the angle ˛ such that cos ˛ D 3=5. The number 3=5 is less mysterious than it seems; we also find it when we attempt to construct discrete subfields of the quaternions fields (recall that quaternions provide an algebraic description of the rotations of E3 , see 8.9 of [B]). We then take an arbitrary point x on the sphere and submit it to these rotations, composed in all possible ways, a certain number of times. We arrange the elements of the group G generated by these rotations in an infinite sequence fi g, the indexing being made for example by the length of the words required for writing an element. The sequences of points considered by these authors are the various orbits of the group G acting on a point x of S2 . With each point x we thus associate a sequence fi .x/g and we can consider, for each function f on the sphere, the absolute value of the difference Z ˇ1 X ˇ 1 ˇ ˇ f .i .x// f .x/ dx ˇ: ˇ N 4 S2 iD1;:::;N

It isn’t every difference that the authors majorize (and here is the first difficulty), but the mean in x of these differences with respect to the norm L2 . The exact result is: Z Z 2 1=2 log N 1 1 X f .i .x// f .x/ dx dx D kf k2 p 4 S2 N S2 N iD1;:::;N R for every numerical function, where kf k2 D . S2 jf j2 /1=2 . That is not to say that the orbits are all good for all functions f ; it says only that, for each given function f , the orbits fi .x/g will be good for this given function f and for the majority of initial points x. The authors have another result, also optimal, which says this: for each function f and each point x we have Z ˇ1 X ˇ 1 .log N/3=2 ˇ ˇ f .i .x// f .x/ dx ˇ 6 C.f / ; ˇ N 4 S2 N1=2 iD1;:::;N where the constant C.f / depends only on the Lipschitz constant of f and on the R integral (Sobolev) norm . S2 jdf j2 /1=2 . Here matters are thus optimal only when the function is well-controlled; this is the second of the difficulties we mention. From this result, with the help of a good grasp of spherical harmonics, the authors deduce, for any initial point x, that its orbit is uniformly distributed, moreover with the following estimate in N, valid for each spherical cap A: ˇ #f1 6 i 6 N W .x/ 2 Ag Area.A/ ˇ ˇ ˇ i ˇ ˇ D O.log N=N1=3 /: N 4 Here are several pictures of orbits, taken from Lubotsky et al. (1987). They have been zoomed in on as indicated. The pictures help convince us that the subdivisions are indeed uniform; also that they have a completely different aspect from what we saw in Slazenger or other



Fig. III.3.18. Number of points: 23 437. At left: area equal to 0:00444 of the total area. At right: area equal to 0:05082 of the total area

figures given previously; this is why we have to distrust our intuition. However, if we know that the “O” for the functions is optimal for the distribution in the caps, as it was in the restricted sense seen earlier, the power N1=3 isn’t optimal; we can rightfully hope one day to have something in N1=2 . Most extraordinary in Lubotsky et al. (1987) are the mathematical tools necessary for proving that things go well, better than the best that can be done, in a sense that we will define. They have shown that, if we proceed with a discrete group, there will be a limit for the optimal efficacy of the good distribution with respect to the L2 norms above, and that this limit is obtained by their procedure. The idea is to analyze the action of such a group G on the sphere by regarding its effect on the various spaces of spherical harmonics (see Sect. III.XYZ). And this finally brings in number theory. In particular, in the end they use Deligne’s proof of the Weil conjectures. These conjectures of André Weil, formulated in 1949, concern arithmetic, i.e. number theory. They are very abstract and involve a mixture of the geometry of curves and arithmetic. The curves in question will be encountered in connection with the error correcting codes in Sect. X.7. These conjectures were the object of longing of very many mathematicians. It was in 1973 that Deligne succeeded in proving them, using a whole arsenal of very abstract mathematics amassed by Grothendieck from 1960 to 1970. The truth of these conjectures implies in particular that of a conjecture of Ramanujan, which is very easy to state. It has to with knowing the number of ways an integer n can be decomposed as the sum of 24 squares of integers xi 2 (i D 1; : : : ; 24) : n D x12 C C x24 . Geometers prefer to view these numbers as points of the lattice Z24 in R24 which are situated on the sphere in R24 with center O and radius n. Here again we are interested in the asymptotic behavior of the number r.n/ when n is large. In fact there exist theoretical formulas for calculating r.n/, 16 but these are not explicit. It is known that the first order of magnitude is 691 11 .n/, where 11 .n/ is equal to the sum of the eleventh powers of all the divisors of n. Between 1910 and 1920, Ramanujan conjectured that the growth in n isn’t very strong,



16 259 precisely that we have r.n/ D 691 11 .n/ 128 .n/ with, for .n/, the precise 691 11=2 11=2 < .n/ < 2n . inclusion 2n We encounter the dimension 24 at least two times: i.e. in the next section, then again with the error correcting codes of Sect. X.7. Readers who likes unexpected relationships will want to know that the enumerations relative to sums of squares are encountered naturally in problems of frequency of vibrations in balls; for this see Sect. IX.2. The technique of the authors makes essential use of modular forms. These are series that are essential in number theory, but the book (Sarnak, 1990) already mentioned shows their many applications, including the one above to points on the sphere and another to the uniqueness of measure on the sphere; see Sect. III.6 below. Finally, there is an application to combinatorial geometry, i.e. to the problem of bipartite graphs. Let us repeat that we do practically no combinatorial geometry in this work, despite its growing importance, but we mention it several times; for this subject see the book Pach and Agarwal (1995). The problem is of practical importance: it deals with connecting two networks of n persons by cables (or any other theoretically comparable means), and this in an economic way. To join persons two at a time requires n2 cables, which is prohibitive when n is large. It is necessary to proceed otherwise. Without entering into the details, let us say that the theoretical existence of optimal graphs has been established (based on a computational calculus), but that the first explicit construction giving optimal results can be found in Lubotsky, Phillips and Sarnak (1988); see the book Lubotzky (1994). Discrete quaternions are used here too and above all the Ramanujan conjectures. The radically different approach of Bourgain and Lindenstrauss (1993) merits special mention; it uses the notion of zonotope, which we won’t have time to discuss in Chap. VII, even though this is an important type in convexity theory. Refer to the end of Giannopoulos and Milman (2001). It is a topic once again full of mysteries; see the index of Gruber and Wills (1993). We find there a striking illustration of the true nature of mathematics: the structures and the concepts that it creates are paradoxically the most natural and, at the same time, the most abstract. This is why its constructions can be applied suddenly to totally unexpected contexts. Not only in mathematics, but in all of science. The platonists don’t lack for evidence to bolster their arguments. As for us, our view is that the nature of mathematics is halfway between Platonism and reductionism, but perhaps such a compromise is not very courageous? For this category of questions, see Berger (2001). In spite of its seductiveness for the mathematician the preceding method is not practically applicable, for the results are not optimal in a probabilistic sense. Just recently double progress has been made, in both theory and practice, this with the physicist’s approach to minimizing the potential energy. We return to the basic reference (Hardin and Saff, 2004) for more details. We begin with some notation. We denote by !N any set of N different points of the sphere: !N D fN different points x1 ; : : : ; xN of S2 g



and define the associated s-energy: Es .!N / D

X i¤j

1 : jxi xj js

For s D 1 this is the Coulomb–Thompson potential, for s D 1 these are the enemy dictators. And for s D 0 we get X 1 log ; E0 .!N / D jxi xj j i¤j

or simply the product (for maximization) Y

jxi xj j:


We recall (see above) the relation between S2 and the solution of polynomial equations P D 0. In Newton’s method what matters is the pair .P.z0 /; P0 .z0 //, but only within a scalar multiple: now the .z W z 0 / defined within a scalar is CP1 D S2 . The two basic questions are: (Q1 ) If Es .N/ denotes the minimal energy Es .N/ D inffEs .!N / W !N S2 g; how are the configurations for minimal energy distributed for large values of N and how can we construct them? (Q2 ) What is the asymptotic of the minimal energy as a function of N and s? Presently we find in Kuijlaars and Saff (1998) that the basic answer is known for Q2 and that, for all the s-energies when 0 < s < 2, we have Es .N/ N!1 C.s/ N2 : For s D 2, the order is in N log N, for s > 2 it is in N1Cs=2 . The constants C.s/ are explicit, for 0 < s < 2. The proof uses analysis, i.e. potential theory and spherical harmonics (see Sect. III.XYZ). But in addition the configurations for minimal energy are asymptotically uniformly distributed. Attention: this says nothing about the question Q1 . As for the case s D 0 of Smale, the conjecture is that an !N can be found in polynomial time (by algorithms to be precisely constructed) such that E0 .!N / 6 E0 .N/ C C log N. The technique of spirals and that of constant areas, seen previously and cited in Rakhmanov, Saff, and Zhou (1994) and Saff and Kuijlaars (1997), go in this direction but haven’t yet provided everything that is desired. They are however good up to N D 12 000 with C D 114. Now Q1 is a totally different story, among other problems that rely on the power of computers. But even with this approach the supplementary difficulty of finding the minimum and the corresponding configurations is considerable, for there is a



number exponential in N of local minima that are extremely close to the true minimum. Apart from the references given above, for the experimental point of view see Katanforoush and Shahshahani (2003). However two types of recent work, working with the N in excess of 1 000 points, contradicts the intuition, which may induce configurations for several hundreds of points from configurations seen above. In those previous configurations, we found twelve pentagons (based on the regular icosahedron), and all the other cells were hexagons. Here on the contrary we see heptagons appear, which are akin to pentagons; they are arranged along the edges of the icosahedron, forming what are called cicatrices. Cicatrices are found starting with N D 300.

Fig. III.3.19. Almost optimal configurations for 1 600 points and for s D 1 (at left) c Ed. Saff and s D 4 (at right). Hardin, Saff (2004)

Fig. III.3.20. A physics experiment on condensed matter. Hardin, Saff (2004) c Ed. Saff

In the physics of condensed matter, there is no mention of cicatrices, but rather of “defects” and rather of “disclinations”. In Busch et al., Science 2003 physical experiments are done with beads of polystyrene (one micron in diameter) attached to drops of water immersed in an oily mixture; “cicatrices” are also found.



Now this has to do with evaluating the uniformity of the above distributions. For the energy method, the method of point counting in the caps is replaced by the evaluation of point energies. These are the X Es .xi / D jxi xj js : j D1:::NIj ¤i

Figure III.3.20 represents the point energies for the figure with s D 1; the hexagons are the dominant sea, whereas the pentagonal points have higher energy and the heptagonal a lower energy. The conclusion of experts is that we will need to wait until configurations with values of N of order 10 000 or more are obtained in order to have really good distributions. Another big problem is to find, for the Es .N/, at least the second term for their asymptotic evaluation. Thus the mission is likely possible, but tens of thousand of points are needed. III.4. The kissing number of S2 , alias the hard problem of the thirteenth sphere The kissing number has to do with knowing how many solid and disjoint balls, all of radius 1, can be arranged in space so as to touch the unit ball. Two balls tangent to S2 will be exterior to each other precisely if their two points of contact p, q satisfy d.p; q/ > =3. The problem here is thus a sort of partial inverse to that of the enemy dictators from the preceding section: we seek the maximum number of points that can be put on the sphere whose distances from one another are at least equal to a given number, but whose value here is fixed at =3. Or again, to stack spherical caps of radius =6. This optimal number is called the kissing number or sometimes the contact number. We can see why it is called Newton’s number on p. 849 (Sect. 5.3, Chap. 3.3) of Gruber and Wills (1993), but ask whether it wasn’t for the sake of propriety that the authors abandoned now classic terminology. Interest in this number is evident when we seek to stack spheres in space in a densest possible way, a problem we treat amply in Sect. X.1. But in each case the process begins around a first sphere. The problem can be posed in all dimensions; we will see some of this in Sect. III.5. For the case of the plane, this number is clearly equal to six, for on the circle S1 things are trivial, because they are somehow linear. In the space of three dimensions we can easily distribute 12 such balls, or 12 points, on the unit sphere. Impinging a bit on Chap. X, we can find them by regarding a bunch of canon balls or seeing how grocers stack round fruit. The geometer, worshiper of symmetries, may prefer the much more regular configuration furnished by the vertices of a regular icosahedron, conditional on verifying that the distance between vertices (to be calculated) is just enough greater than =3, i.e. 63ı 440 . We can thus still move the spheres, and enough to be able to permute them among each other (always without interpenetration) in an adept way; it is the planetarium of Figure 1.7 of Conway and Sloane (1999); see below Fig. III.4.1 below. We thus arrive at stacking configurations, even though these configurations have a gridlocked appearance:



it is surely thus for the spheres one at a time, but not if we move them all simultaneously in order to free up a bit of space and arrive at the regular icosahedron; see the details of the proof on p. 29 of Conway and Sloane (1999). We will return to this exasperating, but unavoidable, situation in Sect. III.7 and above all in Sect. X.2.

c Springer Fig. III.4.1. How to permute the planets. Conway, Sloane (1999)

But the problem we started with was the maximum number: can we add a thirteenth ball? An evident motivation being to obtain better stackings than those that are known. The problem was the object of an animated controversy between Newton and Gregory around 1694. Gregory assured Newton that it was possible to place a thirteenth sphere by an argument from the calculus of areas: the total area of the sphere is 4 while the area of a spherical cap of radius =6 equals 2.1 p1 /, 3

thus the quotient 4=2.1 p1 / D 14:9280::: is not only greater than 13, but 3 also 14. Newton was too good a mathematician to believe Gregory: a measure argument isn’t sufficient for an essentially metric problem. There then came several purported proofs of the impossibility of a thirteenth sphere, but all contained errors; spherical geometry is treacherous enough, as we will see again in connection with Cauchy’s theorem in Sect. VIII.6. The first presumably correct proof that at most twelve spheres are possible is in Schütte and van der Waerden (1953). It is long and in German; it isn’t clear to this author whether anyone has really read this proof in its entirety, and whether it is complete. There is, let us say, an epistemologicalsociological reason for this and what follows. Publishing a complete and careful proof won’t bring glory to its author, for the result is “evident” or “well known”. Here are some references: first (Leech, 1956), an extremely short text which was long thought to be the obligatory reference. Our colleague Marjorie Senechal wanted to achieve clarity and organized a seminar to complete Leech, but after some time the participants abandoned. A sad part of the history, but which may be consoling to those who believed themselves infallible and didn’t recognize the traps of spherical geometry. In the first edition of Aigner and Ziegler (1998) the proof given rested on a lemma that was clearly false. In spite of their efforts the authors weren’t able to set up a complete (correct) proof, and in the second edition of this remarkable book the theorem of the thirteenth sphere no longer figured. Subsequently this



author has received several new purported proofs, but admits to not having verified any in detail. Here are a few references on the subject: Pfender and Ziegler (2004), Casselman (2004), Anstreicher (2004), Krauth and Loebl (2004). Why is this so difficult? The initial idea is so simple: we triangulate the sphere with the given points as vertices. By hypothesis, all these triangles have three sides of length > =3. The naivety consists then of believing that these triangles have an area greater than that of an equilateral spherical triangle of side =3. Then 13 or more triangles would have a sum of areas more than the total area of the sphere. But it is grossly false for triangles that are too flattened, since the area can be very, very small, indeed nil. It is thus necessary to choose a triangulation, or a tiling, by spherical polygons for which the angles aren’t too small, and to thus obtain a reduction of their areas. Matters are simple enough for the elimination of 14 vertices, but for 13 it is much more difficult. We refer to the cited texts to see the various attacks. It seems that there can’t exist a truly conceptual proof; the reason is that the twelve spheres aren’t rigid contrary to the case of the plane (6 disks, trivial), which is a rigid situation. It is interesting to know that the attempts at conceptual proofs, i.e. using linear programming and spherical harmonics, don’t allow the conclusion in dimension 3 (14 spheres on S2 ) but “easily” allow the conclusion in dimension 8 (with 240 spheres on S7 ) and in dimension 24 (with 196 560 spheres on S23 / for the proof shows moreover that the situation is rigid, unique (within isometry of course), i.e. we can’t move anything at all; see Sect. X.2 in this regard. III.5. Four open problems for the sphere S3 Two open problems for S3 concern the volume of its tetrahedra, i.e. the figure formed by four points not in the same plane, its faces being spherical triangles. The total volume of S3 is equal to 2 2 (see Sect. VII.6.B). In contrast to the case of dimension 2, where we have two very simple formulas for calculating the area of a triangle: D ˛ C ˇ C as a function of the angles, and tan 4 D p sin p sin.p a/ sin.p b/ sin.p c/ as a function of the sides, for the tetrahedra of S3 there does not exist any simple formula. Here are two problems on them without solution up to the present. The first is that of the isoperimetric inequality: to find the tetrahedra of S3 for which the volume is fixed and the area (i.e. the sum of the areas of its four faces) is minimal. The problem is solved for Euclidean space E3 : the minimum is attained by regular (equal sides) tetrahedra, and for these only. We will see in Sect. VIII.7 what the solution is more generally for the isometric inequalities of polyhedra. In fact, the same conclusion is valid in hyperbolic space, but more difficult to show: Fejes Tóth (1963). On the other hand, the problem is open in S3 . We merely note for enthusiasts for algorithms and complexity that this problem is subject to Tarski’s principle: each problem that can be written as the solution of a finite number of equalities and (real) polynomial inequalities may be entered into a computer; but



this isn’t sufficient for our problem. In Sect. X.5 we will encounter other problems of this type, involving the combinatorics of polyhedra in Sects. VIII.10 and VIII.12 and also involving Hermite’s constant. The second problem begins just as simply: the formula D ˛ C ˇ C implies that, if a triangle of S2 has angles that are rational multiples of , then its surface area too is rational to , or again rational relative to the total area of the sphere, which equals 4. Intuitively, this is not unreasonable: if we think about angles of the form =n, where n is an integer, we then think of being able to tile the sphere with copies of our triangle, and thus the result is evident. In the case of rationals more general than the 1=n, we might think of an argument by a covering the sphere. The same question thus in dimension 3: if a tetrahedron of S3 has each of its dihedral angles (six in number) rational to , is its volume also rational to 2 2 , the volume of S3 ? We must be very clear at the outset that these aren’t the solid angles at the four vertices, as in Sect. III.3, that are interesting here, but rather the dihedral angles, for it is these that control the geometry of the tetrahedron and enter also, by the way, into problems on tilings. Unfortunately, even if the dihedral angles have the form =n, n an integer, and even if tiling proceeds well about the edges, the catastrophes occur at the vertices; we will encounter this difficulty in Sect. XI.11. We must definitely mention the surprising fact: this question of rationality has come up recently, and in an essential way, in the theory of the secondary characteristic classes of Chern, Cheeger and Simons. There is an historical explanation of this ascent of the ladder in Simons (1998). It can be seen there that this rationality hypothesis is, underneath its simple appearance, actually extremely profound and still open. The kissing number of S3 is now known to be exactly equal to 24; this is completely recent; see Musin (2004). The proof is based on the technique explained in Sect. III.7 below. Lawson’s conjecture concerns the minimal surfaces in S3 : certainly the surfaces which are equators are minimal, but are they the only minimal surfaces of the topological type of the sphere? The answer is affirmative, a result that we might now call classic. The general idea is the same as that of Hopf in Sect. VI.8: to exploit the complex structure of the sphere. But then, in a topology that is just a bit more complicated, we can ask for the tori that are minimally embedded in S3 . We know them, they are the Clifford tori, i.e. those introduced in Sect. II.6; here we use the name only for tori made up of points equidistant from two orthogonal circles of S3 . Lawson’s conjecture is that there aren’t any others that are minimal. For a generalization, due to Calabi, from the study of minimal spheres in spheres of higher dimension, see the informal text (Berger, 1993b) and the references mentioned there, in particular Michelsohn (1987). See also the nice figure by Herman Gluck: Fig. VII.6.3.



The Poincaré conjecture does not fit the spirit of this book very well; we mention it to allow our ignorance of dimension 3 to be perceived once again. Moreover, it is trivial to state, once the notions of manifold and simply connected are known. For the notion of abstract manifold, readers can consult any work on differential geometry, e.g. Berger and Gostiaux (1987). In fact, the conjecture is stated in the same way for manifolds that are only topological. Simply connected means that every closed curve is contractible to a point; see also the end of Sect. V.XYZ. The circle is not simply connected, but in any dimension greater than 1 a sphere is always simply connected, although we must pay special attention to the proof when the closed curve isn’t differentiable. Poincaré’s conjecture maintains that every compact and simply connected manifold of dimension 3 is in fact the topological sphere S3 . It forms, with “Fermat’s theorem” and the Riemann hypothesis on the zeros of the function, the set of three great mathematical conjectures. Now that Fermat’s theorem has been proved, there remain no more than two “very great” conjectures. As for seniority, the (merely “great”?) conjecture of Kepler was older, but we will see in Sect. X.2 that it has just very recently been proved. Perelman has announced a proof of the Poincaré conjecture. To learn more, read the end of Sect. X.3 for the evidence, as well as Milnor (2003), Anderson (2005), Morgan (2005). There thus remains today only the Riemann hypothesis. It should also be mentioned that the sphere S3 is an important object for comprehending the geometry of singularities of an algebraic curve in the complex plane (end of Chap. V): we cut the curve by small spheres centered at the singularity to be studied; these spheres are in a space of real dimension 4, thus spheres S3 . This study is perfectly motivated and presented in the book (Brieskorn & Knörrer, 1986). III.6. A problem of Banach–Ruziewicz: the uniqueness of canonical measure It can be shown easily enough by analysis (by performing a convolution to make things regular in the sense of measure, i.e. an integral of translation with an infinitely differentiable function) that the canonical measure on the sphere is the only measure (within a scalar multiple) that is invariant under the group O.3/ of isometries. For, since O.3/ is transitive on the sphere, if the measure is given by a function in terms of canonical measure, this function must be constant. Recall that a Lebesgue measure must be additive for denumerable unions of disjoint open sets; in 1920 Ruziewicz posed the question of knowing what happens if we only require the measure considered be additive for finite unions: does there exist, under this weaker hypothesis, a noncanonical measure invariant under the group O.3/? Since Banach in 1921 we know that such measures exist abundantly (which is clearly appalling) on the circle S1 , even if they are very pathological. For their construction, it seems best to consult the book (Sarnak, 1990), since we have seen that



there he also treats the problem of Sect. III.3; but there is also Lubotzky (1994), which is quite similar. We also find out what happens in higher dimensions. The answer is that, starting with dimension 2, there is no longer another invariant measure that is both finitely additive and canonical, but the proofs are all very involved and very abstract. Uniqueness itself is not so surprising, for the group O.3/ on S2 , then O.nC1/ on Sn , is very active, much more than just transitive; the subgroup called an isotropy group that leaves a point fixed, and which is O.n/, is a good evaluator of this “abundance”. What is surprising: the difficulty of all these proofs. Numerous mathematicians have attacked this very natural problem. Margulis in 1980 and Dennis Sullivan in 1981 showed the desired uniqueness by using that O.n/, starting with n D 4, is a really large group; the conceptual tool was the notion of a group having Kazdan’s property .T/, a powerful tool invented in 1967. The whole book (de la Harpe & Valette, 1989) is dedicated to property (T), and its Chap. 7 to the present problem. But for S2 the group O.3/ isn’t all that big, the isotropy group is just the circle S1 (doubled, to be precise), which is commutative. In 1984 Drinfeld proved the conjecture for S3 and S4 , but this necessitated not only a quite conceptual use of automorphic forms, but above all the truth of the Ramanujan conjectures established by Deligne (see above in Sect. III.3), and we might well ask what number theory has to do with all this. The proof given in Sarnak (1990) is new and the first that runs through all dimensions at once. But it too uses Deligne’s solution of the Ramanujan conjectures. The basic tool is the construction on the sphere of appropriate sets, a bit like in Sect. III.3. III.7. A conceptual approach for the kissing number in arbitrary dimension The spheres of higher dimension (volumes, isoperimeters, etc.) will be amply studied in Chap. VII on convex sets. However, we treat here the problem of the “kissing number” (number of contacts) in arbitrary dimension, as much for reasons of equilibrium between chapters as for giving an interesting conceptual ascent. The notion of the kissing number .n/ is defined in all dimensions, in addition to the dimension 2 which was treated in Sect. III.4. The kissing number in dimension n, for the sphere Sn1 , is the maximum number of solid spheres of unit radius that can touch the unit ball of the sphere Sn1 without interpenetration. It is thus the maximum number of points that we can put on Sn1 such that the distances between pairs is never less than =3. Which conceptual approach can we expect for this problem? If we ask physicists, they will say right off: “why not use spherical harmonics?” We will speak about this briefly in Sect. III.XYZ. The fundamental idea is that the spherical harmonics are the basic functions on the sphere; every other function on the sphere can be expressed as a convergent series of them, just as any periodic function on the line (i.e. on the circle S1 ) possesses a Fourier series composed of sines and cosines. It is thus natural to treat a problem (of distances or otherwise) on the sphere by using these harmonics, all the more since the first spherical harmonics are precisely the cosines of distances to a point, in fact restrictions to the sphere of linear functions. We give this method



of attack in some detail, first because it is conceptual, but above all because it yields the optimal answer in dimensions 2, 8 and 24 (and better; see below). This approach, and the results it furnishes, date from Odlysko and Sloane (1979); it also figures in Chap. 13 of Conway and Sloane (1993). Here are the essential stages. We need to adeptly find a suitable polynomial f .t /, for t 2 Œ1; 1, that satisfies the hypotheses to come. We should remark that here in fact t represents the cosine of distance on the sphere, which goes from 1 to 1 as the angle goes from 0 to . Then we have the result that is key to all: (III.7.1) Suppose that the polynomial f .t / satisfies first f .t / < 0 for t 2 Œ1; 1=2 P and then that, when expressed as a sum f D i fi Z.n/ of zonal spherii cal harmonics, its coefficients satisfy f .0/ > 0 and fi > 0 for positive i . Then the kissing number .n/ for dimension n satisfies the inequality .n/ < ff .1/ . .0/ If we make the change of variable t D cos d , we see that the underlying idea is to find a function f of the distance on the sphere that is negative for those pairs of points for which the distance is less than =3, points that don’t count in the enumeration with the help of the function f . These are the points that have no right to exist as points of contact with S2 of the exterior balls that touch it without interpenetration. Here is the proof: we let A.t / denote the distribution equal to the sum, in the sense of distributions, which equals u.t/ ı.t/ , where ı.t/ is the Dirac N distribution in t and u.t/ is the number of pairs .x; y/ such that cos.d.x; y// D t . In the calculations that follow, the integrals will be ultimately extended in the sense of distributions. Then the kissing condition of (Sect. III.7.1) says that f is negative or zero for all pairs for which the distance is greater than or equal to =3, that is for a distribution of “kissing” type, i.e. that A.t / D 0 for t 2 1=2; 1Œ. And of course A.1/ D N, the total number of points of the distribution considR1 R1 ered. Then: 1 A.t / dt D N2 . Next: 1 A.t / dt 6 A.1/f .1/ from the precedR1 R1 P R1 P .n/ ing; but 1 A.t / dt D 1 i fi Z.n/ i>0 fi A.t /Zi .t / i dt D f0 1 A.t / dt C and a property of the spherical harmonics is that of having a positive kernel, so that for each i > 0 and for each distribution of points fxk g we always have P .n/ k;h Zi .xk :xh / > 0. We thus have Z 1 A.t / dt > N2 f0 ; QED. f .1/N > f0 1

Now we need to find effective polynomials f .t /. Here is what has been discovered up to the present. For the dimensions 8 and 24 we take the polynomials .t C 1/.t C 12 /2 t 2 .t 12 / and f .t / D 1490944 .t C 1/.t C 12 /2 .t C f .t / D 320 3 15 1 2 2 / t .t 14 /2 .t 12 / respectively. These polynomials are excellent, for they provide 4 the exact value of .8/ and .24/. In fact, they yield the inequalities .8/ 6 240 and .24/ 6 196560. But we will see in Chap. X that these numbers are attained with the aid of spheres centered at points of the lattice denoted E8 and ƒ24 (the Leech lattice). We thus finally have .8/ D 240 and .24/ D 196560.



It is very important to know the following: when we follow the path to equality in the proof, we can show that there are as many relations satisfied in the spherical harmonics as arise from the points of contact and that the positions of these points are completely determined within an isometry of the space, i.e. that the configuration of spheres is given completely: it is a rigid. Rigid here means that any two configurations of spheres can be derived one from each other by an orthogonal transformation. The details are given in Chap. 14 of Conway and Sloane (1993); it is seen there that the rigidity is still stronger. For the case of the plane and .2/ D 6, we leave it to readers to see if an effective polynomial can be found following this method so as to determine the contact number 6, even though geometrically matters are trivial. The drama here is that this method seemingly perfect theoretically and in practice actually so in dimensions 2, 8, 24 goes so badly wrong for the other dimensions that it doesn’t even furnish, at least for the polynomials f .t / currently known, more than the inequality .3/ 6 13, whereas we have seen that .3/ D 12 and .4/ D 24. III. XYZ The spherical harmonics provide a perfect Fourier analysis on spheres, even though they don’t allow, as we have seen, for the solution of all problems there. They are found in practically all the books on mathematical physics, for example Courant and Hilbert (1953), but also in Berger, Gauduchon, and Mazet (1971), as well as in more specialized books such as Andrews, Askey, and Roy (1999) or Takeuchi (1994). A starting point for finding them is to think of “vibrations of the sphere” or “water waves on the sphere”. On the circle, viewed as the periodic line, the functions that give Fourier series sin.nt / and cos.nt / (n any integer) are in fact solutions of the equation f 00 C f D 0, which yields simultaneously the eigenfunctions and eigenvalues (and thus the frequencies): D n2 . The thing that plays the role of the second derivative on the sphere Sn1 is the Laplacian . We get an infinite sequence of eigenvalues and the associated eigenfunctions, and every reasonable function on the sphere can be expressed as a convergent series in terms of them. These functions are very simple, being the restrictions to the sphere of harP 2 monic polynomials f on the space Rn , i.e. of polynomials f such that i @@xf2 D 0. i

The first are thus the linear functions, and their restriction to the sphere are, when normalized, exactly the functions f .q/ D cos.d.p; q// for each point p. Then come the quadratic forms with trace zero, etc. In fact, we can generate, by linear combinations, all the spherical harmonics using only zonal functions. For each nonnegative integer i there exists a unique polynomial Z.n/ of degree i such that, for each point p of the sphere, the function i .n/ Zi .cos d.p; // is a spherical harmonic. The complete sequence of these functions for dimension n is denoted fZ.n/ i g.



There exists an isoperimetric inequality for the sphere S2 , which says that among all domains of given area, the spherical caps (and only these) attain the minimal perimeter. The proof is the same as for domains of arbitrary surfaces with the powerful technique of GMT (geometric measure theory); see Sect. VI.11. We won’t speak about this further. Bibliography [B] Berger, M. (1987, 2009) Geometry I,II. Berlin/Heidelberg/New York: Springer [BG] Berger, M., & Gostiaux, B. (1987). Differential Geometry: Manifolds, Curves and Surfaces. Berlin/Heidelberg/New York: Springer Aigner, M., & Ziegler, G. (1998). Proofs from the book. Berlin/Heidelberg/New York: Springer Andrews, G., Askey, R., & Roy, R. (1999). Special functions. Cambridge: Cambridge University Press Anstreicher, K. (2004). The thirteen spheres: A new proof. Discrete & Computational Geometry, 31, 613–626 Berger, M. (1992). Les placements de cercles. Pour la Science, 176, 72–79 Berger, M. (1993a). Encounter with a geometer: Eugenio Calabi. In P. de Bartolomeis, F. Tricerri, & E. Vesentini (Ed.), Conference in honour of Eugenio Calabi, Manifolds and geometry (Pisa) (pp. 20–60). Cambridge: Cambridge University Press Berger, M. (1993b). Les paquets de cercles. In C. E. Tricerri (Eds.), Differential geometry and topology. Alghero: World Scientific Berger, M. (1998). Riemannian geometry during the second half of the century. Jahrbericht der Deutsch. Math.-Verein. (DMV), 100, 45–208 Berger, M. (2001). Peut-on définir la géométrie aujourd’hui? Results in Mathematics, 40, 37–87 Berger, M. (2003). A Panoramic introduction to Riemannian geometry. Berlin/Heidelberg/New York: Springer Berger, M., Gauduchon, P., & Mazet, E. (1971). Le spectre d’une variété riemannienne. Berlin/Heidelberg/New York: Springer Berger, M., & Gostiaux, B. (1987). Géométrie differentielle: variétés, courbes et surfaces. Paris: Presses Universitaires de France Bourgain, J., & Lindenstrauss, J. (1993). Approximating the sphere by a Minkowski sum of segments with equal length. Discrete & Computational Geometry, 9, 131–144 Brieskorn, E., & Knörrer, H. (1986). Plane algebraic curves. Boston: Birkhäuser Casselman, B. (2004). The difficulties of kissing in three dimensions. Notices of the American Mathematical Society, 51, 884–885 Colin de Verdièere, Y. (1989). Distribution de points sur une sphère (a la Lubotzky, Phillips and Sarnak). In Séminaire Bourbaki, 1988–1989. In Astéerisque, 177–178, 83–93 Conway, J., & Sloane, N. (1999). Sphere packings, lattices and groups (3rd ed.). Berlin/Heidelberg/New York: Springer Courant, R., & Hilbert, D. (1953). Methods of mathematical physics. New York: Wiley Croft, H., Falconer, K., & Guy, R. (1991). Unsolved problems in geometry. Berlin/Heidelberg/New York: Springer de la Harpe, P., & Valette, A. (1989). La propriété (T) de Kazdhan pour les groupes localement compacts, Astérisque 175, Société mathématique de France Delsarte, P., Goethals, J., & Seidel, J. (1977). Spherical codes and designs. Geometriae Dedicata, 6, 363–388 Deschamps, A. (1982). Variétés riemannienne stigmatiques. Journal de Mathématiques Pures et Appliquée, 4, 381–400



Fejes Tóth, L. (1963). On the isoperimetric property of the regular hyperbolic tetrahedra. Mag yar Tud. Akad. matematikai Kutato Intez. Közl, 8, 53–57 Fejes Tóth, L. (1972). Lagerungen in der Ebene, auf der Kugel und im Raum (2nd ed.). Berlin/Heidelberg/New York: Springer Giannopoulos, A., & Milman, V. (2001). Euclidean structure in finite dimensional normed spaces. In W. B. Johnson & J. Lindenstrauss (Eds.). Handbook of the geometry of banach spaces (Vol. 1). New York: Kluwer Gibbs, J. (1961). The scientific papers of J. Williard Gibbs (Vol. 2). New York: Dover Gromov, M. (1988b). Possible trends in mathematics in the coming decades. Notices of the American Mathematical Society, 45, 846–847 Gruber, P., & Wills, J. (Eds.). (1993). Handbook of convex geometry. Amsterdam: North-Holland Habicht, W., & van der Waerden, B. (1951). Lagerungen von Punkten auf der Kugel. Mathematische Annalen, 128, 223–234 Hadamard, J. (1911). Le´cons de géométrie élémentaire. Paris: Armand Colin, reprint Jacques Gabay, 2004 Hardin, D., & Saff, E. (2004). Discretizing manifolds via minimum energy points. Notices of the American Mathematical Society, 51(10), 1186–1195 Helgason, S. (1980). The radon transform. Boston: Birkhäuser Katanforoush, A., & Shahshahani, M. (2003). Distributing points on the sphere, I. Experimental Mathematics, 12, 199–210 Krauth, W., & Loebl, M. (2006). Jamming and geometric representations of graphs. http://www. Kuijlaars, A., & Saff, E. (1998). Asymptotics for minimal discrete energy on the sphere. Transactions of the American Mathematical Society, 350, 523–538 Leech, J. (1956). The problem of the thirteenth sphere. The Mathematical gazette, 40, 22–23 Lubotsky, A., Phillips, R., & Sarnak, P. (1987). Hecke operators and distributing points on the sphere, I, II. Communications on Pure and Applied Mathematics, 39(40), S149–S186 Lubotsky, A., Phillips, R., & Sarnak, P. (1988). Ramanujan graphs. Combinatorica, 8, 261–277 Lubotzky, L. (1994). Discrete groups, expanding graphs and invariant measures. Boston: Birkhäuser Melissen, H. (1997). Packing and Covering With Circles. Ph.D. Thesis, Universiteit Utrecht Michelsohn, M. L. (1987). Surfaces minimales dans les sphères. Astérisque, Théorie des variétés minimales et applications, Astérisque 154–155. Société mathématique de France, 131–150 Morgan, F. (2005). Kepler’s conjecture and Hales proof – a book review. Notices of the American Mathematical Society, 52, 44–47 Odlysko, A., & Sloane, N. (1979). New bounds on the number of unit spheres that can touch a unit sphere in n dimensions. Journal of Combinatorial Theory. Series A, 26, 210–214 Pach, J., & Agarwal, P. (1995). Combinatorial geometry. New York: Wiley Pfender, F., & Ziegler, G. (2004). Kissing numbers, sphere packings and some unexpected proofs. Notices of the American Mathematical Society, 51, 873–883 Rakhmanov, E., Saff, E., & Zhou, Y. M. (1994). Minimal discrete energy on the sphere. Mathematical Research Letters, 1, 647–662 Robinson, R. (1961). Arrangement of 24 points on a sphere. Mathematische Annalen, 144, 17–48 Saff, E., & Kuijlaars, A. (1997). Distributing many points on a sphere. The Mathematical Intelligencer, 19, 5–11 Sarnak, P. (1990). Some applications of modular forms. Cambridge: Cambridge University Press Schütte, K., & van der Waerden, B. (1953). Das problem der dreizehn Kugeln. Mathematische Annalen, 125, 325–334 Shub, M., & Smale, S. (1993). Complexity of Bezout’s theorem III: Condition number and packing. Journal of Complexity, 9, 4–14 Simons, J. (1998, June). Interview with Jim Simons. The Emissary, MSRI Berkeley, 1–7 Smale, S. (1998). Mathematical problems for the next century. Mathematical Intelligencer, 20(2), 11–27 Stewart, I. (1991). Circularly covering clathrin. Nature, 351, 103



Stolarski, K. (1973). Sum of distances between points on a sphere. Proceedings of the American Mathematical Society, 41, 575–582 Takeuchi, M. (1994). Modern spherical functions. Providence: American Mathematical Society van der Waerden, B. (1952). Punkte auf der Kugel. Drei Zusätze. Mathematische Annalen, 125, 213–222 Wagner, G. (1993). On a new method for constructing good point sets on spheres. Discrete & Computational Geometry, 9, 111–129 Walker, J. (1979a). More on Boomerangs, including their connection with a climped golf ball. Scientific American, 240, Walker, J. (1979b). The amateur scientist, column: More on boomerangs, including their connection with the dimpled golf ball. Scientific American, 1979, 134–139

Chapter IV

Conics and quadrics IV.1. Motivations, a definition parachuted from the ladder, and why Even though there isn’t to our knowledge any important open problem concerning the conics for quadrics it’s a different story we are going to stay with them for a long time, but talk about the quadrics only very briefly. We hope, however, that the chapter will please many readers. More knowledgeable but not necessarily omniscient readers may skip all the beginning material and just look at Sects. IV.8 and IV.9. Here are our motivations: we have already stated how much the teaching of geometry, however useful it is nowadays, has almost completely disappeared from instruction, whether in middle or upper schools, or in the university. If a few circles remain, the other conic sections are gone, even though they are an integral part of many things in our everyday lives. Here are a few examples, to which readers may append their own. The summit of the Eiffel tower describes an ellipse (of diameter about a meter in high wind), but also that of every analogous object (the Montparnasse tower, your skyscraper of choice). Even though the amplitude may very, it is never zero. More important is the fact that when you look at a circle, you see it from most viewpoints as an ellipse. In every drawing, in every painting, circles generally appear as ellipses. The shadow cast on a wall by a circular lamp shade is an hyperbola (or in reality one piece of an hyperbola the Greeks incidentally didn’t always work with both pieces), the trajectories of thrown objects (military projectiles or otherwise) are approximately parabolas. The trajectories of the planets of the solar system are approximately ellipses, while those of comets may be any of the three types: ellipse, parabola, hyperbola. Architects frequently use conics, but also sometimes quadrics. For example, when a roof must cover a non-rectangular trapezoid, they construct it in the form of an hyperbolic paraboloid. We also know that ellipses are found in physics (e.g. harmonic oscillators), parabolas and hyperbolas in optics (large and small mirrors of telescopes and automobile headlights), even though in fact paraboloids and hyperboloids of revolution are involved. But once again triality, the triptych (ellipse, parabola, hyperbola) invades the language and, more profoundly, in mathematics it is the structure and the behavior of the objects concerned. The ellipse and the hyperbola are classic figures of style, whereas the word “parabola” is used with different meanings in different contexts. Gromov likes to compare the triad (elliptic, parabolic, hyperbolic) with the three physical states of matter: solid, liquid and gaseous.

M. Berger, Geometry Revealed, DOI 10.1007/978-3-540-70997-8_4, c Springer-Verlag Berlin Heidelberg 2010




Here now is a picture of these respective realms. We note here the double label “parabolic, subelliptic”. In fact, in numerous cases, the intermediate realm left over from the elliptic and the hyperbolic cases is not well defined, not well understood, even completely unknown. It’s typical in the case of discrete groups. Gromov again likes to compare finite groups to ellipses, the hyperbolic groups (a notion he introduced in 1987 and which dominates the present theory of discrete groups) “to themselves”. Like the gases, the hyperbolic groups are expansive: they spread out everywhere. In contrast typically there still does not exist today a good definition of what a parabolic group might be. And in one sense there exist/be: what remains between finite groups and hyperbolic groups will require a (likely very subtle) sub-classification. We also know that in physics the solids and liquids are rather badly understood. The case of partial differential equations in analysis is exemplary: elliptic equations (very rigid), parabolic equations (e.g. the heat equation), hyperbolic equations (e.g. the wave equation). For the Riemannian manifolds, the case of positive curvature and the more subtle one of strictly negative curvature correspond well to the terms ellipse and hyperbola, whereas to the contrary the parabolic case is not well defined, for it cannot be solely the case of the manifolds with nonpositive curvature nor that of the manifolds with nonnegative curvature. Last but not least: elliptic geometry is that of the sphere or that of the real projective spaces; hyperbolic geometry is that of the hyperbolic spaces of Chap. II. The name “elliptic curve” that we encounter in various places does not enter the realm of the triptych, since for example in the complex domain they are tori; the name is explained solely by the fact that “elliptic” functions required for computing the arc length of an ellipse, hence their name are the very functions that are needed for parameterizing the planar cubic curves; see Sect. V.14. In Strichartz (1987) there is a rather elaborate table (requiring more than a little theoretical knowledge) of three-fold partitions in several mathematical domains. For the three-fold aspect, e.g. the trinity of fields R, C, H, we have already seen the references Arnold (1999, 2000) at the end of Sect. I.8. We will now persecute our too impatient readers a bit for wanting to know “what is a conic really”? Here is the right answer: A conic is a curve of second degree in the complex projective plane, i.e. the set of the .x W y W z/ satisfying an equation P.x; y; z/ D 0, where P is a polynomial of second degree in the three variables x, y, z that is not identically zero. (The notation .x W y W z/ was introduced in I.5 and denotes triples of complex numbers that are not all zero, taken modulo multiplication by a scalar.) Readers may ignore this definition for the moment; it will appear again in Sect. IV.7. In this chapter we pursue three goals more or less simultaneously:



(i) To state the essential properties of the conics and the most beautiful and profound concerning them; (ii) to explain why and how mathematicians, in studying (i) in the course of history, have been naturally and necessarily led to the doubly abstract but final definition given in the above statement. Doubly abstract in fact, by the introduction of the projective plane with its points at infinity and by the introduction of complex numbers with their “imaginary” aspect. It’s thus a climb of three rungs up the ladder: the first already accomplished by Descartes replaces metric geometry by an equation (here of second degree), the second projectifies the real affine plane, the third complexifies the real projective plane into a complex projective plane, and all this just for the good definition of a conic. For the two theorems we have chosen as being principal, we need to climb several rungs higher still. (iii) to say a little about quadrics. The conics and many of their often subtle properties were known to the Greeks. An ideal historical reference is, among others, Coolidge (1968) and Van der Waerden (1954–1975), Vol. I, p. 247; but for a complete inventory of what was known in that epoch, the chapters dedicated to the conics in the Encyclopedia of the Mathematical Sciences (Dingeldey, 1911) (which is an elaborated translation of the original work in German) is unavoidable, even though it only gives the statements or even just references. This is why, as it seems to us, few recent works are as detailed and complete for a relatively elementary exposition – as the collection formed by the five chapters 13–17 of [B], to which we can turn for more details about what is treated here. The other references are not always easy to find in libraries, and Dingeldey is of an incredible density. IV.2. Before Descartes: the real Euclidean conics. Definition and some classical properties We give at the outset the historical definition, and an etymological translation, of the conics. These are all the plane curves that can be obtained as sections of cones of revolution in Euclidean space. Here are some drawings in space: Which yield in the plane:

Fig. IV.2.1.



Fig. IV.2.2. 2

In the Euclidean plane E , the simplest curves other than lines are the circles (simpler than lines, by the way, to trace mechanically). We can’t stay with circles, for the images of circles under affine/projective transformations are generally not circles. In E2 there are four types of conics (later we will add the qualification “proper” or “non-degenerate”), specifically: – circles, the set of which is denoted by C ; – ellipses, the set of which is denoted by E; – parabolas, the set of which is denoted by P ; – hyperbolas, the set of which is denoted by H . The union of these preceding sets is denoted by K D C [ E [ P [ H . The Greeks knew more or less (often a half hyperbola only), that the union of these four types was exactly all the sections by planes not passing through the apexes of cones of revolution in the Euclidean space E3 , as shown in the above figures. So by leaving the plane and going into space the lines, a circle and a plane suffice for finding all the conics. Note that for the ellipses E there are two still simpler ways of finding them: by projecting a circle onto a plane or by cutting a cylinder of revolution by a plane not parallel to the axis of the cylinder.

Fig. IV.2.3.

Of course our preference is to find the elements of K without leaving E2 . Here are the methods:





m f x



C Fig. IV.2.4.

(i) In E2 let a point f and a line D not containing f be given. The curve P D fm 2 E2 W d.m; f / D d.m; D/g formed of points at equal distance from D and f is a parabola; and conversely each parabola can be obtained in this way. We say that f is the focus of P and that D is its directrix. (ii) Same given f , D as in (i) but in addition a real e 2 0; 1Œ. Then the curve E D fm 2 E2 W d.m; f / D e d.m; D/g is an ellipse in E2 , and reciprocally every ellipse can be obtained in this way. We say that f is a focus of E and that D is the associated directrix for f ; in fact there exists (this needs to be shown) another pair .f 0 ; D0 / yielding the same ellipse (for the same e), for ellipses always have a center of symmetry. We say that f and f 0 are the foci of E. The number e is called the eccentricity. (iii) Same data as in (ii) but here e > 1. We then obtain all hyperbolas. Same definitions of foci and directrices. (iv) Let f , f 0 be two distinct points of E2 and a a nonzero real such that 2a > d.f; f 0 /. Then the set E D fm 2 E2 W d.m; f / C d.m; f 0 / D 2ag is an ellipse whose foci are f and f 0 . Moreover, every ellipse can be obtained in this way. (v) Same data as in (iv), but here 2a < d.f; f 0 /. Then H D fm 2 E2 W jd.m; f / d.m; f 0 /j D 2ag is an hyperbola, with foci f; f 0 . Moreover, every hyperbola can be obtained in this way. Note that fm 2 E2 W d.m; f / d.m; f 0 / D 2ag is only a demi-hyperbola. The Greeks knew these propositions; the definitions (i), (ii), (iii) are referred to as monofocal and (iv), (v) as bifocal. The property of equal angles (see the figures) for the tangents with the rays emanating from the foci explains the optical properties of conics and the uses indicated below in the section on quadrics. It remains to learn how to show the equivalence of these definitions; we will return to this problem in






a2/c m


y m

m f



c C





C a2/c

Fig. IV.2.5. Here are shown, for the ellipse and for the hyperbola, the two foci but only one directrix

Sect. III.3. The bifocal definition is generalized thus: gardeners use a rope that goes around two stakes to trace ellipses. But Graves’ theorem (see Fig. IV.2.11) says that if a rope goes around an ellipse instead of two points, then the pointer again describes an ellipse (with the same foci as the initial ellipse).

Fig. IV.2.6. The schoolboy’s parabola. The gardener’s ellipse. II [B] Géométrie. c Nathan Édition Nathan (1977, 1990) réimp. Cassini (2009)

Readers must begin to be somewhat unnerved by this separation of K into the four parts C, E, P , H and must feel a desire for unification (without leaving E2 ). Actually, two partial unifications are immediate. The first is that P [ E [ H can be obtained with the same monofocal definition by letting e take on all positive real values: P for e D 1, E for e < 1 and H for e > 1. But here the circles escape hopelessly: it is in fact the case that the “directrix” of the center f D 0 is at infinity, or again that the two foci are merged. This can be demonstrated in an appropriate sense by limits. We begin to sense here the necessity of infinity, i.e. of the projective plane. A second unification allows putting together C , E, H by the same bifocal definition. To obtain P , it is necessary to transfer the focus f 0 to infinity. The Greeks knew they even had the notion of coordinates but in the complicated language which for them played the role of our modern algebraic language (see Coolidge, 1968) that the circles, ellipses, hyperbolas and parabolas might be defined by equations of the form



x2 y2 ˙ 2 D1 2 a b for C [E [H and y 2 D 2px for P . The equation of ellipses shows easily that these are the harmonic oscillators of physics: x D a cos !t , y D b sin !t . For the movement of the planets and comets it suffices to see (and this is easy) that, in polar coordinates, if e is the eccentricity, then every conic can be written D

p 1 C e cos

.p > 0/

(the circle is obtained for e D 0). We then need to know that when Newton’s law of universal attraction is expressed in polar coordinates, the second derivative of 1 satisfies the relation 1 00 1 C D const. The calculation takes several lines. We need at first to write everything as a function of time: .t /, .t /, next apply the law of universal attraction (or gravitation), then eliminate t. We use the fact that, as the acceleration is central, the quantity 2 d remains constant. The law in 1=r 2 had been presented before Newton, but the dt above calculation had only succeeded with him and for its time was an absolute tour de force of differential calculus. We note that, in the same epoch but before Newton, Hooke had constructed a laboratory machine for verifying if this law indeed furnished “elliptoid” curves and also if it was susceptible to numerical integrations. For the details of this history and the very violent relations between Hooke and Newton, see Arnold (1990), to be read with some caution because the author has an inclination toward the spectacular to the detriment of a completely well documented history, even though he surely is one of the great mathematicians of our time. The angular properties of tangents are indicated in the figures below. They explain the use of conics in optics, automobile headlights and numerous lamps and for large distances telescopes. In a large telescope, the principal mirror is a paraboloid of revolution, but the small mirror which reflects everything to the eye of the astronomer is a piece of an hyperboloid of revolution (see Figs. IV.10.1 and IV.10.2). An acoustic property has been attributed to ellipses, permitting persons situated at the foci to communicate so that no one else can hear (old usage by the ticket punchers of the Paris Metro, older still for confessing lepers). This is true only in small part. The true explanation of such a property, for example that of the circular wall of the Temple of Heaven in Beijing, is the fact that the vibrations of higher frequencies in the interior of a flat space are concentrated near the boundary. In more learned



language, the Bessel functions the harmonics of the planar disk of a somewhat elevated order are very, very flat close to the center and don’t really manifest themselves except very close to the boundary.

m m







Fig. IV.2.7. In these figures, we see the angular property of the bisectrix for the tangents

A very important figure, encountered in numerous contexts, is that of hom*ofocal conics. Analytically the family is written . /

x2 y2 C D1 a2 C b2 C

.a > b/:

We obtain ellipses for 2 b 2 ; 1Œ and hyperbolas for a2 ; b 2 Œ. In mathematical physics, this family provides a coordinate system called elliptic, by associating with a point the pair f; g furnished by the ellipse and the hyperbola that pass through this point. This is useful for example in studying the proper frequencies (or “eigenfrequencies”) of a vibrating elliptic plate; see Morse and Feshbach (1953). The hyperbolas give interference curves when the two foci emit waves. This was already used for radio tracking for some time before GPS. In studying dynamic geometry in Sect. XI.9 we will encounter the hom*ofocal conics while examining



Fig. IV.2.8. Distribution of intensity for the vibrations of the disk associated with the c American Physical Society 40-th Bessel functions. McDonald, Kaufman (1988)

Fig. IV.2.9. hom*ofocal conics

the trajectories of a billiard ball (or of light) on the interior of an ellipse and we will see revealed a completely atypical phenomenon.

Fig. IV.2.10. Trajectories of a billiard ball in an ellipse (reflections following Descartes’ law), programmed by John Hubbard



This property of the billiard ball is equivalent to the fact that the tangent at the point describing the second ellipse is a bisectrix of the two tangents issuing from that point to the interior ellipse. It thus implies the following spectacular consequence: if we trace, on the interior of an ellipse, the trajectory of a light ray (or a billiard ball), this trajectory will always remain tangent to a fixed ellipse (or an hyperbola if we start at a point on the segment joining the foci). This is to say, in the language of dynamics, that these trajectories will never be everywhere dense since a whole portion of the elliptic domain eludes them. Whereas, for a convex set taken at random, P. Gruber has just shown everywhere density (for almost all the trajectories): see Sect. XI.10.C. We can furthermore say that the ellipses have a continuous family of caustic curves that fill their whole interior. The problem remains open, at the present moment, as to whether only ellipses have this property; a recent reference is Section 2.4 of Tabachnikov (1995), where a variety of more or less definitive forms that this conjecture (called Birkhoff’s) might take are presented. See also Sect. II.4 and, for what happens if the trajectory closes, Sect. IV.8 below. A final property is Graves’ theorem mentioned earlier: if we stretch a cord about an ellipse, the pointer will describe a hom*ofocal ellipse. The analog for quadrics has been awaited for a very long time and remains very difficult; see the result of Staude in Sect. IV.10.

Fig. IV.2.11. Graves’ theorem

Here are some remarks, geometric proofs and rather easy calculations that permit us to obtain quite rapidly most of the equivalences between the different definitions of conics. To obtain the monofocal properties we take coordinates .x; y/, where the directrix D is parallel to the 0y axis and the focus f is the origin. Then the relation x 2 C y 2 D e 2 .y c/2 becomes, after a change of variable x ! x C d , one of the 2 2 equations xa2 C yb 2 D 1 that we will obtain below. For the bifocal definitions, the calculations requires a small trick. If we take 0 f D .c; 0/, p f D .c; 0/ andpattempt to get rid of the radicals in the equation of definition .x c/2 C y 2 C .x C c/2 C y 2 / D 2a, we may well risk going in



a metaphorical circle. The starting point consists in calculating mf and mf 0 while noticing that, if m D .x; y/, then mf 02 mf 2 D 4cx D .mf 0 C mf /.mf 0 mf / D 0 mf /: Thus

mf 0 mf D

2cx ; a

mf 0 D a C

cx ; a

mf D a

whence mf 02 C mf 2 D 2x 2 C 2y 2 C 2c 2 D 2a2 C 2

cx ; a

c2x2 ; a2

and whence the equation x2 y2 C 1 D 0: a2 a2 c 2 The above calculation there is an analogous one for the hyperbola furnishes just as rapidly, in polar coordinates, the equation D 1Cepcos . This then clearly shows what was said above concerning planetary motion; for, 1 being linear in cos , we have that 1 C . 1 /00 is constant. Finally a very simple trick: if we start with a focus and a directrix, and if we can show (left to readers, but it isn’t obvious) that there is a center of symmetry, then there is a second focus and associated directrix, parallel to the first. Thus mf C mf 0 D e.d.m; D/ C d.m; D0 // equals the constant distance between D and D0 . The very brief calculations above have shown very quickly the equivalence of the monofocal and bifocal definitions. The author has never understood, in the country of Descartes, why for so many years and until at least 1950, degree candidates in mathematics had to know how to prove in a purely geometric way (no coordinates they were forbidden) the equivalence of the monofocal and bifocal definitions. One of the topics of the preparation classes for the “agrégation” (examination for privileged secondary school teaching positions) at this time there was no choice of subject was “equivalence of the definitions of a conic”. Incredulous readers can consult the book (Lebesgue, 1942) which provided instruction in preparation for the “agrégation” and gives several rather elegant ways of showing this geometric equivalence. Of course Lebesgue didn’t have the courage to stop there: after 40 pages of submission to the establishment we find lots of things that we have encountered or will encounter: the focal lines of circles (see Sect. II.7), Poncelet’s theorem (see Sect. IV.7), Villarceau’s circles (see Sect. II.7), Morley’s theorem (see II.1) and Poncelet’s polygons of Section 8. In fact, one reason was an attachment to the romance of “pure”, i.e. coordinate-free, geometry. Many books speak about this; see Berger (2005) in regard to Poncelet. We will also want to obtain and unify these metric properties in a still more conceptual way, in light of Cayley’s philosophy heralded in Sects. II.1 and II.XYZ: each




t o

f m


t′ o′ S′

Fig. IV.2.12. Bifocal definition of the ellipse according to Quetelet and Dandelin: mf D mt (tangential distance to S), mf 0 D mt 0 , mf C mf 0 D mt C mt 0 D t t 0 D const

geometry arises from complex projective space. In Sect. IV.7 we indicate how this is done; it’s also the philosophy ascribed to Plücker. It wasn’t until the nineteenth century that an elementary and elegant and purely spatially geometric way was found for showing that the plane sections of cones satisfy the mono- and bifocal definitions without using the equations of the conics. This was done by two Belgians, Quetelet and Dandelin. The idea is that a cone of revolution, in which a sphere S has been inscribed tangent along a circle in the plane P, can be defined as the locus of points for which the tangential distance (the tangential distance of a point to a sphere is the common length of the tangents from the point to the sphere) to S is proportional to the distance to P, the ratio being sin ˛, if ˛ is the angle at the vertex of the cone. The proof is left to readers as a drawing exercise. They should pay close attention to the circ*mstance that, even if it is obvious that the points of the cone satisfy this property of distance, it is a little more subtle to see that they are the only ones that do; for the bifocal definition makes use of the fact that the cone is the locus of points for which the sum or difference of the tangential distances of two spheres inscribed in them is a constant. See Figs. IV.2.12 and IV.2.13 below and consult 17.3 of [B] for more details as needed. The Greeks knew other things that are more difficult, for example that the locus of points for which the distances d , d 0 , d 00 to three given lines D, D0 , D00 satisfy d 2 D kd 0 d 00 (k real positive) is always a conic. But they never succeeded in resolving the “locus for four lines”, i.e. to prove that the set of points that satisfy d d 0 D kd 00 d 000 , for the distances d , d 0 , d 00 , d 000 to four given lines D, D0 , D00 , D000 , is a conic. No more than did their successors until Descartes, whose solution to the problem will appear in the next section; see pp. 20, 21 of Coolidge (1968).




S o t




o′ S′

Fig. IV.2.13. Bifocal definition of the ellipse according to Quetelet and Dandelin: jmf mf 0 j D jmt mt 0 j D t t 0 D const

Fig. IV.2.14. The monofocal definition of the ellipse following Quetelet et Dandelin: mf D mt , mt=mh D const. for all points of the cone, mh=md D const. for all points of the plane P, thus mf =md D const

The Greeks knew plenty more things, in particular the diametral properties: when a conic is cut by parallel lines, the midpoints of the segments so obtained are collinear on a line called a diameter. We see however that we only get segments, and almost never an entire line, and would thus like a general context. This property is purely



Fig. IV.2.15.

affine, but we might also be interested, in the Euclidean case, in the lengths of the diameters associated with an ellipse. Appolonius knew this result.

Fig. IV.2.16. Diametric properties of conics. The diameters are the lines passing through the centers for the ellipses and hyperbolas, the lines parallel to the asymptotic direction for a parabola

Let us put matters differently: in the affine plane an ellipse is symmetric with respect to all directions of the plane, as are the circles for the Euclidean symmetries. The diametral properties can thus be seen very quickly analytically, whereas in “pure” geometry we see them for ellipses by transferring to them the diametral property of circles by an affine transformation of the plane. For the hyperbola, readers can try it for themselves. Do there exist other curves having this property? The answer is no. It is easy enough to put together a proof for the conics, but for the general quadric (in any dimension) it is true, but difficult to prove; see 15.6.9 of [B]. The proof uses the so-called John-Loewner ellipsoid that we will explain in Sect. VII.10.C. Pascal’s theorem is exceptional in the history of the conics by its novelty and its proof not being instantaneous: the three pairs of opposite sides of an arbitrary hexagon inscribed in a conic intersect in three points that are collinear. Pascal’s theorem is never easy to show, nor is it so very difficult. It poses a problem for historians: Pascal states that it reduces to the case of circles thanks to the “Greek



definition” but really mostly created by Desargues, of whom Pascal was a disciple, see Berger (2005) since each conic, being the section of a cone of revolution, is thus obtained by the spatial projection of a circle. Such a projection preserves lines and their intersection properties. But Pascal never said how he proved the theorem for a circle. Readers can try it for themselves. Here is such a proof, but it is not at all purely Euclidean. We easily see that we can, by a projective transformation preserving a circle, ensure that ab and de on the one hand, bc and ef on the other, are parallel. Then (by an “angular” exercise use for example Fig. II.2.13) cd and f a are parallel! For details on the history of Pascal’s theorem see p. 33 of Coolidge (1968). Note finally that if the conic is degenerate in two lines, then Pascal’s theorem becomes that of Pappus; see Sect. I.4.

Fig. IV.2.17.

We can use Pascal’s result to construct, point by point with a ruler, the conic that passes through five points of the plane that are in general position. Why does a well defined conic pass through five points of the plane? We will see the answer in Sect. IV.7, where a proof is again furnished by Pascal’s theorem. This existence is of first order importance in positional astronomy, for to calculate the trajectory of a planet, it is assumed that it is a conic (at least for an astronomically brief time); it then suffices to know just five of its positions to determine all the elements of its orbit. In practice, matters are admittedly more complicated and, moreover, computational resources have become considerably better over the centuries, from tables of logarithms (astronomy was the only profession to use six place tables, as opposed to the five places of normal tables) to computers. In the nineteenth century the duality polarity associated with a conic came to be well understood. Here we extract some essential facts from what was given in detail in Sect. II.XYZ. We begin with a circle and its visual properties. At each point m of the plane other than the center O of the circle C (whose radius we take to be 1), we associate its polar line m , which is the line that is perpendicular to the line Om and is at a distance d.O; m / to O such that Om d.O; m / D 1. We need to show that this is really a duality. Each line not passing through O originates from a unique point called its pole. All the lines that pass through m have a pole located on m and thus the pole of the line joining two points m and n is



Fig. IV.2.18. Duality interchanges the point of intersection of the polars with the line passing through the poles

nothing other than the point of intersection m \ n of the polars of m and n. None of this would get us very far if we didn’t know that, if a line D passing through m is such that it cuts the circle in two points p and q, then the four points m, m \ D, p, q are in harmonic ratio: Œm; m \ D; p; q D 1. This furnishes a geometric construction of the polar with the aid of a complete quadrilateral.

Fig. IV.2.19. At left: polar of a point exterior to a circle. At right: how to construct the polar of an arbitrary point with the help of the harmonic property of the complete quadrilateral

Before 1950 the pitiable students of the terminal class of the French lycées had to know how to prove all that “by hand with pure geometry”. An embellishment had been invented between the two world wars: for two points m and n to be such that n 2 m (we say that m and n are conjugate with respect to C) it is necessary and sufficient that the circle of diameter mn be orthogonal to the circle C. All this was torture due to the prohibition of the scalar product. In fact, let us suppose that the plane is vectorized about O and that the circle has radius 1. Then all the above properties amount to the bilinearity of the scalar product: m, n are conjugate if and only if their scalar product equals 1; the polar of m is the line of the equation mn D 1, etc. We will have noted several complications: it’s necessary to remove the center of the circle and the lines that pass through it (projectifying is the remedy for



that); then all the lines passing through m don’t intersect C (we need to complexify to remedy that). For all this see Sects. IV.4 and IV.6. We have already stigmatized this prohibition in Sects. III.1 and IV.4, but we are sad to have to note that, as recently as 1999 students in the graduating class could (or had to) mention the scalar product but were forbidden to use its bilinearity!

Fig. IV.2.20. The points m and n are conjugate with respect to the circle C

Like the circle, in the affine context any conic C furnishes a duality between points and lines of the plane: at a point p of the plane, we associate the line (called the polar of p) containing the harmonic conjugates of p with respect to the sectional segment mn (if it exists). We need to show that these conjugates are all collinear. We see in the preceding array of figures all the exceptions that need be made: the center of an ellipse yields nothing except the conclusion that the polar is the line at infinity. We have a good polarity (duality): the polars D and F of two points p and q have a point of intersection, if it exists, for which the polar is nothing other than the line pq (with some exceptions). Each line (with the exasperating exceptions) is the polar of a well determined point, called its pole. And there are all the exceptions, including the intersection at infinity for parallels: it thus becomes intolerable to stay in the affine context: we need to projectify. For example, the diameters will be the polars of the points at infinity. A celebrated example of the use of the duality is Brianchon’s theorem: the diagonals of a hexagon circ*mscribed about a conic all pass through a common point. This result is nothing other than the dual of Pascal’s theorem.

Fig. IV.2.21. Brianchon’s theorem

All of the preceding is more or less difficult geometrically; we will see in Sect. IV.4 that algebraically and in the projective plane it borders on being child’s play.



IV.3. The coming of Descartes and the birth of algebraic geometry Descartes’ basic idea is that what is common to the elements of K D C [ E [ H [ P is their capability of being defined by an equation of second degree: 2 x2 ˙ yb 2 1 D 0 or y 2 D 2px. We remark that an equation of second degree in a2 x, y . /

ax 2 C 2b 00 xy C a0 y 2 C 2b 0 x C 2by C a00 D 0

remains one in any affine system of coordinates of the plane E2 . We of course restrict ourselves to equations that are truly of the second degree, i.e. a, b 00 , a0 aren’t all zero. Descartes showed that each equation of type ( ) can, in an appropriate system of orthogonal coordinates for E2 , be written ax 2 C a0 y 2 C a00 D 0


ax 2 C 2by D 0:

In modern language, this is simply the theorem of linear algebra that permits the diagonalization of real quadratic forms with respect to a positive definite quadratic form, the latter defining the Euclidean structure (and translations of axes). Thus every equation ( ) represents an element of K or else one of the “aberrant” cases that follow (interpreted for the two reduced forms): – the empty set, if a, a0 , a00 are all nonzero and of the same sign – one point only, if a, a0 are nonzero and of the same sign, and a00 D 0; – two distinct lines, if a, a0 are of different signs, and a00 D 0; – a single line (called in the sequel a double line or two merged lines), if a is nonzero and a0 , a0 zero. This leads us to realize (a classic phenomenon) that this unification demands a price in fact a double price that need be payed. If we exclude the single point and empty set cases, it is still required to include pairs of lines and the double lines. We will see in Sect. IV.9 that this is necessary for certain problems. We might note, by the way, that the two cases are obtainable geometrically, the first in (iii) of Sect. IV.2 for f 2 D, the second in case (i) and f 2 D again. If we want to remove the aberrant cases of the empty set and a single point (these also can be obtained geometrically by taking the constant 2a to be negative or zero and f D f 0 ) in (iv) of Sect. IV.2, it is necessary to introduce the complex numbers (see Sect. IV.7). There is a second reason for complexificaton, coming from the connection between the geometric object “conic set of points in the plane” and its equation. We see in ( ) that there are six parameters, whereas the conics obviously depend on five only. This has simply to do with the fact that we get the same conic if the six coefficients of ( ) are multiplied by the same nonzero real. In spite of that the correspondence isn’t totally satisfactory, for in the case of the empty set we have much more choice than just “within multiplication by a scalar”; in effect we have a continuous infinite choice.



In modern language, that of quadratic forms, the equation ( ) is written Q.m/ C L.m/ C c D 0, where Q is a quadratic form, L a linear form and c a constant. This somewhat cumbersome expression will be proved in the projective context of the following section. As must be the case for truly innovative concepts, Descartes’ discovery was not just a new interpretation for the conics, but brought with it a whole harvest of new results. We mention three of them: the first is that every affine (or projective) transformation, when applied to a conic, yields a conic. The second is that every plane section of a cone of second degree (of revolution or not) is a conic. The third is the solution of the problem of the “locus for four lines”, mentioned at the end of Sect. IV.2. In fact, the distance of a point to a line is, in arbitrary coordinates, the absolute value of a linear form jpx C qy C zj. Thus the locus of four lines is defined by an equation of the form jpx C qy C zjjp 0 x C q 0 y C z 0 j D kjp 00 x C q 00 y C z 00 jjp 000 x C q 000 y C z 000 j: Whence we have two conics (in the broad sense above, the one being possibly empty, etc.). We make two pleasant observations. First, Descartes wrote that this locus, that had vainly been sought by his predecessors, is a single conic, even though in fact there are two of them (because of having to get rid of absolute values). This circ*mstance gave rise to a bitter discussion between Descartes and Roberval; in fact Descartes always took great care with signs, for he knew that the sign of px Cqy Cz depends on which side of the line px C qy C z D 0 the point .x; y/ is located. Next, even for the circle, readers will see that it is difficult to show in an elementary way this property of the points m of a circle that passes through four points; for evidently the four intersection points (if they exist) D \ D00 , D \ D000 , D0 \ D00 , D0 \ D000 always satisfy d d 0 D kd 00 d 000 (for any k). Besides the circle, there is an hyperbola, which in the case of the square degenerates into the pair of lines formed by the diagonals of the square (when k D 1).

Fig. IV.3.1. At left: the “locus of four lines”. At right: for k D 1 and the four sides of a square, the locus of four lines consists of the circ*mscribed circle for the square and the hyperbola degenerated into the two diagonals of the square



Much more important was that Descartes’ study was for him a particular case of consideration of curves with equation f .x; y/ D 0, where f is a polynomial in x and y, recognized as being the cradle of algebraic geometry (see p.74 of Coolidge 1968), of which we will see a very little bit in Sect. V.13. For the essence, vision and history the reference Dieudonné (1985), despite its density, stands out most strongly. IV.4. Real projective theory of conics; duality By unifying the efforts of Sect. I.5 and Sect. IV.3, we now have all the elements needed for handling what follows. We place ourselves in RP2 . Recall that, in nonintrinsic fashion, RP2 is the set of triples .x W y W z/ of reals, not all zero, considered modulo multiplication by a nonzero scalar (see Sect. I.5). We know from Chap. I that the real plane E2 has a natural completion in RP2 , in which it is embedded by .x; y/ 7! .x W y W 1/. A conic with equation . /

ax 2 C 2b 00 xy C a0 y 2 C 2b 0 x C 2by C a00 D 0

is thus extended naturally to the conic .


ax 2 C a0 y 2 C a00 z 2 C 2byz C 2b 0 zx C 2b 00 xy D 0

of RP2 , which is often called the hom*ogenization of the equation ( ). But then (

) represents a (general) quadratic form on R3 and we will thus very naturally call conic each portion of RP2 of the form p.Q1 .0//, where p designates the canonical projection p W R3 n 0 ! RP2 , where Q is any quadratic form. Things depend on the sign and rank of Q. If Q is only of rank 1, we find a double line (a projective line); if Q is of rank 2, we find two distinct lines or a point, according to the sign of Q. In these two cases we say that the conic is degenerate. Finally, if Q is of maximum rank 3, we say that the conic is proper: we get the empty set if Q is of constant sign (i.e. positive definite or negative definite, in the language of quadratic forms), and otherwise a single conic. Here is some more about hyperbola, parabola, ellipse: all have the topology of a circle and even possess an interior (in RP2 ), which corresponds to the convex pieces associated with the affine plane (one piece for the ellipses and parabolas, two pieces for the hyperbolas). Modulo an hom*ography (a projective and bijective transformation), there is but a single proper conic. Note that the projective lines in RP2 are also topological circles; but they don’t have interior, for a line of the real projective plane does not divide it into two regions. Surely readers will understand how to revert to the affine plane: we trace the conic C and the line at infinity D. If C\D consists of two distinct points, a hyperbola remains in the affine plane; if C \ D reduces to a single point, we find a parabola; and finally we have an ellipse if C \ D is empty. The majority of the affine and projective results on the conics stem from the double fact in the following chain: first we define a bijection f W C ! D between the points



Fig. IV.4.1. A projective conic always has an interior

Fig. IV.4.2.

of a proper projective conic and some projective line D (which isn’t tangent to it) by the following figure. We fix m 2 C and the bijection is f W n 7! f .n/ D mn\D 2 D. When m D n the line mm is, by convention, the tangent to C at m. The essential fact is that this provides a parameterization of C by the points of D (the simplest typical case is .sWt / 7! .s 2W stWt 2 /, which parameterizes the conic xz D y 2 ), but above all the two parameterizations of the type f W D ! C, g W E ! C (where E is a second projective line) are always such that the bijection g 1 f W D ! E is an hom*ography. A principal consequence of what precedes is that each invariant of the projective line carries over to the conics. In particular, with four points a; b; c; d of a conic C we can associate their cross ratio, denoted Œa; b; c; d C . It is defined, for any parameterization f W D ! C whatever, as Œa; b; c; d C D Œf .a/; f .b/; f .c/; f .d /D . Note that we thus have, by definition, Œa; b; c; d C D Œma; mb; mc; md , where the last cross ratio is that of four lines of RP2 passing through m, an arbitrary point of C. Thus we can define a conic as the locus of points m such that Œma; mb; mc; md D k, where a; b; c; d are four points in the plane such that no three of them are collinear and k is a nonzero real. When the four points are fixed and k varies, we obtain the whole family of nondegenerate conics passing through the four points (a pencil, see Sect. IV.6).



With the exception of the most elegant theorem of all on conics (Poncelet polygons, see Sect. IV.8) we can derive from the preceding practically all the properties of conics (real or, later, complex), the projective properties as well as the affine properties, and also even the metric properties of Sect. IV.2, this by complexifying and using the cyclic points introduced in Sect. II.XYZ. We restrict ourselves here to some important classical examples and some commentary.

Fig. IV.4.3. Pascal’s theorem. At left, general case; middle, case where two of the six vertices are merged; at right, case where the six vertices are reduced to three, each counted twice

We first of all prove Pascal’s theorem stated in Sect. IV.2. The proof is rapid: if we set x D bc \ ed , y D cd \ ef , z D ab \ de, t D af \ dc, we have Œz; x; d; e D Œba; bc; bd; be D Œa; c; d; eC D Œf a; f c; f d; f e D Œt; c; d; y (see Fig. IV.4.3) so that zt, xc, ey are coincident. Note also that Pascal’s theorem remains true when two vertices such as a and b are merged if we define aa (initially undefined) to be the tangent to the conic at the point a. We may regard Pascal’s theorem as the generalization of Pappus’s theorem, encountered in Sect. I.4: it has then to do with a conic degenerated into two distinct lines. The conics can be generated as the intersection of two lines that turn: if m, n are two points of RP2 and f an hom*ography between the pencil of lines passing through m and that of lines passing through n, then the point D \ f .D/ describes a conic as D turns about m (passing through m and n, proper if f .mn/ 6D mn). By duality we deduce that if m and n describe two lines D and E of RP2 while staying in projective correspondence, then mn envelops (that is to say, makes up the family of tangents to) a conic tangent to D and E. Otherwise expressed, if f is an hom*ography (projective correspondence) between two lines D and E of RP2 , then the lines xf .x/, as x traverses D, envelop a conic tangent to D and E. For example, if two points traverse two given lines of the plane R2 with constant speeds, then the line that joins them envelops a parabola.



Fig. IV.4.4. Construction of the center of the conic passing through a, b , c , d , e . II [B] c Nathan Édition Géométrie. Nathan (1977, 1990) réimp. Cassini (2009)

f (D) D x f (x) m

n D


Fig. IV.4.5. When two lines D and f .D/ describe two pencils while staying in hom*ographic correspondence, their point of intersection describes a conic. When two points x , f .x/ traverse, while staying in hom*ographic correspondence, the respective lines D and E, then the line xf .x/ that joins them envelops a conic

The above figure (right side) appears to “be in space”. There is a good reason for that, which will be explained in Sect. IV.10 with the aid of ruled quadrics. See Fig. IV.10.4, where we see the projection the apparent contour of a ruled quadric. We can easily prove Poncelet’s theorem for triangles (Figs. II.2.11 and IV.4.6 below): we take two conics C and C0 and seek a triangle inscribed in the one and circ*mscribed about the other. The answer is surprising: either there isn’t any, or there are just as many as there are points on C (we have however encountered the same phenomenon for circles that are tangent in succession to two circles; see Fig. II.2.7). The proof is trivial; we have: Œb; d; e; c D Œb; b 0 ; c 0 ; cC D Œd 0 ; b 0 ; c 0 ; e 0 . We will



see in Sect. IV.8 that the result is true for polygons with an arbitrary number of sides, but vastly more difficult to prove.

Fig. IV.4.6.

The duality (polarity) with respect to the conic C D p.Q1 .0// simply translates the fact that a quadratic form arises, by definition, from a symmetric bilinear form P, i.e. that Q.x/ D P.x; x/ for each x. Then the polar line m of a point m D p.u/ will be the projective line resulting from projection onto RP2 of the set fv W P.u; v/ D 0g of R3 . The word duality is sacred in mathematics; we find it in Sects. I.1, I.7, IV.4 and a large portion of Chap. VII, where it is unavoidable. As for figures, see those given for circles; there is no fundamental change. All the properties of points and their polars arise simply from the fact the P is a symmetric bilinear form. Don’t forget that, if m 2 C, then its polar line m is nothing other than the tangent to C at m; in particular, m 2 m . In fact, the points of C are characterized as those that belong to their polar. All the properties stated in Sect. IV.2, diametral and otherwise, are now trivial. The polar of a focus is the associated directrix. If P.u; v/ D 0 we also say that the associated points are conjugate with respect to the conic: if the line that joins them cuts the conic in two real points, we then have four points in harmonic ratio. There exists another duality, no longer between points and line of RP2 , but between lines (resp. points) of RP2 and points (resp. lines) of .RP2 / , that depends neither on a basic conic nor on a choice of coordinates (in what follows, we could replace RP2 D P.R3 / and .RP2 / D P..R3 / / by P.Q/ and P.Q /, where Q is a real vector space of dimension 3 and Q the dual). With each line d of RP2 is associated canonically a point d of .RP2 / . Geometrically: if d arises from the canonical projection p of a subspace of dimension 2 of R3 , then d arises from the line of .R3 / that is orthogonal to this subspace. In coordinates: for each system of projective coordinates in RP2 , there is a unique system of projective coordinates in .RP2 / such that the point d corresponding to the line d with equation ux C vy C cz D 0 has coordinates .u W v W w/. This duality extends to conics in the following manner. When d ranges over the set of tangents to the conic C of RP2 , the corresponding point d describes a



conic C in .RP2 / , called the dual conic. This is easily proved by using the matrix representation of the equation for conics (for the details, see Sect. IV.XYZ or 14.6 of [B]). Regarded as a family of lines of RP2 , the conic C envelops C. This is why we say that it is a tangent conic. We haven’t mentioned the duality between points of RP2 and lines of .RP2 / . It is easily derived from the fact that the lines of .RP2 / are called pencils of lines in another context and thus define a point of RP2 . We then show that as q ranges over the set of tangents to C , the corresponding point q describes the conic C in RP2 . In other words, we have .C / D C. Apparently then there is no difference between point conics and tangential conics. But pay attention that this is only true in the case of proper conics. It is thus that if the first case of degeneracy of a conic is a pair of distinct lines, for a tangential conic this degeneracy will become the set of lines passing through one or the other of two distinct points. This fact will be fundamental in Sects. IV.8 and IV.9; we will see there that the good notion that needs to be used for a conic is neither a set of points nor a set of lines, but rather the union of its set of points and its set of tangents. It’s a good rung up the ladder. By way of an example, in considering C we can define, for a proper conic C, the cross ratio of four tangents to C and show that it is equal to the cross ratio of the four points of contact.

Fig. IV.4.7.

More generally, if C and C0 are two proper conics, we may ask for the set of polars m of the points m of C, but with respect to C0 . The answer is that these polars envelop a third conic C00 , called the dual conic (or polar of C with respect to C0 . In fact, the preceding case of C , when RP2 is identified with .RP2 / by the mapping .a W b W c/ 7! .ax Cby Ccz D 0), corresponds to the case where the conic C0 is that with equation x 2 C y 2 C z 2 D 0, which as a set is empty and exists only in an imaginary fashion a good reason for complexifying everything in Sect. IV.6. IV.5. Klein’s philosophy comes quite naturally If a conic C and a projective line D are thus put in biunique correspondence, we can legitimately ask ourselves: what, mapped on C, are the hom*ographies (see Sect. I.6) of the projective line D? These are interesting transformations; they



form a group, and as this group (as is easily seen) is independent of the parameterizations used, we call it the group of the conic C. This should inspire us: we have succeeded in attaching, in canonical fashion, a group to a conic. We say no more, for three reasons: first a minor reason, which is that it is not our purpose in this book to treat all the properties of conics; for that see 16.3 of [B]. But beyond that there are two fundamental reasons. First, this presentation isn’t the right one it’s better to say that this group is that formed by set theoretic restrictions to C of the set of hom*ographies of the projective plane that preserve the set of points of C. We need to show (which is easy) that these two groups coincide. But, if our proper conic is defined by the quadratic form Q, then this group is clearly the group consisting of transformations of RP2 that arise from bijective linear mappings of E3 that preserve Q. Already it can be said that the duality with respect to a conic is completely described, coded, by the quadratic form Q; but now we must dare to say that the whole geometry of the conic is coded by the group of (linear) automorphisms of Q (only take care to quotient by multiples of the identity to be in the projective space). Now all the quadratic forms that we have to consider are isomorphic to Q D x 2 C 2 y z 2 . The linear group that preserves this form is denoted in general GL.2; 1I R/; here we have its projective version PGL.2; 1I R/. But in Sect. II.XYZ we already encountered quadratic forms of this type when we defined the hyperbolic geometries. In particular, here it’s the case of the hyperbolic plane and thus the group of a conic is isomorphic to the group of the hyperbolic plane, which we have denoted Möb.S1 / and called the Möbius group of the circle (see Sect. II.4). The action on the circle corresponds here to the actions on the conic Q D 0. The essential remark of Felix Klein from 1872 (when he was but 23) is that a geometry is characterized by its group of automorphisms. In particular, we don’t have to redo a theory every time: if we know hyperbolic geometry well, we automatically know the geometry of a conic well. Of course we must make up a dictionary, but subsequently the translation is automatic. Here is an element of it: the transformations that preserve the unit disk of E2 were generated by the inversions centered at the exterior of that disk. For the conic C, the dictionary translation of an inversion is what is called a hom*ology of a conic. A general hom*ology of a projective space is the transformation associated with a pair fp; Hg composed of a point and a hyperplane not containing that point (see harmonic hom*ology in Chap. II). At each point m distinct from p we associate the point of the line f .m/ which is such that the four points p; pm \ H; m; f .m/ are in harmonic ratio. If we have a conic, it will be preserved by all the hom*ologies of the form fp; p g where p is the polar of p with respect to C. The fixed points of this hom*ology are exactly the base point p and all the points (individually) of H. A consequence, for example, is that the group of a conic is generated by its hom*ologies. There exist numerous geometric analogies of this sort; it’s what had driven Felix Klein in his famous Erlanger Program in 1872 (see Klein, 1974, in French) to reduce in some way any sort of geometry whatever to the theory of groups, considering only those results interesting that are an “automatic translation” of a theorem of this or that “geometry”. We will amply see in our work that geometry in a still



Fig. IV.5.1. hom*ology conserving a conic. General definition of hom*ology

broader sense remains a fascinating workplace. But let us illustrate Klein’s principle with situations encountered in the course of Chaps. II and IV. The projective group PGL.3; 1I R/ of automorphisms of the quadratic form x 2 C y 2 C z 2 t 2 appear there three times: the first time in the hyperbolic geometry of dimension 3; then in the group Möb.S2 / of the sphere; or again the group of all the inversions of the Euclidean plane, made sufficient by adjoining a point at infinity; but it’s also the group of the space of circles in the plane (see Sect. II.6); finally, in the projective space RP3 , the proper quadric with equation x 2 C y 2 C z 2 t 2 D 0 will have an associated group. All four geometries are, however, isomorphic. Readers will be able to compose a dictionary, most likely incomplete, and use it to prove the things that interest them. This systematic association of a group with a geometry explains why the theory of Lie groups has been an essential subject for mathematics from the end of the nineteenth Century to the present day. It also explains the importance accorded the groups that preserve a nondegenerate quadratic form. It is why, in [B], the study of conics and quadrics is preceded by a chapter dedicated only to quadratic forms. A large part of the geometry of these forms, and thus of geometric objects that have this as their automorphism group, results from two essential theorems associated with the names of Witt and Cartan-Dieudonné. The latter theorem states that the group O.Q/ of a quadratic form in a space of dimension n is generated by hyperplane symmetries; better yet, each of its elements is the product of at most n hyperplane symmetries ([B], 13.7.12). Let us see what this theorem says in some particular instances. For Euclidean geometry, each isometry (preserving an origin) is the product of at most two symmetries with respect to lines. For the conformal group Möb.S1 /, each isometry of the hyperbolic plane is the product of at most 3 inversions (preserving the disk); but for a conic, this will be the product of at most three hom*ologies. For a quadric of RP3 , it will be a product of at most 4 hom*ologies; for the conformal plane it will be a product of at most four inversions and of five for the conformal group of three-dimensional space. All these results of products in precise number are helpful in the geometries studied, but their proof in the context of Klein is reduced to that of a single theorem, that of Cartan-Dieudonné. The refusal to see that we have an



explanation that is systematic and quick and that it is absolutely necessary to make use of it, and to want at all costs to do things by pure geometry, leads to the sort of aberrations already mentioned in Sects. III.1 and IV.2. For the conformal group, for example, we can refer to the long Note L of Hadamard (1911), already mentioned in Sect. II.7. It’s the moment to quote Yuri Manin: “a good proof is a proof which make us wiser”. And yet it turns out the theory of Klein with its groups doesn’t exhaust the possible geometries. If the examples above seemed to indicate that it does, it’s because the geometries considered have “big” transformation groups, whether the linear group for projective geometry or the entire group preserving a quadratic form. These groups act in a very overdetermined fashion; in particular they act transitively. Recall that the action of a group G on a space X is called transitive if, given p; q 2 X, there exists g 2 G that takes p into q, whereupon we say that X is a hom*ogeneous space of G. We then have X D G=H, where H is the isotropy group of a given point of X, i.e. the group leave this point invariant (if we change points we obtain an isomorphic subgroup). The affine and projective spaces, spheres, the hyperbolic plane, etc. are all hom*ogeneous spaces of their natural symmetry groups (for spheres this is the larger isometry group, or Möbius group). On the other hand, there exist on the sphere S3 and on RP3 which are groups in their own right geometries invariant only under the group, but with isotropy group reduced to the identity. Within a change of scale, these geometries depend on two parameters and can be very different, in particular from the canonical geometries on S3 and on RP3 (for which the isotropy group is O.3/ and thus has six parameters). The much greater portion of our book will study topics “beyond Klein” outside the Erlanger Program where the groups are most often nonexistent. IV.6. Playing with two conics, necessitating once again complexification We have seen that modulo a hom*ography there exists but one proper conic. What should we say of an attempt to classify pairs of conics? Let C and C0 be two proper conics of the (let’s say projective) plane: how do they intersect? We are interested only in those cases where there are points in common. To study them we solve simultaneously the two equations of C and C0 thus: we can always parametrize C as in Sect. IV.4 by .s 2 W st W t 2 / and substitute these coordinates into the quadratic form which is the equation of C0 . The result is thus a polynomial P, hom*ogeneous and of degree 4 in .s W t /. We call a pair .s W t / a root if it is a zero for P and we are interested for the moment only in the case where all four roots are real (note that this implies that if there is, for example, just a single root, then it must be or order 4). If they are distinct, we obtain below the anticipated figure; but we have in fact five possibilities for these four roots and their multiplicities: f1; 1; 1; 1g, f2; 1; 1g, f2; 2g, f3; 1g; f4g. What is the geometry of these situations? It is clearly seen in the figures; but why would we want to draw all these conics anyway, instead of just two? The reason amply justified in various contexts, of which we will see several is that for studying the pair of conics fC; C0 g there is great interest in considering the one



parameter family of conics whose equations are all the linear combinations of the Q and Q0 of C and C0 . Specifically, the conics p.Q C Q0 / for the pairs of reals f W g are the ones drawn below. Such a family is called a (linear) pencil of conics. All of these conics intersect two at a time in the same way, with the exception of those that are degenerate; see below.

Fig. IV.6.1. At right: the type f1; 1; 1; 1g; at right: the type f2; 1; 1g. II [B] Géométrie. c Nathan Édition Nathan (1977, 1990) réimp. Cassini (2009)

The case of double roots is easy to interpret geometrically; it signifies exclusively that the conics are tangent at the points in question. But the two cases f3; 1g and f4g are more subtle. In fact, these have been omitted in numerous textbooks that we will be so kind not to name. In the case f3; 1g we say that the conics are osculating at the point of order 3, and in the case f4g, we call them superosculating. In the language of differential calculus, this is to say that at the point in question the two conics have contact of order 3 (resp. 4) for mere tangency, the order would be 2. Geometrically, if we endow the affine plane with a Euclidean structure, being osculating is equivalent to the radii of curvature at the point being equal. For superosculation, it is necessary to use projective transformations; see 16.5 of [B] for all this. We extract figures as shown below, where we can read osculation and superosculation with collinearities. The next figure represents the trajectories of a small ball on the interior of a circle situated in a vertical plane, when these balls are launched from the lowest point (some readers may prefer the language of the simple pendulum, but with a string and not a rigid rod). The trajectory always begins with the circle on which the ball rests; then, at a point which depends on the initial velocity (if the speed is not too large), the ball leaves the circle and describes a parabola in free fall (in the figure we have subsequently forgotten the circle, but if the ball bounces on it, it’s a source of many amusing problems to know what happens at moments of successive rebounds; see Problem 17.9.2 of [B]. Where the ball leaves the cir-



Fig. IV.6.2. The types f2; 2g, f3; 1g and f4g. II [B] Géométrie. Nathan (1977, 1990) c Nathan Édition réimp. Cassini (2009)

Fig. IV.6.3. Different trajectories of a ball launched in a circle from the lowest point, with different initial speeds

cle is precisely where the circle and the parabola osculate; because the accelerations (which are of second order) coincide at this point, there will be third order contact. In algebraic language, osculation and superosculation are easy to describe: C and C0 will be of type f3; 1g if a quadratic form Q0 defining C0 is of the form Q0 D k QC D D0 , where D and D0 are two linear forms defining two lines. For superosculation, this will be Q0 D k Q C D D, where D is an equation of the tangent at the point in question. We can do unusual things with pencils of conics, but we will opt to stay with our guiding point of view, giving special mention to difficult results that require more abstract concepts or lead us to open problems. For more than what we can



say briefly, see [B] or Coolidge (1968). But the next item will serve us well in the sequel: the most important thing is first to look for degenerate conics in the pencil of the pair fC; C0 g. They can be seen in the figure: there are three pairs of lines, thus three such conics. Algebraically, these will be (within a scalar) the pairs f; g such that the quadratic form Q C Q0 isn’t of maximum rank 3. But that is to say that the determinant of Q C Q0 is zero (the determinant being zero is an invariant, even though the determinant itself isn’t). We thus find a hom*ogeneous equation in f W g of third degree and the figure shows that, in the case f1; 1; 1; 1g, it has three distinct roots (guess what it is in the other cases in the figures). In the case of four distinct points, we have a nice picture: C′

C p



Fig. IV.6.4. Pencil with distinct base points. The three degenerate conics of the pencil and the autopolar triangle

An important and even useful remark is that the figure shows the existence of an autopolar triangle simultaneously with respect to the two conics, i.e. a triangle such that the polar of each vertex is the side opposite the vertex. Notice that this autopolar triangle is common to each of the conics of the pencil generated by these two conics. The figure shows it in the case f1; 1; 1; 1g, where the triangle is p; q; r. Algebraically it’s easy; the geometric proof is the construction of the polar of a point with the complete quadrilateral. We can’t but mention, because of its elegance, the fact that the cross ration of four points common to one of the conics, is equal to the cross ratio of the four tangents to the other conic. We can also investigate how the conics of a pencil intersect a fixed line D. The answer is that these pairs of points are in involution, i.e. that there exists an involutive hom*ography of the line that sends the one onto the other; see Sect. I.XYZ. In



particular there exist two conics of the pencil that are tangent to D (two at most in the real case). All this can be seen by replacing, in the equation of the conic, the coordinates by a (linear) parametric representation of the line; the pair f; g defines a conic tangent to D when this equation has a double root, which is a condition of second degree. We see right away what is the right interpretation of a pencil of conics. Finally, we can ask about the morphology of the pencils of (real) conics that aren’t of the types studied above, i.e. where the points of intersection aren’t all real. Such a classification can be found in Levy (1964). IV.7. Complex projective conics and the space of all conics We have seen in several places the necessity of complexifying; we thus now define a conic by the definition that was parachuted in Sect. IV.1: A conic is a curve in the complex projective plane, i.e. the set of .x W y W z/ that satisfy an equation P.x; y; z/ D 0, where P is a hom*ogeneous polynomial of degree 2 in the three variables x, y, z (and not identically zero). But over the complex numbers, there is but a single invariant for quadratic forms, their rank. Thus there aren’t more, modulo a projective transformation, than three types of conics, specifically those defined by the equations: x 2 C y 2 C z 2 D 0, x 2 C y 2 D 0, x 2 D 0. i.e. the proper conics, the conics that degenerate into two lines, and those that degenerate into one double line. Of course, this has to do with complex projective lines. Here we may seek to “see” things, but CP2 is actually an object of real dimension equal to 4, which furthermore has a complicated topology. Intrinsically, a conic is a surface (in the real sense). We generalize all of what was said in Sect. IV.4; for example, each proper conic is in bijection with a line and we have a notion of cross ratio for four points of a conic. Since a projective line has the topology of the sphere S2 , a proper conic has thus the topology of a sphere. We always have parameterizations of second degree, such as for example .s W t / 7! .s 2 W st W t 2 /. But all this is not the most important issue: we always work with essentially geometric language, but visualizing isn’t always necessary or possible; we must think by analogy. All this, by the way, in fact depends on ones psychology. Here then, in cascade, are two essential facts. The first is that the set of points p.Q1 .0// in CP2 determines the quadratic form Q within a scalar multiple. Of course, two proportional quadratic forms define the same conic, but the converse is totally false in the real case, where all the positive definite forms correspond to the empty set. The proof is practically trivial for conics. Its essence is that an equation of second degree (with coefficients in C) is determined within a scalar multiple by the values of its roots. This easy uniqueness is in fact a particular case of the very general theorem that is basic for algebraic geometry, namely, the “Nullstellensatz”; see Sect. V.13.



But “within a scalar multiple” makes us think of projective spaces. Now, the set of all quadratic forms Q on the vector space C 3 in fact forms a vector space over C under addition and scalar multiplication; its dimension is 6, since there are exactly six coefficients: Q.x; y; z/ D ax 2 C a0 y 2 C a00 z 2 C 2byz C 2b 0 zx C 2b 00 xy. Finally, the preceding can be reformulated as follows: the set of all the conics is in bijection with the complex projective space of dimension 5; we use the notation CC P D CP5 (space of complex projective conics). We will see in Sect. IV.9 that this structure is unavoidable for certain problems, and we can already put it in place, install it on the present rung of Jacob’s ladder. For example, the condition that a conic pass through a given point is linear, but the set of conics passing through a given point in CCP is a projective hyperplane. It’s hardly astonishing that there is one and only one conic that passes through five given points (in general position). Pay attention, though, that such hyperplanes are special in C C P , if only because they depend on two parameters (since they are points of CP2 ) and therefore the hyperplanes depend in all generality on 5 parameters. The geometric nature of the conics of a general hyperplane of C C P is described in 14.5.4 of [B]; see Sect. IV.9 below for the condition of “being tangent to a given conic”. Now a pencil of conics, if written p.Q C Q0 /, will thus be nothing other than a projective line of C C P . If for example we seek degenerate conics, we can say that they are the points where this line intersects the subset C D of C C P formed by the degenerate conics. As the determinant (within a scalar multiple) of a quadratic form is of third degree (in its coefficients, for example), in the language of algebraic geometry we have that C D is a hypersurface of third degree of C C P , and thus a line intersects it in three (generally distinct) points; we will see a good bit more in Sect. IV.9. The classification of the pencils in C C P is identical to that given in Sect. IV.6: morphologically there are exactly five types, but clearly the five figures given below are but a nice support for the mind, and indeed also an aid for certain reasoning. We must take care not to fall into the following trap: the classification of pairs of conics is finer than that of pencil types. In the case of four common distinct points a, b, c, d a pair fC; C0 g is known modulo a projective transformation if the two cross ratios Œa; b; c; d C and Œa; b; c; d C0 are given (quite an old result). To prove it, we remember from Sect. I.XYZ that the group of hom*ographies is transitive on quadruples of points. But since a conic passing through four points is exactly described as the locus of a point such that the pencil of lines joining that point to four given points has a given cross ratio, we’re done. The author has not failed to fall into the trap of false classification by type, but was saved in time by Jacques tit*. The assertion 16.5.1 of [B] is true, for it is addressed to the set of conics of a pencil and not to the pair that generate it. There is moreover a good reason encountered in the next section for two pairs of conics not to be projectively isomorphic: it’s the existence or nonexistence, for each pair, of polygons of N sides (N fixed) inscribed in the one and circ*mscribed about the other.



In the other direction for the types f3; 1g, f4g there isn’t any invariant and two pairs of the same type are always projectively “the same”. For the types f2; 1; 1g and f2; 2g, there is a single invariant, which is not immediately identified from C and C0 . On the other hand, this invariant on the projective line that defines in CC P the lattice constructed with fC; C0 g is nothing other than the cross ratio of points represented by C, C0 and the points represented by two degenerate conics of the pencil. We can thus find more easily the geometric definition of this invariant. When C and C0 are transformed projectively into two circles of the Euclidean plane E2 , then in the case of type f2; 1; 1g we obtain two tangent circles and the invariant sought is the ratio of their radii; whereas for the type f2; 2g we obtain two concentric circles and the sought-for invariant is the square of the ratio of their radii! As promised in Sects. II.1, II.XYZ and IV.2 we now explain the principle that permits recovery of all the metric geometry of the Euclidean conics, which is the philosophy of Plücker . Recall that (see Sect. II.XYZ) we can embed the Euclidean plane E2 in CP2 by first projectifying first and then complexifying. There is the canonically attached pair fI; Jg of points of CP2 , called the cyclic points of E2 (even though they are in CP2 ). We also note that a conic C in E2 can be embedded canonically in a complex projective conic C in CP2 . With this notation, we interpret first the circles: they are exactly the conics passing through the cyclic points, i.e. C is a circle if and only if fI; Jg C . The center of the circle is the pole of the line at infinity; two circles will thus be concentric if and only if they are bitangent in ŒI; J. But above all the angular property of the points of a circle (seen in Sect. II.1) result similarly from Laguerre’s formula: angle.D; D0 / D

1 j log.Œ1D ; 10D ; I; J/j 2

and the constancy property of the cross ratio seen in Section 4: Œa; b; c; d C D Œma; mb; mc; md : The drawings below are just an absurdity that can be helpful in certain arguments.

Fig. IV.7.1.



Fig. IV.7.2.

For general conics, we fist need to find the foci: these will be the points (again an “absurd” figure) f such that the pair of tangents to C emanating from f contain the pair fI; Jg. There are thus four foci, but only one pair “is real”. For an a priori object in CP2 to be real means that it is in fact in RP2 . It can be seen in 17.4 and 17.5 of [B] how to recover all these metric properties of C by regarding C CP2 and by utilizing the projective properties and the duality for conics. The directrix of a focus is its polar (if necessary, restricted to the Euclidean plane).

Fig. IV.7.3. At left, pencil of conics passing through 4 cocyclic points. At right, construction of the intersection of the axes of the parabolas of the pencil; the directions are obtained by considering one of the degenerate conics of the pencil, formed by two lines

With this sort of consideration using cocyclic points and using appropriate pencils we obtain some nice Euclidean theorems, such as: through four cocyclic points of a Euclidean plane E2 there pass two parabolas with orthogonal axes; moreover, these axes pass through the center of mass of the four points (see [B], 17.5 for the details and for other results). More generally, the points of intersection of two conics are cocyclic if and only if these conics have parallel axes of symmetry: this is seen by regarding how the points where these conics intersect the line at infinity of CP2 are situated with respect to cocyclic points. We stop here; the preceding shows how the great majority of classic results on conics can be obtained in a unified way



with the three rungs of the ladder: projectify, complexify and finally introduce the projective space of all conics. In return, we are going to dwell further on two theorems, simultaneously for their elegance, their difficulty, their historical importance and the fact that they are going to force us to climb still higher on the ladder. IV.8. The most beautiful theorem on conics: the Poncelet polygons

Fig. IV.8.1.

We consider first the case of two real affine conics C and C0 such that C0 is entirely interior to C. The result of Poncelet is of the type “all or nothing”, like Steiner’s porism in Sect. II.2: For each integer N > 3 either there exists no polygon with N sides inscribed in C and circ*mscribed about C0 , or else there exists an infinity of them; furthermore, we can take any point of C as initial vertex. It is natural to believe to the contrary that only certain points, chosen carefully, could be the vertices of such a polygon. This is what happens for pairs of “ordinary” curves: first it never works for all N; next if it works at all it’s only for a finite number of polygons, but not with the right to choose an arbitrary point of C for a vertex. We have already encountered such a theorem Steiner’s in Sect. II.3. But we remarked in Sect. IV.1 that there does not in general exist a projective transformation which takes the pair fC; C0 g into a pair of concentric circles; if such were the case, we’d be through, for such a transformation preserves lines and properties of tangency. And now we know, after the preceding section, the profound reason: two concentric circles are bitangent, while a general pair fC; C0 g of conics intersect in four points (more precisely here, the complexified-projectified pair fC ; C0 g). This theorem of Poncelet for polygons has a double attraction, which is why numerous great mathematicians have been interested in it, right up to the present as we shall see and this interest is likely to persevere. The first attraction comes from the elegance of the result, but the second is linked to the first, i.e. the difficulty of its proof (in contrast to Steiner’s porism, which is proved by an inversion). Next, there is the challenge of finding, among the equations of conics, the condition as a function of the integer N for the existence of Poncelet polygons with N sides. Since we too have been seduced by it, and for a longish time, we are going to dwell a while



on this topic. Readers will have noticed that there is no reason to limit ourselves to the real case and that all the complex conics, affine and projective, are also amenable to such a result, likewise even the conics over arbitrary commutative fields, since the notions of conics, lines and tangency remain valid (for tangency, the purely algebraic language of double root of an equation applies). As for references beyond those that will be mentioned here and there in the sequel, the basic one is Bos et al. (1987). Just a bit of history however: above we proved the result for triangles, a result which for circles dates to William Chapple in 1746 this for the statement of the “all or nothing” type but the proof is inadequate; see more in Bos et al. (1987), and likewise for the case of quadrilaterals. This last case is easily treated with polarity and pencils of conics. The interest in the Poncelet polygons appears, for example, as a typical application in treatises on elliptic functions: Appell and Lacour (1922) mention them only briefly, whereas (Halphen, 1886, 1888), in Chap. X of Volume II, treats the problem in great detail, including the formulas of Cayley that we will see further on. But in fact history connects elliptic functions in a more essential fashion with Poncelet’s theorem, i.e. that Jacobi by 1828 had confirmed a direct link between it and the addition and associativity formulas for elliptic functions; see Sect. IV.5 of Bos, Kers, Oort, and Raven (1987), also Hrasko (1999) and Hrasko (2000). Here first is a proof in the real case a Euclidean proof in fact. We show first, without much difficulty, that a pair of conics as above can be transformed projectively into a pair of circles, one interior to the other. We can do it by hand with the geometry of E3 and conic sections, but also, by the way, in CP2 using the cyclic points. It remains then to prove the theorem for two circles C and C0 . To our knowledge there doesn’t exist any relatively short proof that uses only what has been said so far about the projective theory even the complex version of conics and pencils. The best proof we know has been taken from Shen (1998), but we know that it was already in the air (in this explicit form) for several years, and in essence known already by Jacobi. To each point m 2 C we attach its tangential distance L.m/ to the circle C0 . Then we endow the circle C with a measure different from the canonical Euclidean measure ds, but which is locally proportional to it, specifically ds . We denote by L F the mapping C ! C of the figure (it needs to be decided in which sense to traverse C). The basic lemma is that F preserves the measure ds (if we want to avoid L the specialized language of “measure”, we can use only the notion of integral). If s designates a parameterization by the arc length of C (see Sect. V.6), then for each interval Œa; b we have Z b Z F.b/ ds ds D : L.s/ L.s/ a F.a/ The verification requires just a little differential calculus, based on two essential facts: on the one hand, the line joining m 2 C to its image F.m/ cuts the circle C0 at equal angles; on the other hand, the two tangents issuing from a point of a circle are always of equal length. In particular, if we reparameterize C with a parameter t



with the aid of this new measure, the mapping F becomes a translation t 7! t C k. But if we denote by ƒ the new total measure of C, we have a polygon of N sides in the pair fC; C0 g if and only if Nk D ƒ. Now this condition is independent of the point (of parameter t) chosen initially, Q.E.D.

Fig. IV.8.2.

Note the ascent of the ladder with the notion of measure. In fact we could do much better, as Poncelet himself did, and we will be able for example to know what happens with the sides of the star polygons associated with a family of Poncelet polygons, or indeed with the diagonals when n is even! Poncelet in fact proved the following lemma, which we call the general lemma in the sequel: Let C00 be an arbitrary circle (interior to C) and belonging to the pencil of circles defined by the pair of circles fC; C0 g. We traverse all these circles in the same sense, and with each m 2 C we associate the point n where the tangent issuing from m to C0 encounters C again; then from n we follow the tangent to C00 which cuts C again at p. Then the line mp, when m traverses C, envelops a circle C000 belonging to the pencil considered.

Fig. IV.8.3. The general lemma. At left: case of circles. At right: case of ellipses, applied to various diagonals of the polygons. II [B] Géométrie. Nathan (1977, 1990) c Nathan Édition réimp. Cassini (2009)



This lemma implies everything we want. First, the result on the polygons of the pair fC; C0 g: for if we apply the result several times in succession, we see that if we start off from a point of C, at the end of N times we have a line mn that envelops a circle. But if the polygon closes once, this circle will then have N points in common with C (in fact 2N because things get counted twice); it is thus C itself. But by applying again the iterated lemma to a Poncelet polygon, we see that the star polygons of arbitrary type fN; kg (N vertices, k turns) remain continuously circ*mscribed about a circle of the pencil in question. In the case where N is even, the diagonals thus pass through a fixed point. Needless to say that all this carries over to pairs, and to pencils, of real conics. As for the proof of the lemma, it suffices to know this: the tangential distances from a point of C to two circles C0 and C00 of a pencil containing C are in a constant ratio. To see this, it’s simplest to remark that the square of the tangential distance of m to C0 is equal to Q0 .m/, the value for m of the quadratic form Q0 that defines the circle C0 (Q0 must be normalized to start with by x 2 C y 2 ). Now membership in a pencil is a linear condition, and if m traverses C, then Q.m/ D 0 if Q is the normalized quadratic form defining C. Then if the tangential distances L.m/ and L0 .m/ are proportional, the associated measures ds and ds , when normalized, will L L0 coincide. So, with the right parameterization, the chord mn will correspond to a translation of the parameter, and each translation generates an envelope which is a circle of the pencil (proceed in the reverse sense and by uniqueness). All this seems to tell us nothing about the case of complex conics. In fact our whole presentation is not geometrical, but purely algebraic and, above all, the operations performed are all algebraic (construct a tangent, intersect a line). We could appeal to the extension principle for algebraic identities. Briefly: the algebraic entities (relations, identities) true over the real field R remain true over the complex field C; see Bourbaki (1981), Chap. IV, § 2, Section 3, Theorem 2. There is here again an ascent of the ladder. In contrast, in the nineteenth century, Poncelet and many others appealed to the continuity principle, but that was more philosophically than solidly founded. Whether with regard to Poncelet’s theorem or to other results, due to Chasles and many other geometers including Plücker and Steiner, the quarrel over the continuity principle raged, sometimes rather violently, throughout a good part of the nineteenth century; see the references and citations in 7.0 of [B]. In any case, we can now consider Poncelet’s theorem as being achieved in its more general form over the complex numbers and in consequence return, as desired, to the real case for circles or, more generally, intersecting conics, e.g. an ellipse and a hom*ofocal hyperbola. The preceding does not answer the question about Poncelet’s theorem for the conics over an arbitrary (commutative) field, finite for example; nor the question of an explicit, and much wanted, condition in N on the equations of conics. In 16.6 of [B] there is an elementary proof using only the projective geometry of conics over an arbitrary commutative field, and thus valid over all commutative fields; but it is rather long.



If we restrict ourselves to the “all or nothing” theorem as it is rightly called for a pair of conics, then there exists an ultra-rapid algebraic proof based on the socalled theory of algebraic correspondences. This method consists of parameterizing C to the second degree with, say, one parameter: let m.t / be the point of C with parameter t and note that the condition for the line m.t / m.t 0 / to be tangent to C0 is of second degree simultaneously in t and in t 0 . More generally, if we construct the broken line starting from m.t / that has N sides successively tangent to C0 , the final point m.s/ is such that the relation between t and s is of second degree in t and in s. The difficult part of the proof is that this iterate Q ı Q ı ı Q of the original correspondence Q is again of second degree. To say that the broken line closes up is to imply that this relation (of fourth degree then in t ) is satisfied for a certain t ; but it is also for N 1 (N > 3) values, to which we need to adjoin the points of C \ C0 . The fact that this relation is algebraic (polynomial) of type f2; 2g and that it is zero for more than four values implies then that it reduces identically to zero. It is thus satisfied for all t . Here now is the proof of Poncelet’s theorem which goes furthest toward getting to the bottom of things; we give a completely modern version; but, in essence, it is that of the theory of elliptic functions, known since Jacobi in the nineteenth century. We partly use the equivalent language of elliptic curves, specifically of cubics (see Sect. V.14). The fact is that these have a group structure and that the elliptic functions provide them with parameterizations (not at all algebraic). Here is the most rapid formulation there is, laid out here for the complex domain. To the extent that the theory of elliptic functions remains valid over other fields, what follows is more generally applicable. We begin with two conics C and C0 for which the four points of intersection are distinct. The subtle idea which follows consists in doing what is necessary for avoiding the double event that from a point of C two tangents depart, and that one tangent to C0 cuts C in two points (this had been avoided in the case of real circles by orienting the circle of departure). We introduce the algebraic set that is the product C C0 , where C0 denotes the dual conic of C0 (i.e. the set of its tangents). In this product the ad hoc object is the subset D f.m; D/ W m 2 Dg C C0 that is formed of pairs of a point of C and of a tangent to C0 such that m 2 D. What is this object? The fact that the tangents to C0 at the points of C \ C0 are distinct implies that is a curve without singularities in C C0 . The first coordinate of the mapping ! C is a covering of two sheets, except at the four points of C \ C0 , which are of simple ramification of order 2. The elementary geometry of Riemann surfaces shows that the topology of is that of the torus T2 (said to be of genus 1). It’s thus what is called an elliptic curve (complex, on C) and it is classic that it possesses a group structure (once an identity element is chosen anywhere on the curve; this is amply treated in Sect. V.14). The group is written in additive form.



Fig. IV.8.4.

Now Fig. IV.8.4 defines two involutions ; ı W ! on , specifically .m; D/ D .m0 ; D/ and ı.m; D/ D .m; D0 /. The construction of a polygon of Poncelet type starting from an .m; D/ consists of iterating the composite operation ı ı . The polygon departing from the pair .m; D/ closes on itself at the end of N times if and only if .ı ı /N .m; D/ D .m; D/. But the elementary theory of elliptic curves assures that an involution having at least one fixed point is of the form x 7! x C a (for an appropriate a), thus the product ı will be a translation x 7! x C k. Thus the polygon closes if Nk D 0. But oh miracle! this condition is intrinsic, it does not depend on the point of departure, Q.E.D. There remains the problem of finding the condition leading to the equations of C and C0 and the integer N which will tell if the pair fC; C0 g admits a Poncelet polygon with N sides (or vertices). It’s Cayley who gave the answer in (Cayley, 1861); here we explain his solution following the exposition of Griffiths and Harris (1978a), where by the way the presentation just given figures too. We need to find a representation of the elliptic curve that is explicit as a function of the equations of C and C0 and where we can also make explicit the operation ı ı . The ingenious coup of Cayley is first to find a cubic associated with a pencil of conics and the connection with its group law. In fact, Cayley introduced two cubics. The first, not really unique (but this is not important), consists specifically of a point p being fixed in the plane, taken somewhere on the initial conic C; the locus of the points of contact of the tangents issuing from this point to the conics of the pencil considered is a cubic (without double point). Let us say it has to do with Cayley’s “concrete” cubic, and the formidable connection between this cubic and Poncelet’s theorem on triangles, which we have called in the case of a circle pencil the general lemma, is going to be precisely this. We recall the general lemma: Let a pencil of conics, generated by the two conics C and C0 , be given. If a triangle inscribed in C has its first side tangent to C0 and its second side tangent to another conic C00 of our pencil, then the third side is tangent to a third conic C000 of the same pencil. The connection discovered by Cayley is that, among the six points of contact of the tangents issuing from p to these three conics C0 , C00 , C000 , at least three are collinear, which gives us the group law of this cubic (see Sect. V.14). And all at once



Fig. IV.8.5. In the figure we see two drawings of the first cubic of Cayley in the case of a pencil of circles. In the first of these we see the cubic and the pencil. In the second we see the triangle inscribed in the first circle, then the three collinear points corresponding to a tangent to three circles, issuing from the point E which defines the cubic, for the three circles of the pencil which are tangent to the three sides of the inscribed triangle ABC considered

there appears the notion of a point of order N on this cubic when we have a polygon with N sides inscribed in C and circ*mscribed about C0 . To see this it suffices to decompose the polygon into triangles, but this does not easily allow the calculations for finding the condition between N and the equations of C and C0 . To find it Cayley introduces a second cubic an abstract cubic specifically p the elliptic curve (Riemann surface) † defined by the algebraic function y D det.x Q C Q0 /, i.e. the curve y 2 D det.x Q C Q0 / of CP2 . Here Q and Q0 are the quadratic forms defining C and C0 , and the determinant can be taken (for example) in the canonical basis of C 3 . Note that det.x Q C Q0 / D 0 possesses three distinct roots fxi g (i D 1; 2; 3) corresponding to the three degenerate conics of the pencil defined by C and C0 ; also we denote by fmi g (i D 1; 2; 3; 4) the four points of C \ C0 . Finally, let Cx be the conic of the equation x Q C Q0 D 0 (with x different from the xi ). Now we define a mapping † ! thus: with an x (different from the xi ) we associate the point m of C different from the m1 , where the tangent at m1 to the conic Cx intersects the conic C. We then choose any point of which projects onto m. Then, since we are on coverings (ramified, to be sure), we extend without difficulty the mapping x 7! analytically to a mapping † ! , complete since the ti correspond correctly to the mi and x D 1 to the point m0 . We verify finally that we have a good isomorphism between elliptic curves.



Fig. IV.8.6.

On † we thus see, if we take the point x D 1 as identity element of the group law, that the operation ıı corresponds to the translation which goes from x D 1 to x D 0. In this universal description, we must thus be able to see whether N D 0 in the single expression for det.x Q C Q0 /. It’s a problem on elliptic curves and we resolve it with a theorem of Abel; it’s reduced to finding the points of hyperinflection of the so-called normal elliptic curve traced in CPN1 with the aid of derivatives up to order N 1 in the parameterization by x. We have thus to write that a determinant of order N is zero. The corresponding derivatives are given by the development in a series det.x Q C Q0 / D a0 C a1 x C a2 x 2 C a3 x 3 C and the conditions obtained by Cayley are thus the vanishing of a determinant: ˇ ˇ ˇ a2 : : : amC1 ˇ ˇ ˇ ˇ :: :: ˇ D 0 if n D 2m C 1 ˇ : : ˇˇ ˇ ˇamC1 : : : a2m ˇ and ˇ ˇ ˇ a3 : : : amC1 ˇ ˇ ˇ ˇ :: :: ˇ D 0 ˇ : : ˇˇ ˇ ˇamC1 : : : a2m ˇ

if n D 2m:

Readers will find it instructive to calculate several of these conditions for the case of two circles. Let R and r be their respective radii and d the distance between their centers. By 1827 Steiner had given the conditions n D 3 R2 a2 D 2rR n D 4 .R2 a2 /2 D 2r 2 .R2 C a2 / n D 5 r.R a/ D .R C a/Œ.R r C a/.R r a/1=2 C.R C a/Œ.R r C a/2R1=2 2 2 4 2 n D 6 3.R a / D 4r .R2 C a2 /.R2 a2 /2 C 16r 4 a2 R2 2 n D 8 8r 2 .R a2 /2 r 2 .R2 C a2 / ˚

2 .R C a2 / .R2 a2 /4 C 4r 4 a2 R2 8r 2 a2 R2 .R2 a2 /2 2 2 D .R a2 /4 4r 4 a2 R2 : For n D 5, 6, 8, Steiner gave no explanation! We therefor don’t know how he obtained these formulas. Moreover his formula is false for n D 8, as may be seen in Pécaut (2000) which is a novel and elementary treatment of Poncelet’s theorem (and where the correct formula for n D 8 and others for higher degree can be found).



But all this says nothing for fanciers of explicit formulas about how to calculate the coefficients ai above. On p. 609, Vol. II of Halphen (1886, 1888) there are double recursion formulas of the “continued fraction” type (see Sect. IX.1.A for this theory), which seem to us very complicated compared to those in the pair f˛; g that are to come below (and which are already complicated enough). The preceding easily permits us to recover the general lemma of Poncelet. In fact it suffices to know how to compare the elliptic curves C \ C0 and 0 C \ C0 corresponding to the pairs fC; C0 g and fC; C00 g, where the third conic C00 belongs to the pencil defined by C and C0 . But in fact we have just seen that is isomorphic algebraic function p to the elliptic curve which0 is the Riemann surface † of thep det.x C C C0 / (and thus is isomorphic to the surface of det.x C C C00 /). But, as C00 D a C C C0 , we see that there is a change of parameter x to be made, whence the general lemma of Poncelet by reasoning as above. There is also something else we might desire (mathematicians being insatiable): a pencil based on C being given along with an integer N, to know how many conics there are in the pencil that furnish Poncelet polygons with N sides. This problem is treated in Barth and Michel (1993). It is in fact a problem in number theory since an elliptic curve is isomorphic to the quotient of R2 by an appropriate lattice ƒ: we need to count how many of the elements v of R2 are such that Nv D 0 (modulo ƒ), but while paying attention to counting only the “primitive” (or “primary”) elements, i.e. to eliminate those of the form kv which correspond to a polygon traversed several times. Readers may have sensed felt a small scandal. If the formulas of Cayley are a technical tour de force typical of the nineteenth century and in the spirit of “analytic geometry” it should perhaps be asked why we shouldn’t look for the existence, for C and C0 , of polygons with N sides inscribed in C and circ*mscribed about C0 as being among the invariants that classify such pairs of conics under the group of hom*ographies, in particular the pair of complex numbers f˛ D Œa; b; c; d C ; D Œa; b; c; d C0 ; see Sect. IV.7. We are thus here more in the style of “geometry” than “analytic geometry”. In Halphen (1886, 1888), Vol. II, p. 377, can be found the formulas with this exact notation in f˛; g that give, for each N, the relation imposed between ˛ and , again where these formulas are obtained through recurrence relations (building on the two preceding terms like the archetypical Fibonacci sequence defined by the relation uN D uN1 C uN2 ). It’s difficult not to give the relation for N D 4, for it is ˛ D 2 , which we can prove in a completely elementary fashion. For N D 3, it is already a lot less amusing: ˛ 2 2.2 2 3 C 2/˛ C 4 D 0. For this case and the general case, we need essentially the theory of elliptic functions. We define first two numbers x and y by: xD

Œ˛ 2 2.2 2 3 C 2/˛ C 4 3 28 ˛ 2 .˛ 1/2 4 . 1/4



and yD

. 2 ˛ 2 /. 2 2 C ˛/. 2 2˛ C ˛/ : 23 ˛.˛ 1/2 2 . 1/2

Then the conditions are: x D y for N D 5, y D x C y 2 for N D 6, .y x/x D y 3 for N D 7 and .y x/.2x y/ xy 2 D 0 for N D 8. The relations between x andy are defined step by step by a recurrence relation. The true space of pairs of conics is not the set of f˛; ˇg C 2 , but precisely the sets f0; 1; 1; ˛; ˇg of five different points of CP1 D S2 , a set that plays an important role in physics. We may well ask whether there exist generalizations, variants in other contexts, for example in the space of three dimensions, for the polygons zigzaging between the generators (see Sect. IV.10) of two quadrics or with certain circles, etc. In Barth and Bauer (1996) there is not only a very long list of theorems of “Poncelet” type,

Fig. IV.8.7.



but a very general scheme. In particular, there is a modern treatment of the result of Emch (1901), which encompasses both Steiner’s theorem and that of Poncelet. This result is the one of the figure below; to deduce from it the result of Poncelet, take a circle reduced to a point; then, by an inversion, all the circles passing through that point become lines and form the desired polygons. In Emch (1900) there is a mechanical realization of all that the manuals for producing these mechanisms with an Erector or Meccano kit or something similar. This fabrication is done with parallelograms (deformable) linked to one another (see the figure). These articulated systems will be encountered briefly in Sects. V.16 and VIII.3. For an elementary treatment of Poncelet and zigzags, see Hrasko (2000) and Pécaut (2005). The Poncelet polygons continue to fascinate mathematicians; they are found in foldings of polygons; see Sect. VII.4 and Benoist and Hulin (2004). See also Barth and Michel (1993). But Schwartz, (2001) (unpublished) seems to us to be an astonishing revival. In Berger (2005) there is an explanation for the magnificent figure of Schwartz below:

Fig. IV.8.8.

IV.9. The most difficult theorem on the conics: the 3264 conics of Chasles The problem is to find the number of tangent conics to five given conics. We won’t repeat again “in general position” after this. The notion can be made precise, but that is not our mission here; we leave it to readers to absorb it and to do more if they are curious. We have seen in Sect. II.1 that there are as many as eight circles



Fig. IV.9.1.

tangent to three given circles, but this may be interpreted in CP2 as the search for conics passing through the two cyclic points (see Sect. II.XYZ) and tangent to three conics that have these two cyclic points in common. From the viewpoint of conics, this is thus not at all a general result. This problem of five conics has a long and fascinating history. It seems to be the only problem on conics that resisted beyond 1900; in particular it’s an implicit part of the 15th Hilbert problem. Moreover, it has been one of the principal motivators in the development of the fundamentals of algebraic geometry, for to be well understood it requires a solid foundation of intersection theory. We will find in Sect. V.13 what needs to be understood and the importance of such a theory in numerous mathematical domains other than algebraic geometry itself. For this reason and for the beauty of the problem in its own right we are going to describe a little of its history. For more details, see Kleiman, 1980) and the introduction of Ronga, Tognoli, and Vust (1997). We place ourselves exclusively in the complex projective domain, i.e. in CP2 see at the end for the real case but we can surely work with other base fields. In 1848 Steiner (always the same one) announced that there exist 7776 conics tangent to five given conics (in general position). It is in some way the most difficult problem that bears on the conics, and it is normal to impose five conditions to determine one conic, since the space they form is of dimension five. We have seen that five points determine a unique conic; and that four points and one line determine two of them, those that pass through these points and are tangent to the line. The reason is that, in the space CCP of all the conics, to pass through a point is a linear condition, having to do with a hyperplane of C C P . To be tangent to a line is an equation of second degree: to see this we substitute for the values of .x; y; z/ into the equation Q.x; y; z/ D ax 2 C a0 y 2 C a00 z 2 C 2byz C 2b 0 zx C 2b 00 xy of the unknown conic by a linear parametric representation in .s; t / of the line. We obtain an equation of second degree in the six coefficients of C. The tangency between the line and C requires that this equation have a double root, which yields a condition of second degree in these coefficients. When the line is given, the set of conics that are tangent to it constitutes a hypersurface S of degree two in C C P , what is called a quadric of CC P (or better here, a hyperquadric; see Sect. IV.10). Finally, the conic we seek will be formed by the points of intersection of four hyperplanes and this quadric S.



Without appealing to general theorems, we will be intersecting S with the line of intersection of the four hyperplanes, which in general yields two distinct points. Steiner generalized this general vision as follows: it has first to do with knowing which set CT contains the conics of C C P that are tangent to a given conic C. We can always write C in the form xz y 2 D 0, and thus parameterize it by .s 2 ; st; t 2 /. We substitute these values in Q.x; y; z/ D ax 2 Ca0 y 2 Ca00 z 2 C2byz C2b 0 zx C2b 00 xy and find thus an equation in .s; t / of degree 4. There will be tangency if this equation has a double root, which is classically expressed by its discriminant, which is a polynomial in the coefficients of degree 4 3 D 12. In fact, the true degree is only 6. Here’s why: it suffices to remark that our two conics with equations Q and Q0 will be tangent if the equation det.Q C Q0 / D 0, of third degree in which furnishes degenerate conics of the pencil that they determine possesses a double root (look at Fig. IV.6.1). This time the discriminant of this equation is of degree 3 2 D 6 (in the coefficients of the equation) and we have only to verify that this discriminant also remains of degree 6 with respect to the coefficients of Q (make the calculation with Q0 D x 2 C y 2 C z 2 ). For example, there will be, in general, six conics passing through four points and tangent to a given conic:

Fig. IV.9.2.

The tangent conics to five given conics Ci (i D 1; :::; 5) will thus be the points of intersection of 5 hypersurfaces CTi of degree 6 of C C P . The general theorem of Bézout (see any fairly recent book on algebraic geometry) states that there are 6 6 6 6 6 D 7776 points common to these five hypersurfaces. In 1859 de Jonquières criticized the reasoning of Steiner and found the right number: 3264. But he didn’t publish his result, for the stature of Steiner’s geometry in the epoch intimidated everyone. Chasles didn’t believe the reasoning of Jonquières and published his own proof in 1864. But a complete proof needed to wait for the XXth Century. The fact of a hypersurface of degree 6 had been made rigorous by Bischoff in 1866, but it was the number 7776 that caused the problem. In fact, the following objection was raised to Steiner’s reasoning: Steiner’s



Fig. IV.9.3.

proof led to finding 25 D 32 conics tangent to five given lines, whereas it is well known that there exists but one of them, thanks to duality (see Sect. IV.4). Where is the contradiction? The reason wasn’t found until 1864, by Cremona: every conic that is a double line must be considered, in the algebraic sense, as tangent to whatever conic; thus all the hypersurfaces of C C P of this type contain the submanifold V of CCP formed of double lines, i.e. all quadratic forms of rank equal to 1. It is interesting to remark that it is a submanifold isomorphic to the set of lines of CP2 , itself isomorphic to CP2 . The embedding of CP2 in C C P by the quadratic forms of rank 1 is nothing other than the complex Veronese surface V , whose real form was encounteredpin Sects. and II.0. We have here the same p I.7 p mapping .x; y; z/ 7! .x 2 ; y 2 ; z 2 ; 2yz; 2zx; 2xy/.

Fig. IV.9.4. Above: a double line is always tangent to any conic; below: a more or less symbolic sketch of the five CT i and of V in CCP



Bézout’s theorem is thus here too crude; we need in some way to know how to deduct the points of intersection for the fact that the five CTi contain all V , by removing V . We thus need a very general theory of intersection. But this did not acquire its definitive form until about 1960; it is the object of the book (Fulton, 1984). But from Cremona’s time until today numerous mathematicians studied the problem, with varying rigor but with good ideas. Chasles, following de Jonquières, was thus the first to innovate, beginning in 1864, and to find the right number: 3264. It remained only to put his ideas into rigorous form, which was attacked by numerous geometers. In reading (Kleiman, 1980) it seems difficult to say who first obtained a truly complete proof; we mention Severi and van der Waerden in the first half of the twentieth century. We find two modern proofs in Griffiths and Harris (1978b) and we will explain one of these, whose idea is that of Chasles. In contrast, a proof based on the rigorous theory of intersection remains too costly, for it isn’t easy to show that the Veronese surface counts, or “has a multiplicity”, for 7776 3264 D 4512. The other method, described in Griffiths and Harris (1978b), is classic in intersection theory. It consists of “blowing-up” V and counting the number of times it then intersects itself; see in Sect. V.13 for a very tiny bit of this and for the Bezout theorem for curves. The idea to in some way get rid of double lines consists of considering complete conics, i.e. that a conic must simultaneously be the set of its points in CP2 and the set of its tangents in the dual .CP2 / , thus reasoning in CP2 .CP2 / . Then the complete conics can degenerate in double fashion: either into two lines (point perspective) or into two points (tangential viewpoint), having then to do with the set of lines that pass through one or the other of two distinct points.

Fig. IV.9.5.

Now we do this: we degenerate in this double way the five conics Ci into the couples f.Di ; Di 0 /; .pi ; pi 0 /g of a pair of lines and of a pair of points, there being six possible cases. The technical work of making rigorous what Chasles allowed



himself to call the “continuity principle” amounts to showing two things during this quintuple degeneracy of the five pieces CTi n V . On the one hand the pieces tend toward the set of (true) conics that are either tangent to one of the lines or contain one of the points; on the other hand these five pieces intersect each other continually transversally, which shows that the number of their points of intersection is constant. There is nothing left but to calculate the number of the five double pairs f.Di ; Di 0 /; .pi ; pi 0 /g. There will always be 25 choices, but six possible types: containing 5 points, containing 4 points and being tangent to one line, containing 3 points and being tangent to two lines, and the three dual cases remaining. The first furnishes a conic, the second two conics (see above regarding pencils of conics) but with five choices; for the third type it can be shown rigorously that there are 4 conics but with C25 D 10 choices. For the three remaining choices the numbers are the same by duality. The total number is thus exactly 25 .1 C 2 5 C 10 4 C 10 4 C 2 5 C 1/ D 3264. To finish we can ask what happens in the real case: certainly 3264 is a maximum, but it is difficult to conceive that there can exist such a number of conics tangent to five given conics in the real projective (or affine) plane. Now such configurations really can be found, the idea being to begin with five well-chosen degenerate conics, then to show that we can deform them continuously in such a way that they have real persistence for the a priori complex solutions. This is very recent: Ronga, Tognoli, and Vust (1998); there was previously a proof by Fulton from 1980, but unpublished. It’s the occasion to point out that real algebraic geometry has been for a long time the poor relative of the complex version, but that there has recently been a role reversal; see the book Bochnak, Coste, and Coste-Roy (1998). The configuration of Ronga-Tognoli-Vust consists of five hyperbolas very close to their asymptotes, said asymptotes being pairs of lines very near the lines supporting the sides of a regular pentagon:

Fig. IV.9.6.



In the spirit of the following section, we can surely also seek to find how many quadrics (in three-space) are tangent to 9 given quadrics (the space of quadrics is of dimension 4 5=2 D 10). Armed nowadays with a complete theory of intersection in any dimension, 666 841 088 have been found, but it can also be done in higher dimensions. For all that see Sect. IV.10.4 of Fulton (1984). IV.10. The quadrics We consider, without going into detail for lack of space the real or complex quadrics: affine, Euclidean or projective. We deal first with the study, in the spaces E3 , R3 , RP3 , CP3 , of the surfaces defined by equations of second degree. We consider only proper (or nondegenerate) quadrics. Secondly,we can consider the same objects, always defined by equations of second degree (a quadratic form), in spaces of arbitrary dimension. Some use the word hyperquadric when the dimension exceeds three. Here is a caricatural selection of questions and answers on quadrics; for more, consult Chaps. 14 and 15 of [B]. In the Euclidean space E3 , quadrics are encountered naturally as the simplest surfaces after planes and spheres.

Fig. IV.10.1. Mirrors of a telescope; the two mirrors are: a piece of the paraboloid of revolution for the principal mirror and a piece of the hyperboloid of revolution for the small mirror

For example it is known that the Earth is, to a first approximation, an ellipsoid of revolution; that solid bodies have an ellipsoid of inertia and that their movement can be interpreted geometrically by rolling an ellipsoid of fixed center on a plane, see Sect. VII.13.C. The paraboloids of revolution are the mirrors of telescopes (and the small mirror is a piece of an hyperboloid of revolution), whereas the hyperbolic paraboloids are fundamental in architecture, just as are the hyperboloids (of revolution) by the way, for they are ruled surfaces, i.e. composed of lines; see Sect. VI.1. The ellipsoids of revolution have two foci; this property is utilized in the lamps of dentists and operating theaters, called by the pretty name scialytic. Until a few year ago, the headlights of automobiles were also paraboloids of revolution; nowadays they are very sophisticated surfaces, composed of several pieces, but paraboloids,



ellipsoids and hyperboloids of revolution remain essential elements. For the possible detection of gravitational waves, an essential problem of general relativity, there is presently being constructed a telescopic mirror whose focal length is three kilometers, which is an extremely elaborate technical problem. Note then here that what is reflected on these surfaces is not light but electromagnetic waves (see the parabolas for picking up satellites). See in [B] several photographs of varied architecture where quadrics are used systematically. The ellipsoids play an essential role in algorithmics and optimization see Grötschel, Lovasz, and Schrijver, 1988 as well as in the theory of normed spaces; see for example Pisier (1989). But we will in fact encounter ellipsoids amply in Chap. VII, in particular in Sect. VII.6.B. There exists a whole array of characteristic properties of ellipsoids; see Gruber and Wills (1993). But we will also see in Sect. V.3 that geometry on an ellipsoid isn’t yet completely well understood. As for the mystery of rolling stones, i.e. the fact that the pebbles on a beach are roughly ellipsoidal, this will be partly elucidated in Sect. VII.13.C.

Fig. IV.10.2. The five types of proper quadrics, discovered and named by Monge (see for example Berger, 2005): hyperboloid of two sheets, hyperboloid of one sheet, elliptic paraboloid, ellipsoid and hyperbolic paraboloid. Rouché, c Elsevier de Comberousse (1912)



The proper quadrics in E3 are: I - The ellipsoids x2 y2 z2 C C 1D0 a2 b2 c2 (of revolution if a D b). II - The elliptic paraboloids x2 y2 C 2z D 0 a2 b2 (of revolution if a D b). III - The hyperbolic paraboloids x2 y2 2 2z D 0: a2 b IV - The hyperboloids of two sheets y2 z2 x2 C C1D0 a2 b2 c2 (of revolution if a D b). V - The hyperboloids of one sheet x2 y2 z2 C 1D0 a2 b2 c2 (of revolution if a D b). The affine classification thus contains but five types, the Euclidean classification consists only in appending the values of a; b; c figuring in the above equations. The projective classification in RP3 has room for two types only: the topology of the first type is that of a sphere; the topology of the second that of the torus T 2 . Another essential matter distinguishes them. The quadrics of the second type contain lines, projective or affine: there are two families of them, F and F 0 for each quadric; their lines are called generators of these ruled quadrics. In the projective context, these are thus topologically circles and we note that those of the same family don’t intersect one another but are enlaced, whereas two generators of different families intersect in a unique point. The first type is formed of convex quadrics and the tangent plane intersects the quadric in one point only; for the second type the tangent plane intersects the quadric precisely in two secant lines at the point of contact, one from each family. The global topological configuration of these two families F and F 0 is the same as that of Villarceau circles, encountered in Sect. II.7; inquisitive readers will ask if we can pass from one situation to the other by an appropriate transformation. Analytically, the generators are found thus: the two types correspond to quadratic forms with signatures .3; 1/ and .2; 2/ (since .4; 0/ and .0; 4/ yield the empty set). In the case x 2 C y 2 z 2 t 2 D 0, the generators are given by the pairs of equations: x z D k.y C t /


x C z D k 1 .y C t /



and x z D k.y C t /


x C z D k 1 .y t /:

This immediately furnishes an affine (and projective) way of “tracing” quadrics. We take three quadrics D, D0 , D00 in the space, no two in the same plane. Then the set of lines of R3 or RP3 that simultaneously intersect D, D0 and D00 form a ruled quadric; and they are all so obtained. This property was used to construct the double six of Schläfli in Sect. I.9. The following is easily shown: whatever the four lines Di 2 F and the line E 2 F 0 , the cross ratio ŒD1 \ E; D2 \ E; D3 \ E; D4 \ E of four points depends only on the Di and not on E.

Fig. IV.10.3.

Whence also this: let D and D0 be two lines not in the same plane and f W D ! D any hom*ography. Then the line m f .m/ describes a ruled quadric. Or again this: let D, D0 be two lines not in the same plane, and f a hom*ography between the planes passing through D and those passing through D0 . Then the line P \ f .P/ describes a ruled quadric. A particular affine case is this: if two points traverse two lines D, D0 not in the same plane with constant speeds, then the line that joins them describes a hyperbolic paraboloid, the shape of many building roofs with non rectangular walls. This is also the explanation of Fig. IV.4.5; we need only remark that the apparent contour of a hyperbolic paraboloid is a parabola. If architects and builders love hyperbolic paraboloids of one sheet, it’s because they have two series of lines, so that concrete can be doubly reinforced, etc. 0

The metric definitions analogous to those of Sect. IV.2 exist but are not very inspiring; see a quite detailed study in Coolidge (1968), §1 of Chap. XI. To find a quadric



D f(P)

D′ P f(P)

f(m) m D′



Fig. IV.10.4.

of E3 with three unequal axes (i.e. not of revolution) we need to assume a point f, a line D, a planar direction P and a constant e. Then we obtain practically all the quadrics of E3 as the set of points m satisfying d.m; f / D e d.m; p.m// where p.m/ is the point of D obtained by cutting D by the plane passing through m and parallel to P. The generation analogous to bifocal generation and even its extension of that of Graves, discussed in Sect. IV.2, is due to Staude and only about 1850!; see below. The ruled quadrics themselves can be obtained much more simply as the set of points for which the ratio of the distances to two given lines is constant. Hilbert is credited with saying that Staude’s result on the generation of the quadrics with string was one of the great mathematical results of the nineteenth century, but the basis for this opinion isn’t clear. In fact, if we go to the bottom of things, it has to do with hyperelliptic functions and their additive properties; see below for references on this subject. The complex quadrics are not so very interesting in CP3 . Pay close attention to what these complex surfaces are: objects in four real dimensions. There is but one type of proper quadric and this in arbitrary dimension, since the quadratic forms P 2 over the complex numbers of maximum rank are all isomorphic to nC1 iD1 zi . The topology of the complex quadrics is interesting for algebraic topologists it’s that of a product S2 S2 of two spheres, but in higher dimensions the discovery of the topology of hyperquadrics commenced only with Cartan (1932). In dimension 2, the complex quadric possesses generators with properties entirely identical to the real case; the equations are the same. The properties announced earlier about generation and the cross ratio remain valid. In higher dimension, generating lines are generalized by projective subspaces, but things depend on the



Fig. IV.10.5. Staude generation of an ellipsoid by a string stretched between an hyperc Springer bola and an ellipse; see also Fig. IV.10.9 Hilbert, Cohn-Vossen (1996)

parity of the dimension. The fundamental result is Witt’s theorem on quadratic forms, valid over an arbitrary field. It goes back to 1936; see 13.7.1 of [B]. This theorem also regulates the major portion of questions concerning the group of isometries of quadratic forms, thus a goodly part of the geometry of quadrics. To show the mental state of geometers not so long ago, we complexify the sphere S2 . It thus has generators, and it is thus that the geometers of the nineteenth century spoke profusely of the “generators of the spheres”; see for example Darboux (1917), also Dieudonné (1985). But the first was the romantic Poncelet; see Berger (2005). To dive into Duporcq (1938) was a rule for the author in his final year at the lycée in 1944, but Koszul informed him at Strasbourg in 1957 that although this book is surely nice to read it is regrettably devoid of any serious foundation for algebraic geometry. The whole theory of duality seen for the conics generalizes immediately to quadrics, because this duality is the geometric interpretation that a quadratic form is, by definition, the diagonal of a bilinear symmetric form, this both in the real and in the complex domain. In the complex case, i.e. in CPnC1 , we have again a Nullstellensatz in each dimension, and in particular we can define the space of all the quadrics (of dimension n) of CPnC1 . It’s a complex projective space of dimension n.nC3/=2. For the “ordinary” quadrics, it is thus of dimension 9. Equipped now with modern techniques, we won’t fail to calculate the number of quadrics tangent to 9 quadrics in general position; see the end of the preceding section. Now we have but little space left for speaking both of properties of quadrics such as the intersection of two or three quadrics, the geometry on a given quadric, etc. and the entities that extend the quadrics to spaces of any dimension n. The second part of this program can be found in Chaps. 13,14, and 15 of [B].



The intersection of quadrics leads to very subtle questions. An important starting point is that these intersections are curves that are parameterized naturally with elliptic functions. Here are four references storied in time: Appell and Lacour (1922), Halphen (1886, 1888), Donagi (1980), and Tjurin (1975). To conclude we choose to speak of a very important configuration of the Euclidean space E3 , called hom*ofocal quadrics, whose interest beyond its intrinsic beauty is, as we will see, that it overlaps quite varied problems that have no immediate relation to quadrics. Here such a family has one real parameter and is made up of quadrics with equations .


x2 y2 z2 C 2 C 2 1 D 0; C b C c C


that generalize Equation ( ) of Sect. IV.2. The case where a; b; c are distinct is the most interesting; we thus suppose from now on that a > b > c.

c Springer Fig. IV.10.6. Hilbert, Cohn-Vossen (1996)

According as 2 1; c 2 Œ, c 2 ; b 2 Œ, b 2 ; a2 Œ, we obtain respectively: an ellipsoid, a hyperboloid of two sheets, a hyperboloid of one sheet. The figure formed by these three one-parameter families enjoys properties that are many and remarkable. Here is a sampling: First, through each point of space located outside the union of the three coordinate planes there passes one and only one quadric of each sort, whence the mapping .x; y; z/ 7! .; ; /. If we restrict ourselves, for example, to the region x > 0, y > 0, z > 0, then this mapping is bijective and provides a parameterization by the so-called hom*ofocals of each of the 8 quadrants of space. These coordinates allow us to study questions rather systematically, whether they relate to the whole collec2 2 2 tion or to a fixed quadric such as the ellipsoid E such that xa2 C yb 2 C zc 2 1 D 0, for example for studying vibrations on or interior to this ellipsoid; see e.g. Morse and Feshbach (1953).



An elementary property is that the three quadrics, at a point that they all contain, have mutually orthogonal tangent planes. The proof can be hard if the calculations are made in .x; y; z/ coordinates and without some finesse. Recall (see Chap. XII) that the geodesics of a surface (or more generally of a reasonable metric space) are curves that provide the shortest paths from one point to another. But the geodesics defined by this local minimum property remain interesting when they are prolonged indefinitely, which is always possible if there is compactness. If we want to study the geodesics on an ellipsoid E, we will then set D 0 in the triple .; ; / and the surface of the ellipsoid will be parameterized by .; /. These are called elliptic coordinates. The geodesics of E are defined by a very simple family of differential equations in .t /, .t /, with the additional condition of speed equal to 1, dependent on one parameter. This permits us to study them almost completely. A fairly recent text is Knörrer (1980); see also the references at the end of Sect. of [BG]. Here are some properties, but also look in Sect. VI.3. With one fixed value of the parameter mentioned above is associated a hyperboloid H (of one sheet or two depending on these parameter values) which is such that the geodesics having this parameter oscillate between the two curves of intersection of E with H. They may, in E3 , be defined geometrically by the condition that their tangents are the lines of E3 constantly tangent to H. But we will see in Sect. VI.3 the unfinished history of some recalcitrant ellipsoids.

Fig. IV.10.7. At left, the tangency property for geodesics on an ellipsoid. At right, a figure discovered by Monge, that of lines of curvature of the ellipsoid Hilbert, c Springer Cohn-Vossen (1996)

The above figures correspond to the case where H has one sheet; as an exercise readers can sketch the case where H has two sheets. The intermediate case is where H is reduced to the hyperbola H (see Fig. IV.10.9) x2 z2 C 1D0 a2 b 2 b2 c2



of the plane y D 0. The intersection H \E is composed of 4 points a, a0 , b, b 0 . They are called the umbilics of E, for they are the points where the surface E in E3 has two equal radii of principal curvature; see Chap. VI or Berger and Gostiaux (1987) for the vocabulary for what follows. But here they have the following extraordinary focalization property, which is nothing other than the limit of the oscillation property mentioned above: each geodesic issuing from a (resp. b) passes through a0 (resp. b 0 ). See more of this in Chaps. VI and XII. We don’t know if the geometry of ellipsoids of dimension greater than two for example what generalizes the focalization of umbilics has been studied; see however Joets and Ribotta (1999) for the focal sheets of these ellipsoids in higher dimensions; see also the end of Sect. VI.3. The curves of the intersections E \ H are all lines of curvature of E (and, moreover, of H also), i.e. the curves for which the tangent always has a direction of principal curvature (see the references given above). It involves a definition that is valid for every surface. With elliptic coordinates, we can easily show that all the lines of curvature are defined as the loci of points for which the sum of the intrinsic distances (see Sect. VI.1) to the two umbilics is constant (compare this with the bifocal properties of conics in Sect. IV.2). Note that this is now the difference which becomes constant if we change an umbilic into its antipodal! We find this nice figure, due to Monge, that will arouse our enthusiasm again in Sect. VI.3.

Fig. IV.10.8. The lines of curvature of an ellipsoid can be defined, just as for ordinary ellipses in the plane, by a bifocal condition where the two foci are the umbilics

Just as the bifocal definition of conics can be extended by Graves’ theorem (Sect. IV.2), so perhaps can the above property d.a; v/ C d.b; v/ D constant be extended to give initial results of Graves type for a given quadric. But they especially long resisted the ardor of geometers, in particular in attempting to generate an entire quadric with a single stretched string. The case d.f; x/Cd.f 0 ; x/ D constant in E3 yields in fact only ellipsoids of revolution. It was necessary to wait for Staude toward the end of nineteenth century to find this generation. The simplest is that of Fig. IV.10.5.



The hyperbola is x2 z2 1 D 0; y D 0 a2 b 2 b 2 c 2 and the ellipse is z2 x2 C 1 D 0; z D 0: a2 b2

c Springer Fig. IV.10.9. Hilbert, Cohn-Vossen (1996)

But Staude extended this result “a la Graves”: the string is stretched between an ellipsoid and a hyperboloid of one sheet which is hom*ofocal with it: the points which tighten the string describe an ellipsoid. For a proof of Staude’s results, see any of the following: p. 450 of Salmon (1874) or Chap. XIII, §3 of Coolidge (1968), or p. 179 of Coolidge (1940–1963), or again Staude (1904–1992).

Fig. IV.10.10. Playing billiards on the interior of an ellipsoid that has been cut open so as to see what happens (drawing by Geoffroy Wagon)

A property of hom*ofocal ellipsoids, that generalizes the one for ellipses what was mentioned at the end of Sect. IV.2, is again a property of light caustics: a ray of light propagates on the interior of the ellipsoid E, reflecting on the interior surface with the symmetry which is imposed with respect to the tangent plane at the point of reflection, remains always tangent to an ellipsoid E0 , hom*ofocal and interior to E.



This if it stays on the interior; if not, it will remain tangent to one of the hyperboloids of the family. A result, by the way, that is purely local and of little difficulty is that: the only Euclidean surfaces that admit a caustic surface are pieces of quadrics; cf. Berger (1995).

Fig. IV.10.11. Represented on the right is what happens along a line of curvature of S c Société Mathématiques de France Berger (1995)

The proof of the “causticity” of the quadrics in fact valid in arbitrary dimension is accomplished very nicely with duality, complexification and the use of what generalizes the cyclic points of the plane (see Sect. II.XYZ) to E3 , specifically the conic in the plane at infinity (called umbilical) defined by the cone x 2 C y 2 C z 2 D 0. For example the spheres are the quadrics for which the complexification contains . This classic proof is found for example in Douady (1982). Recently an extraordinary and unexpected relation was discovered among the geodesics of the ellipsoid (more precisely the obvious extension of the hom*ofocal quadrics to En ) and the solutions “a la Peter Lax” of the KdV (Korteweg-de Vries) equation, solutions called solitons. The KdV partial differential equation describes the movement of water in a channel. A connection is made by isospectral deformations of matrices. Several other problems are directly connected to it: that of the movement of a point that moves on a sphere under the influence of a quadratic potential, and that due to H. Knörrer which generalizes the fact that the intersection of two quadrics of CP3 is parameterized by an elliptic function and which involves the manifold formed of subspaces of dimension n 1 of the intersection of two quadrics of CP2nC1 . For all the preceding consult (Knörrer, 1980; Moser, 1980) and the quite recent (Audin, 1995). IV.XYZ Conics over arbitrary fields. Conics and quadrics too can clearly be defined, in any dimension, over commutative fields other than the reals or the complexes. When the field is no longer commutative in the case of the quaternions H for example the



notion of quadratic form becomes more difficult. A very little bit of the study of quadrics over H can be found in Porteous (1969). Over finite fields, we are led to problems in number theory and combinatorics. Geometric intuition must then be handled with great caution. Here are some examples that illustrate this remark. It can be shown (but is not obvious) that over a finite field no conic is ever empty. If the field has and even number q of elements, then a conic has at most q C 1 points (because it must be isomorphic to a projective line) and all the tangents pass through the same point! But if q is odd, then through every point not belonging to the conic there are either two tangents to the conic or else no tangent to the conic passes through the point; see Lidl and Niederreiter (1983). Matrix representation of the equation for conics and duality. We represent the ! x 2 point m D .x W y W z/ of RP by the column matrix X D y (defined within a z constant multiple). We also represent the line with equation ux C vy C wz D 0 by ! u the column matrix U D v , in such a way that the equation of the line is written w t UX D 0. We can then consider U as the column matrix with the coordinates of d in the dual projective plane .RP2 / . Now let C be a conic in RP2 , with equation F.x; y; z/ D ax 2 C a0 y 2 C a00 z 2 C 2byz C 2b 0 zx C 2b 00 xy D 0: This equation can be rewritten t

XAX D 0;

where A is the symmetric matrix 1 a b 00 b 0 A D @b 00 a0 b A : b 0 b a00 0

We assume in what follows that A is regular, i.e. of rank 3. (i) Two points X and Y are conjugate with respect to the conic if and only if t

XAY D 0:

We prove the case where X and Y are distinct. The line joining these two points is parameterized by .s W t / 7! sX C t Y, or 7! X C Y if we allow the value 1 for . The points of intersection P and Q of the line with C are given by the roots 0 and 00 of the equation t

XAX C 2 t XAY C 2 t YAY D 0:

We have ŒX; Y; P; Q D Œ0; 1; 0 ; 00 D 0 =00 . This cross ration is equal to 1 if 0 C 00 , the sum of the two roots, is zero, i.e. if t XAY D 0. (ii) It follows that the polar of the point X is the line represented by U D AX, and that the pole of the line U is X D A1 U.



(iii) A pointX belongs to C if it is conjugate to itself, or again if it belongs to its polar. In the latter case, the polar X is the tangent to C at X. We in fact show that the coefficients of the equation for the tangent to a0curve 1 with hom*ogeneous equation F0x F.x; y; z/ D 0 are given by the column matrix @F0y A, which here yields 2AX. F0z (iv) Consequently a line U is tangent to C if and only if we have t

UA1 U D 0:

This is the equation of the dual conic C (in the dual projective plane). Involutions. An involution of CP1 is a hom*ography f W CP1 ! CP1 such that f 2 D Id, f ¤ Id (we treat only the complex case readers can carry out the adaptation to the real case). The hom*ography f .x/ D

ax C b cx C d

.ad bc ¤ 0/ is an involution if and only if a D d . The fixed points of the involution are the roots of the second degree equation cx 2 2ax b D 0. By hypothesis, a2 C bc ¤ 0. An involution therefore has two distinct fixed points (if c D 0, one of them is 1). If we denote the fixed points by ˛ and ˇ, and if x is distinct from these, then we have Œx; f .x/; ˛; ˇ D Œf .x/; x; ˛; ˇ D Œx; f .x/; ˛; ˇ1 , whence the relation Œx; f .x/; ˛; ˇ D 1; since a cross ratio can’t equal 1 for distinct points. From this relation, symmetric incidentally in x and f .x/, we can extract f .x/ as a hom*ographic function of x, which shows that an involution is determined by the specification of its two fixed points. Finally, we frequently define a mapping f W CP1 ! CP1 such that f 2 D Id, f ¤ Id, but without being able (or even wanting) to compute f .x/ explicitly. Once we know that f .x/ is a rational function of x, we are assured of having an hom*ography (we don’t need the condition f 2 D Id; the fact that f is injective suffices. Equations of second degree and involutions. The solutions of an equation of second degree that is linearly dependent on a parameter, e.g. .E /

.a C a0 /x 2 C .b C b 0 /x C c C c 0 D 0

are “in involution”. In modern language this means that there exists an involutive hom*ography f such that x 00 D f .x 0 /;

x 0 D f .x 00 /

where x 0 D x 0 ./ and x 00 D x 00 ./ denote the two roots of E . In applications, the parameter and the unknown x are taken in CP1 , but we avoid using hom*ogeneous



coordinates: if the coefficient of x 2 vanishes, one of the roots is infinite, and if D 1, the equation is simply a0 x 2 C b 0 x C c 0 D 0. The proof is very simple. This sort of computation incidentally is the has been the delight of the fifth form (eleventh grade) students for scarcely more than a generation. We assume that the E don’t have a common root. The three coefficients of the equation are necessarily related by a relation of the type A.a C a0 / C B.b C b 0 / C C.c C c 0 / D 0: In view of the properties of the sum and product of the roots of a second degree equation, we then have A B.x 0 C x 00 / C Cx 0 x 00 D 0; whence we find x0 D

Bx 00 A ; Cx 00 B

x 00 D

Bx 0 A : Cx 0 B

The fixed points of f are the roots ˛ and ˇ of the equation A 2Bx C Cx 2 D 0. According to the preceding, they are distinct and we have Œx 0 ; x 00 ; ˛; ˇ D 1. A fixed point of f is also a double root of E . Since there are two distinct fixed points, the equation E has a double root for exactly two values of . Application 1. (Desargues-Sturm Theorem.) If the line D doesn’t intersect the base of the pencil, the conics of a linear pencil determine an involution on D. Two distinct conics of the pencil are tangent to D, the points of contact being the fixed points of the involution. Application 2. (Frégier’s Theorem.) Let C be a proper conic and p a point not belonging to C. The mapping that takes each point m of C to the point where the line pm intersects C is an involution. Every involution of a conic has this form. The forward implication is immediate in view of what has already been said. For the converse, we indicate that, an involution of C being given, we obtain p as the intersection of the tangents at the fixed points of the involution. Bibliography [B]Berger, M. (1987, 2009). Geometry I,II. Berlin/Heidelberg/New York: Springer [BG]Berger, M., & Gostiaux, B. (1987). Differential geometry: manifolds, curves and surfaces. Berlin/Heidelberg/New York: Springer Appell, P., & Lacour, E. (1922). Fonctions elliptiques. Paris: Gauthier-Villars Arnold, V. (1990). Huyghens and barrow, Newton and Hooke. Basel: Birkhäuser Arnold, V. (1999). Symplectization, complexification and mathematical trinities. In E. Bierstone, B. Khesin, A. Khovanskii, & J. E. Marsden (Eds.), The Arnoldfest (pp. 23–28). Providence, RI: American Mathematical Society Arnold, V. (2000). Polymathematics: Is mathematics a single science or a set of arts? In V. Arnold, M. Atiyah, P. Lax, & B. Mazur (Eds.), Mathematics: Frontiers and Perspectives (pp. 403–416). Providence, RI: American Mathematical Society



Audin, M. (1995). Topologie des systèmes de Moser en dimension quatre. In H. Hofer, C. Taubes, A. Weinstein, & E. Zehnder (Eds.), The Floer Memorial Volume (pp. 109–122). Basel: Birkhäuser Barth, W., & Bauer, T. (1996). Poncelet theorems. Expositiones Mathematicae, 14, 125–144 Barth, W., & Michel, J. (1993). Modular curves and Poncelet polygons. Mathematische Annalen, 295, 25–49 Benoist, Y., & Hulin, D. (2004). Itération de pliages de quadrilatères. Inventiones Mathematicae, 157, 147–194 Berger, M. (1995). Seules les quadriques admettent des caustiques. Bulletin de la Société Mathématique de France, 123, 107–116 Berger, M. (2005). Dynamiser la géométrie élémentaire: introduction à des travaux de Richard Schwartz. Atti della Accademia Nazionale dei Lincei. Classe di, Ser. 25, 127–153 Bochnak, J., Coste, M., & Coste-Roy, M.-F. (1998). Real algebraic geometry. Berlin/Heidelberg/ New York: Springer Bos, H., Kers, C., Oort, F., & Raven, D. (1987). Poncelet’s closure theorem. Expositiones Mathematicae, 5, 289–364 Bourbaki, N. (1981). Algèbre, chapitre IV. Masson Cartan, E. (1932). Sur les propriétés topologiques de la quadrique complexe. Public. math. Uni. Belgrade, ou OEuvres complètes, I-2, 1227-1246, 55–74 Cayley, A. (1861). On the Porism of the in-and-circ*mscribed polygon. Philosophical Transactions of the Royal Society of London, CLI, 225–239 Coolidge, J. (1940, 1963). A history of geometrical methods. Oxford: Oxford University Press, Dover reprint Coolidge, J. (1968). A history of the conic sections and quadric surfaces (1st ed. 1945). New York: Chelsea, Dover reprint Darboux, G. (1917). Principes de géométrie analytique. Paris: Gauthier-Villars Dieudonné, J. (1985). History of algebraic geometry. Monterey, CA: Wadsworth Dingeldey, F. (1911). Coniques et systèmes de coniques. In Encyclopédie des sciences mathématiques pures et appliqués. Paris et Leipzig: Gauthier-Villars and Teubner Donagi, R. (1980). Group law on the intersection of two quadrics. Annales scientifiques Ecole norm. sup., 7 Douady, R. (1982). Applications du théorème des tores invariants. Thèse Paris VII Duporcq, E. (1938). Premiers principes de géométrie moderne. Paris: Gauthier-Villars Emch, A. (1900). Illustration of the elliptic integral of the first kind by a certain link work. Annals of Mathematics, 1, 81–92 Emch, A. (1901). An application of elliptic functions to Peaucellier link-work (inversor). Annals of Mathematics, 2, 60–63 Fulton, W. (1984). Intersection theory (2nd ed. 1998). Berlin/Heidelberg/New York: Springer Greenberg, M. (1974). Euclidean and non-Euclidean geometries. New York: Freeman Griffiths, P., & Harris, J. (1978a). On Cayley’s explicit solution to Poncelet’s porism. L’enseignement math., 24, 31–40 Griffiths, P., & Harris, J. (1978b). Principles of algebraic geometry. New York: John Wiley Grötschel, M., Lovasz, L., & Schrijver, A. (1988). Geometric algorithms and combinatorial optimization. Berlin/Heidelberg/New York: Springer Gruber, P., & Wills, J. (Ed.). (1993). Handbook of convex geometry. Amsterdam: North-Holland Hadamard, J. (1911, 1988). Leçons de géométrie élémentaire (Reprint Jacques Gabay). Paris: Armand Colin Halphen, G. (1886, 1888). Traité des fonctions elliptiques, I, II. New York: Gauthier-Villars Hilbert, D., & Cohn-Vossen, S. (1999). Geometry and the imagination. Providence, RI: American Mathematical Society Hrasko, A. (1999). Letter to A. Shen. The Mathematical Intelligencer, 21(3), 50 Hrasko, A. (2000). Poncelet-type problems, an elementary approach. Elemente der Mathematik, 55, 1–18 Joets, A., & Ribotta, R. (1999). Caustique de la surface ellipsoïdale à trois dimensions. Experimental Mathematics, 8, 57–62



Kleiman, S. (1980). Chasles’s enumerative theory of conics: An historical introduction, in studies in algebraic geometry (pp. 117–138). Washington, DC: The Mathematical Association of America Klein, C.F. (1872). Erlangen Program. htm#Introduction Knörrer, H. (1980). Geodesics on the ellipsoid. Inventiones Mathematicae, 59, 119–143 Lebesgue, H. (1942, 1987). Les coniques (Reprint Jacques Gabay). New York: Gauthier-Villars Levy, H. (1964). Projective and related geometries. New York: McGraw Hill Lidl, R., & Niederreiter, H. (1983). Finite fields. Cambridge,UK: Cambridge University Press McDonald, S., & Kaufmann, N. (1998). Wave chaos in the stadium: Statistical properties of shortwave solutions of the Helmholtz equation. Physical Review A, 37(8), 3067–3086 Morse, P.M., & Feshbach, H. (1953). Methods of theoretical physics. New York: McGraw-Hill Moser, J. (1980). Geometry of quadrics and spectral theory (pp. 147–188). In The Chern symposium. Berlin/Heidelberg/New York: Springer Pécaut, F. (2005). Équivalence du grand théorème de Poncelet pour deux cercles et du théorème des zigzags. Quadratures, 58, 13–18. EDP Sciences, Les Ulis Pisier, G. (1989). The volume of convex bodies and Banach space geometry. Cambridge, UK: Cambridge University Press Porteous, I. (1969). Topological geometry. London: Van Nostrand-Reinhold Ronga, F., Tognoli, A., & Vust, T. (1997). The number of conics tangent to five given conics: The real case. Revista Matemática Complutense, 10, 391–421 Rouché, E., & de Comberousse, C. (1912). Traité de géométrie (2 vols.). New York: Gauthier-Villars Salmon, G. (1874). A treatise on the analytic geometry of three dimensions (Reprint Chelsea). Dublin: Hodges Schwartz, R. (2007). The Poncelet grid. Advances in Geometry, 7, 157–175 Shen, A. (1998). Mathematical entertainments. The Mathematical Intelligencer, 20, 31 Staude, O. (1904, 1992). III 22. Quadriques. Encyclopédie des Sciences mathématiques. J. Molk, Teubner, trad. Gauthier-Villars, reprint Jacques Gabay, III, 1–162 Strichartz, R. (1987). Realms of mathematics: Elliptic, hyperbolic, parabolic, sub-elliptic. The Mathematical Intelligencer, 9(3), 56–64 Tabachnikov, S. (1995). Billiards. Paris: Société mathématique de France Tjurin, A. (1975). On intersection of quadrics. Russian Mathematical Surveys, 30, 51–105 Van der Waerden, B. (1954–1975). Science awakening I. Groningen: Noordhoff

Chapter V

Plane curves V.1. Plain curves and the person in the street: the Jordan curve theorem, the turning tangent theorem and the isoperimetric inequality Here are three examples of “facts” but rather more of “results”, of “theorems” that illustrate the difference between the mathematician and the “person in the street”. They are excellent for explaining the nature of mathematics to a nonprofessional. The first is the Jordan curve theorem: (V.1.1) In the plane we trace a closed curve that doesn’t intersect itself. Then it has an interior, a region that it encompasses.

Fig. V.1.1. The interior of a simple closed curve

The instinctive reaction is that it’s obvious! but the mathematician on the contrary requires a proof; but first and this is also important a definition, formalized from the notion of interior. It’s about the same for the turning tangent theorem known in German as Hopf’s Umlaufsatz: (V.1.2) We trace a smooth closed curve that doesn’t intersect itself and look at how much the tangent to the curve turns as we make a complete circuit: it turns 2 in all. The second result is surely already a little more mathematical in its statement, but it remains nonetheless very intuitive. It was, by the way, long considered evident by practically all mathematicians. Another typical example of a result that was initially considered obvious, then was the object of more or less rigorous proofs before being truly completely proved, is what figures in the beginning of Sect. V.11, that is to say the isoperimetric inequality:

M. Berger, Geometry Revealed, DOI 10.1007/978-3-540-70997-8_5, c Springer-Verlag Berlin Heidelberg 2010




Fig. V.1.2.

Fig. V.1.3. To be able to specify what it means to turn, a mapping ˛ from the curve to the circle is introduced, while the geometric velocity is normalized as a 0

f vector of length 1. If f 0 is this velocity, we set ˛ D kf 0 k . If an axis is chosen, 1 we can identify ˛ , an element of S , with the angle that f 0 makes with this axis

(V.1.3) Among all plane curves of a given length, the circle is the one which encompasses the largest area possible.

Fig. V.1.4.

This last result is the one that in folklore is the most ancient; this sort of thing was known to the Greeks. Moreover, its practical importance is considerable, for example for plane silaging problems. By symmetry, the same result can be deduced for a semi-circle among all curves of given length which are attached to a fixed line at its extremities. Now, to convince our ordinary person that things are not so evident as they appear, we need to show more complex drawings and appeal to the imagination. In such drawings it becomes more difficult to know what is the actual interior and still more difficult to believe that this time the tangent ends up turning only 2, i.e. that all its contortions have been finally cancelled out to leave room in the end for only a single turn. From the conceptual point of view and that of Jacob’s ladder, it is much the same with our two first “obvious” theorems. The isoperimetric inequality will be the subject of Sect. V.11 and V.12.



Fig. V.1.5. In the two figures it can be seen that the notion of interior isn’t so very clear!

First and foremost we need to define curve and, in addition, “closed and not selfintersecting”. If we read the definitions given in Sect. V.2 or in V.XYZ, then we don’t at all see anymore what the thing called the interior is supposed to be. In Sect. V.4 we sketch a proof of the Jordan curve theorem that is relatively elementary when the curve considered is differentiable, but requires in any case the introduction of the concept of index of a point with respect to a curve; but the result is in fact still true when the curve is only required to be continuous. A classic proof consists of identifying R2 with C (see as needed Sect. I.XYZ) and applying Cauchy’s theorem of the theory of functions of a complex variable, the one that figures in many works as a geometric application of this theorem, for example in Dieudonné (1960). Jordan’s proof appeared in the second edition of his famous Cours d’Analyse in 1893. This work had a very important global impact, the more profound reason being that the proof was done by algebraic topology, of which it is one of the very first applications; a very lucid exposition can be found for example in Greenberg (1967). The theorem that is proved is more precise than affirming only that there is an interior. It says this: the complement in R2 of a simple (the technical term for “not self-intersecting”) closed curve is composed exactly of two connected components, one of which is bounded (the interior), the other unbounded (the exterior). This approach moreover furnishes de facto the extension to arbitrary dimension. The proof is carried out on the sphere S2 , on which a simple closed curve C is considered. The result is then that the complement of C in S2 has exactly two connected components. Now, taking the north pole on the interior of one of them, the associated stereographic projection furnishes two connected components in the plane, one of which is bounded and the other, originating from the north pole, unbounded. For the turning tangent theorem or Umlaufsatz, there is at first the problem of correctly defining what is meant by “how much the tangent turns”. This is accomplished by putting in place what is needed in order to say what is meant by “keeping track by continuity of the angle formed by the tangent line with a fixed direction” when the curve is traversed. To do this, an orientation of the plane and a direction of reference ı are fixed. At a point of parameter s of the curve, parameterized by



arc length; see Sect. V.4. The problem is that the angle ˛.s/ of this tangent with ı is an element of the unit circle S1 and not a real number. This mapping is continuous if the curve is continuously differentiable (see Sect. V.XYZ). But the real line rolls onto the circle S1 by the mapping t 7! .cos t; sin t / and no more is needed than to lift the mapping s 7! ˛.s/ to a mapping s 7! ˛ .s/ in R. This lifting is done in small pieces, subsequently joined (or “glued” by continuity. We could say that “we have pursued one of the determinations of ˛.s/ within 2k by continuity”. If now the curve has length L, the mapping ˛ takes Œ0; L into R, and to obtain the theorem we need to prove the equality ˛ .L/ D ˛ .0/ C 2. All this will take form in Sect. V.5.

Fig. V.1.6. In this figure, we suppose that the velocity f 0 D is always equal to a unit vector, giving a (partial) example where this mapping reverts onto itself

This theorem was long considered obvious, even by numerous “professional” mathematicians, Euler first, but also Rolle, Watson (see Watson, 1916) and Riemann. In numerous rather old works that treat curves, the turning tangent theorem was considered “settled”, or “evident”. Mathematicians often use the term “folklore” in reference to such a result. What to our knowledge are the first two complete proofs appeared in the same issue of the journal Compositio mathematica: Ostrowski (1935) and Hopf (1935). Ostrowski proceeded by approximating the curve by polygons, the resulting proof being elementary. Hopf gave a direct version of it, without approximation by polygons, but his proof is intricate; we will illuminate its essential ideas in Sect. V.5. We know of no conceptual proof that is truly simple, even at the cost of a rather steep ascent of Jacob’s ladder. Historical remark. Question relating to Jacob’s ladder: in spite of its astonishing simplicity and in contrast to other objects of this book such as e.g. conics or surfaces, the notions of curve and plane curve in particular have waited until just recently for mathematicians to invent the concepts needed for progressing in their study, as is indicated by the dates given in the present chapter. Our book is organized mainly under the aegis of differentiable curves, and we single out notions which arise from affine geometry, let us say in R2 and those that require a presence in the Euclidean plane E2 , without completely omitting the projective spaces, treated very briefly in Sect. V.9. In the three examples seen above, the first (Jordan) is affine, the



second (turning tangent) is half Euclidean, whereas the third (isoperimetric inequality) makes sense only in E2 . There will also be an essential difference between the simple closed curves (without self intersection) and arbitrary closed curves. The following section will clarify what is meant by that. Our presentation will be brief; we will refer to Chaps. VIII and X of [BG] for an exposition that was rather complete at the time of its publication in 1987. There exist plenty of books on differential geometry that treat curves (and surfaces), but there are few that explain in good detail the notions of geometric curves and of kinematic curves; we will see why. We will have need of the language of differentiable mappings of which we will sketch the definitions and properties in Sect. V.XYZ and also notions of genericity, of robustness; see for example Sects. V.2 and V.10. Beyond the three results that we have used in the way of introduction, we will give other global results. But certain among them, typically the four vertex theorem and its generalizations, have been the object, since about 1990, of a revolution due to Arnold, that replaced these results in a variety of very general conceptual contexts with vast generalizations, that is the very justification for introducing a new concept. This revolution has unleashed publication of very numerous studies, for which we will be able to give useful introductory concepts, as well as some applications. The theory itself is presently in full flight. In spite of this prolificness, in order to stay within the spirit of this book we will ultimately devote but few sections to algebraic curves. In fact we will see that early on, starting with Descartes and Newton, algebraic curves were the object of numerous and profound studies, to the extent that we can say that by the end of the nineteenth century almost everything was known about them, it being stipulated that this has to do with plane curves over the real or complex fields. We will see however an exemplary exception, the subject of a problem of vigorous current research, i.e. the topology of real algebraic curves. In contrast, for the curves that are defined over finite fields, the entire area of number theory comes into play and is always in full swing, the quite recent proof of Fermat’s theorem being a foremost example. We now risk a few explanations of this historical difference between algebraic curves and merely differentiable (smooth) curves. In the algebraic context, the curves of given degree form a space of finite dimension (5 for the conics; see Sect. IV.7) a nice projective space, but metrizable. A whole collection of algebraic tools is needed. On the other hand, the merely smooth curves form a space of infinite dimension, essentially a space of functions, but the classic tools of analysis can’t be directly adapted to their study, as we shall see; we have to be much more creative: we must use algebraic topology and recent notions of differentiable manifolds. Note also that algebraic curves appear as truly mathematical objects, whereas the smooth curves are more objects of kinematics. But contrary to our own discipline kinematics is not much concerned with global problems. It is typical that, until very recently, the abundant instructional material implicitly considered every curve to be algebraic; see a critical analysis for the case of duality in Berger (1972). Also, for those interested in pedagogy and history, Dombrowski (1999) is a remarkable book; we will mention it several times.



Important remark. Of course, the notion of curve, both geometric and kinematic, exists in three dimensional space as well, and just as well in arbitrary dimension. However, the literature is not so abundant and important results are rare. We have chosen not to speak about these matters, as few as they are, so as not to have to expand the present book. But, although progressing slowly, it is an area that is very much alive. We give a recent reference, that will allow readers to review previous contributions to the extent they want, i.e. Uribe-Vargas (2004b). V.2. What is a curve? Geometric curves and kinematic curves For the person in the street a curve is something that is drawn with a pencil on a sheet of paper, or else the trajectory of a car, etc. For secondary school students, it may also be a graphic representation of a function, and how they were made to suffer, until very recently calculators assumed much of the task. What connection is there between these two very different definitions? (V.2.1) A geometric curve is a subset C of R2 which is locally with well chosen axes the graph of a numerical function of class C1 .

Fig. V.2.1. In these figures, we have zoomed in twice on a point, so as to be able to realize that, with suitable axes, what we see of the curve becomes a graph. See also Figs. V.2.2 and V.2.3

This may necessitate us occasionally having to zoom in considerably on particular points of the curve. Note also that by zooming in more and more, what we see of the curve resembles more and more a straight line segment along the tangent at this point (this because the function that comes into play is differentiable). Moreover, an equivalent definition consists precisely in saying that, about each point of the curve, we can find a small piece of the plane that is diffeomorphic to a piece of plane where the corresponding piece of curve is an open segment of a line. The essential lemma is:



(V.2.2) Let f W I ! R2 be a mapping of class C1 defined on an open interval I R and t 2 I such that f 0 .t / 6D 0. Then there exists " > 0 such that the image of t "; t C "Πis a geometric curve in R2.

Fig. V.2.2.

We thus see appear here the notion of tracing a curve. Which brings us then to the definition: (V.2.3) A kinematic curve (or parameterized curve) is that given by a pair .I; f / formed by an open interval I of R and a mapping f W I ! R2 of class C1 such that f 0 .t / 6D 0 for each t 2 I. (If f 0 .t / 6D 0, we say that the point is not stationary for the parameter value t .) In analytic language, we write x D f1 .t /, y D f2 .t /, with the condition that f10 .t / and f20 .t / are nowhere simultaneously zero. In formal language, we call such an object an immersion in R2 of an open interval of R. Important note on methodology. The definition (V.2.3) may be considered too restrictive because there is in fact no reason not to study the curves where the vector f 0 .t / is allowed to be zero at certain points. Even though kinematic motions trajectories may actually have this more general nature, it seems that there do not exist any profound or geometrically spectacular results relating to it that pertain to Jacob’s ladder. Numerous questions now arise, turning mainly about the question: Proposition (V.2.2) furnishes a local equivalence between two categories of objects: geometric curves and kinematic curves. What is this relationship globally? An essential difference is that a kinematic curve can have multiple points (i.e. self-intersections), but not so a geometric curve. But this remark is quite far from answering the question completely from a formal point of view. That is done in [BG], but for us what is said in the next section will suffice. The essential caveats are perhaps more important than the presentation itself. They concern the natural connection between geometric and kinematic curves, specifically the entirely natural consideration of the image f .I/ of a kinematic curve. There are three caveats. The first is the classic pitfall: it is not because the immersion f W I ! R2 is injective that the image is actually a geometric curve: in the figure below the point of “virtual crossing” is the necessary counter example: in any neighborhood of this point there will be three parts, the normal part plus one coming from C1 and another from 1. In formal language, this phenomenon is



due to the fact that the mapping considered is not proper: a mapping is proper when the inverse image of every compact set is again a compact set in the source set for the mapping.

Fig. V.2.3.

The second caveat is that the image, even if it is a compact set, does not determine the immersion even within reparameterization. The two pairs of figures below show this well. We note for Sect. V.10 that this is due to the existence of points where the image is tangent to itself, i.e. the same point with the same tangent line is obtained for different values of the parameter t, but the curve does not actually cross itself. We now explain reparameterization. It is clear that if g W I ! I is a bijection of class C1 , then the images of f and of f ı g will be identical. This leads to an inevitable definition if we want to study kinematic curves and images, namely to consider curves obtained by reparameterization as above as equivalent, and “pass to the quotient”; see [BG] for the presentation. But it is important to remark that there are two classes of diffeomorphisms of an interval, those with positive derivative and those with negative derivative: we are dealing with oriented curves. The third caveat is that the image of a general kinematic curve can be pathological. Some examples appear in Fig. V.6.7; but readers will be able to imagine worse. Conclusions: first, it is useless to seek a classification of these curves by their images. Next, for getting results, we need to limit ourselves to categories of (kinematic) curves having additional properties. These will be the generic curves of Sect. V.10, specifically those which have at worst double points and which are never self-tangent. In particular they don’t have any triple points. These two exclusions are natural when thinking of robust properties, i.e. those that persist under small defor-

Fig. V.2.4. Examples of paths not determined by the image



mations. Triple points and self-tangencies are eliminated right away; see the figures of Sect. V.10. In Fig. V.2.3 it is the points of self-tangency that allow a change in direction for traversing the curve. Moreover, here the image of curve, as a plane set, determines the curve within orientation. V.3. The classification of geometric curves and the degree of mappings of the circle onto itself The classification of geometric curves is really quite simple, especially as regards the statement of the result, but the complete proof requires some care. We always assume that they are connected; then there are but two possible topological types the line and the circle. In the first case, the geometric curve C may be obtained as the image of a kinematic curve, necessarily injective. In the second case, C can be obtained as the image of a periodic kinematic curve, that is an f W R ! R2 for which there exists T > 0 such that f .t CT/ D f .t / for each real t . It is necessary that we have injectivity on the interval Œ0; T. We always stipulate that T is in fact the smallest period (to preclude the curve being described redundantly); and we also pay attention that the images of periodic kinematic curves are not geometric curves in general. In fact, the preceding permits us now to use alternative and somewhat simpler language for the two types of curves that we are going to consider. (V.3.1) By a simple closed curve is meant a geometric curve (or one of its periodic parameterizations) that is of circle type (thus compact). By a closed curve is meant a periodic kinematic curve. In practice, (kinematic) closed curves are considered within a change of parameter, i.e. the change has to be global and always with nonzero derivative. In more formal language, this means that we form the quotient with the group of diffeomorphisms of the circle, in the same spirit as reparameterization above. The curve is called oriented when we only form the quotient with diffeomorphisms with positive derivative. In Sect. V.6 an essential point will be that if the curve is traced in the Euclidean plane, it can be reparameterized by arc length. The above classification of geometric curves into only two topological categories, circles and lines, isn’t difficult in any astonishing way, as seems quite evident; there is, however, something to prove. This classification results from the more general one of differential manifolds of dimension 1. Briefly these are objects which are unions of intervals of R, pieces to be fastened together using local diffeomorphisms, which for curves means specifically a change in parameterization with derivative always nonzero. These parameterizations are those furnished by definition (V.2.2). We briefly follow 3.4 of [BG]: the basic idea is that if I and J are two such intervals and if I \ J is connected, they can be joined in a new interval I [ J. This is continued as far as possible until a complete interval is obtained, i.e. R topologically, this since intersections of intervals are connected. There remains the case where two intervals I, J are such that I \ J has two connected components; but then the union is topologically a circle. To carry out the preceding, we reparameterize the intervals so that the



joins are all translations (changes in orientation are easily excluded), which is possible since a local diffeomorphism of R with positive gradient is always conjugate to a translation precisely because a real function with strictly positive derivative on an open interval always has an inverse function. These considerations by themselves show that we construct, in the case of a circle, a periodic parameterization.

Fig. V.3.1.

Here now is the first concept (rung of Jacob’s ladder) necessary for advancing in our understanding of plane curves. We consider an arbitrary continuous mapping f W S1 ! S1 of the circle S1 into itself. We orient the circle and consider the mapping (universal covering of the circle) exp W R ! S1 given by t 7! e it D .cos t; sin t /. The basic fact is that f is lifted to a mapping f W R ! R that

Fig. V.3.2.



commutes with the two exponentials: exp ıf D f ıexp. This comes from f being .0/ continuous and exp being locally a diffeomorphism. Now the quantity f .2/f 2 is an integer if exp.f .2// D exp.f .0// and is called the degree of f W S1 ! S1 . Heuristically it represents “the number of times that the circle is covered (globally) by the mappingf ”. We can calculate it at a single point if f is well-behaved about this point; we count the inverse images of this point under f with coefficient equal to C1 or to 1 according as to whether the orientation is preserved or changed. In particular the degree is always equal to zero if the circle isn’t covered completely by the image f .S1 /. An essential property of degree is that it is invariant under hom*otopy of mappings, i.e. under continuous deformations; see Sect. V.XYZ. Since the degree is an integer, this isn’t very surprising. See 7.3 of [BG] for the details. V.4. The Jordan theorem Here is the proof of Jordan’s theorem for simple closed curves of class C2 , with the aid of the notion of index of a point with respect to a closed plane curve C (oriented and just continuous for the moment). For each point m of the complement R2 n C we define the index of m with respect to C as being equal to the degree of the mapping f W S1 ! S1 defined thus: we choose a periodic parameterization s.t / s.t /m 2 S1 . The invariance of the degree on Œ0; 2 of C and define f by f .t / D ks.t /mk under hom*otopy shows that this index is constant on each connected component of the complement R2 n C. Moreover, if m is sufficiently far from C then the degree is equal to zero, since the image f .S1 / doesn’t cover all S1 (i.e. not all the directions). We next need to find how the index behaves as the curve is traversed. A figure allows us to guess that it changes from C1 (or 1) because very near the curve we have nearly on the one side and on the other; see [BG] for the full argument, which uses the differentiability of C. Now R2 n C has but two connected components; it is here that we need to use of the properties that C is simple and closed. We see this by taking a tubular neighborhood T of C; then T n C clearly has two connected components, and by continuity

Fig. V.4.1. A tubular neighborhood of a (simple closed) curve is, by definition, obtained by moving at each point of the curve along the normal to the curve (i.e. along the line perpendicular to the tangent at that point) a given distance " in both directions. It is easily seen that the simplicity of the curve indicates that the neighborhood is just like that of the figure if " is chosen sufficiently small



each connected component of R2 n C intersects just one of these components. Thus we have finally shown: If C is a simple closed curve, then R2 n C has exactly two connected components, the one (the exterior of C) made up of points whose index with respect to C is equal to zero, the other (the interior of C) made up of points whose index with respect to C is equal to C1. We could also characterize them by the fact that one is bounded and the other isn’t. What was said in Sect. V.3 for calculating the degree of a mapping S1 ! S1 provides the following criterion given in the figure:

Fig. V.4.2. In this figure, in order to find whether a given point is interior of exterior to the curve, a half-line is traced emanating from the point which always cuts the given curve transversally (for such a line to exist, reasonable conditions are needed for the curve, which readers can find for themselves). Then the point is on the interior if the number of points of intersection is odd, on the exterior if it is even. [BG] Berger, c Presses Universitaires de France Gostiaux (1987)

V.5. The turning tangent theorem and global convexity In this section we consider simple closed curves of class C1 and parameterize them periodically by g W Œ0; 2 ! R, where the derivative g 0 .t / is continuous and nonzero. We take any Euclidean structure and thus have a natural mapping 0 f W S1 ! S1 defined by f .s/ D kgg 0 .s/ . The turning tangent theorem states that .s/k the degree of f is equal to ˙1, the sign being dependent on the orientation chosen. That is, globally the tangent will turn exactly 2 when we traverse the curve a single time. This theorem, indeed visually evident when the curve is strictly convex, is a lot less so if it “turns some in each sense”, and in any case the proof is tricky. Here is an outline, the details are in [BG] and also in the excellent book on curves, (Dombrowski, 1999), in portions of 1.3.



The process consists of finding a deformation f which will clearly be a mapping of degree 1. But we have seen in Sect. V.3 that the degree is invariant under (continuous) deformation. The deformation depends on a parameter k; it can be seen clearly in the figure for small k. We replace the tangents by the segments associated with the arcs corresponding to parameters distant from k D t s, i.e. precisely g.t/g.s/ , which we define on a domain formed by the s; t with 0 6 s 6 t 6 2. kg.t/g.s/k g.2/g.s/ , for which the set constitutes At the end of the deformation we find the kg.2/g.s/k all the unit vectors with origin the point g.0/ D g.2/, which turns thus through 2 as we traverse the curve. But we need to pay attention to some details in order to have a good deformation that is continuous over the entire domain of definition of s and t right up to the boundary. After having chosen the lowest point of the curve as the origin, supposing that g moves from there to the right, we define the deformation fk as the path traversed by the vector g.t / g.s/ g 0 .t / ; ou si s D t kg.t / g.s/k kg 0 .t /k

when s, t increase from 0 to 2 in the following way: – we let t increase to t D k while s remains at 0; – then s and t increase equally, i.e. s D t k until t D 2; – then t remains equal to 2 as s increases to the same number. We can describe this path by introducing two functions sk .u/ and tk .u/ of the same parameter u, ranging over the interval Œ0; 2 C k. The fact that the interval changes with k isn’t important. The essential thing is that fk doesn’t pass through the origin (because g is simple and g 0 is always ¤ 0), and because the extremity of fk coincides with its origin g 0 .0/. The index of fk is thus conserved under the deformation. Now f0 is the path t 7! g 0 .t / (0 6 t 6 2), and f2 consists of the path g.t/g.0/ (0 6 t 6 2), which makes a half turn about 0, from 0 to , and t 7! kg.t/g.0/k g.2/g.s/ (0 6 s 6 2), which makes a second half turn about of the path s 7! kg.2/g.s/k 0, from to 2. The index of f0 is thus, like that of f2 , equal to 1.



f k (k small)



2π S1

f 2π

c Presses Universitaires de France Fig. V.5.1. [BG] Berger, Gostiaux (1987)



Here is a first application of the turning tangent theorem, for which it is indispensable: “local convexity versus global convexity”. We consider a closed curve C of class C1 and one of its points m where it does not intersect itself. We say that C is locally convex at m if there exists a neighborhood U of m such that in U the image of C is entirely in one of the half-planes determined by the tangent to C at m. A closed curve can be everywhere locally convex without having to be the boundary of a plane convex set; see Chap. VII. In contrast it seems evident that the curve must be globally convex if, in addition, it is simple. Such is actually the case and is an immediate consequence of the turning tangent theorem: Every locally convex simple closed curve is globally convex, i.e. it is the boundary of a convex and compact subset of the plane.

Fig. V.5.2. The (non simple) closed curve of the figure, although locally convex, is not convex globally; moreover its index equals 2. In the graph of f (argument of f 0 ), the dotted portion is impossible because of the local convexity. [BG] c Presses Universitaires de France Berger, Gostiaux (1987)

It suffices to show that the curve is entirely on one side of each tangent, for it is classical in the theory of convexity (see Chap. VII) that the intersection of all these half-planes will yield a convex set. Let m thus be a point where there are points on both sides of the tangent T at m; by compactness there are points of tangency parallel to T on both sides. Of the three parallel tangents thus obtained, two have 0 definitely the same sense. We then look at the graph of the lifting into R of kgg 0 k , which is denoted by f in the caption for Fig. V.5.2. The local convexity implies that the graph is that of a function that is nondecreasing on the whole interval Œ0; 2.



The turning tangent theorem states that the final value in 2 of this graph equals exactly 2. Thus our tangents that are parallel with the same direction correspond to a single value in Œ0; 2 and the graph will be horizontal (constant) between the two tangents; thus the curve must be linear over the interval and the tangents must be identical, which is the desired contradiction. V.6. Euclidean invariants: length (theorem of the peripheral boulevard) and curvature (scalar and algebraic): Winding number We are now essentially in the Euclidean plane E2 and we consider a kinematic curve f W I ! E2 , of class at least C1 . We can thus speak of velocity for f , precisely the velocity vector at t of f is f 0 .t / (recall that we are considering only kinematic curves for which f 0 .t / 6D 0). The (scalar) velocity of f at t is by definition kf 0 .t /k. Also of importance is the distance traveled, or the arc length, of the curve f from R t0 t to t 0 , defined as the integral t kf 0 . /k d . We justify this abstract definition by showing, without difficulty, that this length is equal to the limit superior of the length (perimeter) of all polygons with vertices ff .t /; f .t1 /; : : : ; f .tn /; f .t 0 /g (t < t1 < < tn < t 0 ) inscribed in the curve between the values t and t 0.

Fig. V.6.1.

The essential thing realizable since the velocity is never zero is that we can reparameterize the curve by the arc length, starting from a fixed origin m0 D f .t0 / Rt on the curve. In fact we define the function s by s.t / D t0 kf 0 . /k d , and then the new parameterization of f by g is such that g.s.t // D f .t /. This time, by construction, the scalar velocity is constantly equal to 1, since kg 0 .s/k D 1 for all s. We say that the curve is parameterized by arc length , and we may always suppose that this is the case for every kinematic curve, by an appropriate reparameterization. Here is a first result, perhaps rather evident, although perhaps not all drivers will be persuaded but which also requires the turning tangent theorem of Sect. V.5 i.e. that the peripheral boulevard theorem. The problem is to know whether there is a significant difference in distance traveled in making a circuit of the town in question by the interior lane of the periphery rather than by the exterior lane. It seems clear that the interior trajectory is always shorter for convex peripheries, but is it really shorter than the exterior trajectory if there are lots of contortions? To make the



calculation quickly and correctly, we identify E2 with C as explained at the end of Sect. II.XYZ; in particular, multiplication by i is the rotation by 2 . Let g be a reparameterization by arc length of the median curve (of class C2 ) of the periphery. We write the unit tangent g 0 in complex form, i.e. in the form g 0 D e i , and let " be the lane width, so that the parameterization of the interior will be g C i "e i . The velocity is .g C i "e i /0 D g 0 .1 " 0 / and the scalar velocity is 1 " 0. The total RL length of the interior is thus 0 .1 " 0 / ds D L " .L/ .0/ . But the turning tangent theorem assures us that, precisely: .L/ .0/ D 2, whence the interior distance sought is L 2". For the exterior it will thus be L C 2", and the total increase is always 4". It is thus constant and ultimately quite minor, regardless of the various twists and turns of the peripheral.

Fig. V.6.2.

We are now necessarily in class C2. The expression g 0 D e i signifies in fact that we have chosen an axis so as to be able to speak of the angle of the tangent to the curve with this axis. And the quantity 0 encountered above is thus the velocity at which the tangent turns for the kinematic curve considered. A sketch shows that, the larger this velocity is in absolute value, the more the curve is curved. For example for a circle of radius R this velocity will be R1 . We thus define the scalar curvature of the curve under consideration to be equal to j 0 j, and denote it by K. If we have oriented the plane, we give 0 a sign; this will be the algebraic curvature, denoted by k; and for the scalar curvature: K D kg 00 k for the second derivative of g. In the science of kinematics it is useful to know how to calculate K for an arbitrary parameterization f. We find K.t / D

f 0 .t / ^ f 00 .t / ; kf 0 .t /k3=2

where ^ denotes the exterior product, which may also be calculated as a determinant in orthonormal coordinates (this determinant depends on the orientation of the plane, but here we can take the absolute value). In thinking of the case of the circle, where K is not zero, we say that R D K1 is the radius of curvature of the curve considered



c Presses Universitaires de France Fig. V.6.3. [BG] Berger, Gostiaux (1987)

at f .t /. In kinematics there is also a need to decompose the acceleration vector of the curve f into two components, one along the tangent, the other along the normal to the curve (the normal is by definition the perpendicular to the tangent). But if we choose a unit vector there in a way that is connected intrinsically to the curve, this will only be possible when the acceleration vector is not parallel to the tangent. 0 We thus define a standard normal f ; g at f .t / by taking D kff 0 .t/ and the unit .t/k 00 vector located on the normal on the same side as f .t /. Thus the said formula of 2 intrinsic components of the acceleration is written: f 00 .t / D v 0 C vR (here v 0 is the derivative with respect to the time t , of the scalar velocity). The centrifugal force 2 is vR ; and we see that it is indeed large if the velocity is large, but also if curvature is large (i.e. if the radius of curvature is small). We see that the curve turns toward the side of the acceleration in the following sense: at a point m where the curvature isn’t zero, we can define a half-plane attached to the curve at that point, denoted convexm C in the figure above. This halfplane is bounded by the tangent to the curve and contains the acceleration vector. If we change the parameterization the new acceleration vector will remain in this

c Presses Universitaires de France Fig. V.6.4. [BG] Berger, Gostiaux (1987)



“convexity half-plane” for our curve. The term convexity is justified because, locally, the curve is situated in the half plane and moreover is the graph of a convex function. The consequences of this formula in practical life are legion, whether we suffer from travel sickness, are marking out roads or are laying train tracks. The problem is to find the markings that connect the various “turns”; it essential that the curvature change gently when the velocity is large. As a straight line has zero curvature and a circle a constant nonzero curvature, if is thus necessary to connect the circles to lines by means of appropriate curves. Our cherished conics are not at all good in this regard, for their curvature is never zero. Numerous curves have been proposed, and the matter is still not settled, all the more so in that the markings have to conform to imperatives of economy, of visibility, etc. For railroads (Alias, 1984), can be consulted; we mention, moreover, for those interested in technical matters, that two successive opposing curves need to be connected by a piece of straight line long enough to permit the train to regain its equilibrium. The tangent gives a first approximation to the picture to the image of the curve; the tangent has contact of order 2 with the curve. The curvature, when it is nonzero, furnishes an approximation for the next order. In Euclidean geometry, we think of “circles”; the circle that has contact of order 3 with the curve is called the osculating circle (from the Latin osculari, to kiss, to embrace, terminology due to Lagrange). It is defined when the curvature is nonzero, has radius R D K1 and is thus the circle centered on the normal at a point situated a distance R from the point considered. Its center is called the center of curvature of the curve. It can be found by considering the envelope curve for the family of normals to the curve; the family that envelops a curve is called the development of the curve. We will scarcely speak of developments there is not the space but say here only that the point of contact of the normal with the development is in fact the center of curvature. In particular, at a point of nonzero curvature, the curve is locally strictly convex: it is entirely on one side of its tangent the side of the acceleration with a single point of contact. How is the curve situated locally with respect to its osculating circle? Things go well if, in parameterizing by arc length, the derivative K0 of the curvature is nonzero (a condition on the third derivative), in which case the curve traverses its osculating circle in the neighborhood of the point considered. We recommend trying to draw a curve and one of its osculating circles, which is very difficult to do in such a way

Fig. V.6.5.



that the circle is really osculating to draw it so that it isn’t, at least locally, either on the interior or the exterior. It is more difficult still to draw the family of all the osculating circles, because this shows (it’s a good exercise) that, in the neighborhood of a point where K0 6D 0, the osculating circles are mutually nonintersecting. In the figure below, which was provided us by Étienne Ghys, it will be observed that the curve itself seems to have been drawn, but in fact it isn’t; it’s an optical illusion. For more on this topic, see the reference given in Sect. V.10 regarding human vision. This figure is also a very good entirely natural example of a continuous vector field in the plane for which the integral curves aren’t unique: we consider the vector field formed by all unit vectors tangent to osculating circles to a given oriented curve; then the trajectories passing through an arbitrary point outside the curve are unique, but at a point of the curve we can either follow the circle or else the curve while remaining tangent to the vectors of the field the whole time. This figure is also a good example for the theory of envelopes: it’s a classical “result” that the point of contact of a curve of a family with its envelope is the limit of the point of intersection of two neighboring curves of the family. This monstrosity is spelled out in many books, along with others concerning duality. Such a result is true in certain special circ*mstances, but certainly not in general since the osculating circles don’t intersect at all! It’s a result that is true for envelopes of lines. For more on this topic, see Berger (1972). It is important to point out that there does not exist any classification of the local form for curves of class C1 , even if the velocity remains nonzero. The tangent can be traversed an infinite number of times in the neighborhood of such a point. And if

c Étienne Ghys Fig. V.6.6.



Fig. V.6.7. Possible shapes of a curve of C1 at a point where the velocity vanishes, and c Presses Universitaires also where it doesn’t vanish. [BG] Berger, Gostiaux (1987) de France

the velocity becomes zero, there may very well not be a tangent. It’s left to readers to imagine really appalling situations. We now consider closed curves C of class C2 in the affine plane R2 , so that readers may want to review the degree theory of Sect. V.3 and its use in Sect. V.6. We choose a Euclidean structure, an orientation and any parameterization f W Œ0; 2 ! E2 , 0 so that the mapping kff 0 k W S1 ! S1 has a degree, called the winding number of f . It’s an integer denoted wind.f /, and because of its invariance under deformation (Sect. V.3), depends only on C and orientation but not on specific parameterizations, whence it is an invariant wind.C/ attached to each closed plane curve. Now the calculation of Sect. V.6 remains unchanged and thus, with some Euclidean structure, we can calculate the winding of a closed curve of length L by RL means of the algebraic curvature k, using the formula: wind.C/ D 0 k.s/ ds. For RL the scalar curvature K D jkj we will thus always have 0 K.s/ ds > j wind.C/j. Readers can study the case of equality. We will return to this winding number very

Fig. V.6.8.



extensively in Sect. V.10, in particular to its connection with the number of points, called (double points), where the curve intersects itself. For a very recent “DNA” inequality, bearing on the integral of the curvature, see Tabachnikov (2001).

V.7. The algebraic curvature is a characteristic invariant: manufacture of rulers, control by the curvature Scalar and algebraic curvature are Euclidean invariants of kinematic curves: they are preserved under isometry (but pay attention to orientation in the case of algebraic curvature). We are interested in the converse: is a curve characterized, within isometry, by its curvature? For example, are circles (resp. lines) the only curves of nonzero (resp. zero) constant curvature? We suppose throughout that the curve is parameterized by arc length. Then for the curvature zero case we have g 00 D 0, so that g is a vector function of the form g D t x C y: and the image is thus linear (in fact, a piece dependent on the interval of definition of a line). Instead of treating the case of circles by hand, we proceed in an oriented plane as follows: If two kinematic curves g and h, defined on the same interval, are such that their algebraic curvatures are equal for each s: k.g.s// D k.h.s//, then there exists a an orientation-preserving isometry I such that I.g.s// D h.s/ for each s. We can thus say that the function k.s/ is the intrinsic equation of the curve. We place ourselves in C with the language of Sect. V.6: in coordinates with 0 g D .x; y/, we have (since the velocity a unit vector) R s x .s/ D cos. .s// and 0 0 y .s/ D sin. .s//. But k.s/ D .s/, so that .s/ D 0 k.t/ dt C C and we have Rs Rs x.s/ D 0 cos. .t // dt C a, y.s/ D 0 sin. .t // dt C b. The curve g is thus determined within a translation and a rotation. In particular the circles are clearly the only curves with nonzero constant curvature. Observe also that the scalar curvature K, which only yields j 0 j, doesn’t suffice: if K becomes zero in places we can’t draw the conclusion. Take for example the graph of x 7! x 3 and reverse one of the two pieces about 0.

c Presses Universitaires de France Fig. V.7.1. [BG] Berger, Gostiaux (1987)



We mention an application (little known to students in schools today, so it seems, but which the author learned from his teacher during his final year) of the fact that the only curves of constant curvature are lines and circles. How are really straight rulers not graduated rulers, but straightedges, called in some instances “runners” by technicians manufactured? One solution would be to use one of the two of the inverters mentioned in Sect. II.3, that of Peaucellier or that of Hart. It may seem astonishing that still today the practical procedure is as follows: three approximate straightedges are taken and rubbed against each other, two at a time. When a rubbing between two curves is perfect, i.e. if there is sliding of one curve on the other, it’s necessary that the curvature of each be constant. Thus three circles (or lines) are obtained. But circles are impossible two at a time: the convexities are opposed for each pair. The only possibility is that they all three be lines. Those interested in groups and more abstract language will have noticed that it amounts to the same thing to say that the only one-parameter subgroups of the group of planar Euclidean displacements are those of rotations about a fixed point and those of parallel translations. In practice a product is obtained that can be checked for its adequacy. This is done by interferometry, where recently lasers have been used. More precisely, defects are measured and corrected by polishing (possibly with the aid of CAD); the result is checked again, etc., until sufficient precision is obtained. We are repeating here in part what was said in Sect. II.XYZ. Even though the laser provides perfect lines of light, it is only used at present for estimating very effectively, to be sure deviations (of a lot of things, for example in installing railroad tracks). We should know that if Michelson was the first to be able to estimate the velocity of light with a precision sufficient for distinguishing c from c ˙ v, where v is the velocity of the earth in its orbit, it was because he was an excellent experimenter, constructor of slides, of rulers that were finely graduated; see Sect. II.XYZ. For the fabrication of surfaces, plane or spherical, see Sect. VI.6. We have already treated, in Sect. II.XYZ, the manufacture of graduated rulers and length measurement, and likewise that of goniometers for the measure of angles. For surfaces, fabrication of spheres, of planes, see Sect. VI.6; for the fabrication of balls, see Sect. VII.13.C.

Fig. V.7.2.



The fact that knowledge of the curvature of a plane curve as a function of arc length determines the curve leads us to anticipate that more can be done. Typically: if we have control over the curvature, can we hope to control the curve? Here are two results, both completely intuitive, concerning two curves where the one is always more curved than the other: (1) (Schur, 1920). If for each s we have k.f .s// > k.g.s//, then for sufficiently small s and t we have, for the length of the chords: d.f .s/; f .t // 6 d.g.s/; g.t //. (2) Under the same conditions, if f and g have a common origin at s D 0 and are tangent at this point, and if moreover the algebraic curvature of g is positive, then for s sufficiently small the image of f is contained in the interior of g.

Fig. V.7.3.

Once the curves are no longer simple, the restriction to small parameters is indispensable. Readers should be wary of these two results which, although not difficult, aren’t entirely immediate either. For the second, the simplest thing is to use Euler’s equation for curves, which will be given in Sect. V.8. We note a corollary for mechanics: let C be a simple closed curve, and m (resp. n) a point where the curvature is maximum (resp. minimum). Then we can roll the osculating circle at m around the interior of C, and we can roll the image of the curve around the interior of the osculating circle at n. V.8. The four vertex theorem and its converse; an application to physics The curvature function of a planar simple closed curve (of class C2 ), as a function of arc length, gives rise to a continuous function on the circle S1 . Is it an arbitrary function? Otherwise expressed, can we always find a simple closed curve for which the curvature as a function of arc length is the given arbitrary continuous function k on S1 ? Certainly not, for if we look at the formulas given in the preceding



section and if L is the length of the curve, it is necessary that the function defined RL Rs by .s/ D 0 k.t/ dt satisfy the two conditions x.L/x.0/ D 0 cos. .s// ds D 0 RL and y.L/ y.0/ D 0 sin. .s// ds D 0, equalities that have no reason for existing in general. But we might gain hope in being less demanding, i.e. in no longer requiring that curvature be given as a function of are length, but only as an arbitrary parameterization of the circle. Formally we seek an embedding (not necessarily parameterized by arc length) f W S1 ! E2 such that the curvature of the image f .S1 / at the point f .s/ is equal to k.s/. The mathematical problem consists thus of reparameterizing the circle, i.e, finding a diffeomorphism g W S1 ! S1 such that the function f D k ı g satisfies the two zero integral conditions above. Now this is not always possible. We will call a point of a curve where the curvature is zero a vertex. The nomenclature comes from the fact that this is what happens at the four vertices of an ellipse, but in general a vertex may be a local maximum, a local minimum or more generally a more complicated point. Now in Mukhopadhyaya (1909) it was shown that: a convex simple closed curve must have at least four vertices, whereas a continuous function on the circle can very well have only a single maximum and a single minimum, the derivative at all other points being nonzero. We are going first give the ideas of two proofs of this result, called the four vertex theorem, the one because it’s very intuitive (while most other proofs are completely obscure), the other because it’s a really good conceptual proof and because we will see lots of generalizations of it later. The result is true whether the curve is convex or not, but it was not until (Kneser, 1912) that the general case was obtained, although the proof remained obscure. The proof preferred by the pure geometer may well be that of Osserman (1985). A detailed history of the four vertex theorem can be found in Dombrowski (1999).

Fig. V.8.1.

Osserman begins by enclosing the curve in a circle of smallest radius R that contains it; this circle will thus be tangent to the curve at no fewer than two points. Moreover Osserman shows that if the circle is tangent at N points, then the curve has



at least 2N vertices; the visually appealing idea that, between two points of contact, the curvature must have a minimum which is smaller than R1 is easily realized by diminishing the radius of the circle until it is tangent; and since at points of contact the curvature is > R1, we are done. The proof can be adapted without difficulty to the nonconvex case. The second proof, more conceptual, is based on a remark about periodic functions on R alias functions on the circle S1 say of period 2. Moreover, the same idea will turn out to be basic in Sect. V.10 for a whole series of recent results (and surely for more yet to come). It seems to be in Blaschke (1916) that this conceptual proof appeared for the first time. However, now it will be necessary to suppose that the curve has nonzero curvature everywhere and thus is strictly convex, as we have seen above. With the notations of Sect. V.7, if g 0 D e i , we thus have K D 0 D dds > 0, which allows us to take as parameter; it varies from 0 to 2. The formulas of R 2 R 2 Sect. V.7 thus become 0 K1 . / cos d D 0 and 0 K1 . / sin d D 0. In the language of Fourier series, this means that the function K1 is orthogonal to the first two harmonics, i.e. to sin and cos , which implies that K1 Pmust have at least two minima and two maxima. The general idea is that f . / D k>2 .ak cos k C bk sin k / (the derivative of K1 is of this form) has at least as many zeros with change of sign as its first term, i.e. 4. The proof is as follows: the function f . / necessarily changes sign, since it has mean zero (except for the case K1 D const.). If f . / becomes zero and changes sign at a finite number of points x1 , . . . , xN , then N is even because of the periodicity. If N D 2, we can find a, b, c such that a cos C b sin C c becomes zero at x1 and x2 and has the same sign as f at all

Fig. V.8.2.



R 2 other points, whence 0 f . /.a cos C b sin C c/d > 0, a contradiction. Thus N > 4. Here are two small variations in the presentation: Euler’s equation for convex curves consists of writing the curve as the envelope of its tangent, the latter being the line that makes the angle with a fixed axis and u denoting its distance to an origin situated on the interior of the curve; see Fig. V.8.3. It is then shown that the radius of curvature equals u C u00 , where u00 denotes the second derivative of u with respect to the variable . We can then reread the preceding in two ways: by R 2 R 2 periodicity the integrals 0 .u C u00 / cos d and 0 .u C u00 / sin d are zero and we have thus at least two maxima and two minima; but we can also look for points where the derivative of the radius of curvature is zero, i.e. u0 C u000 D 0. R 2 R 2 Again, we have both 0 .u0 C u000 / cos d and 0 .u0 C u000 / sin d equal to zero. But, just as functions orthogonal to the first two harmonics have at least four extrema, we see also that they have at least four zeros. We will encounter this point of view of zeros of functions orthogonal to harmonics in Sect. V.10. More geometrically, and this will also serve for the sequel, we can consider K1 as a mass distribution on the circle S1 , so that the fact that the two integrals above are zero says that the center of gravity of the total mass is at the origin. This is clearly impossible if K has only one maximum and one minimum, so consider a line passing through the origin such that K.t / D K.t C /, which always exists. Then all the density is larger on one half of the circle and smaller on the other, thus the center of gravity is not on this line. The simplicity condition is essential; the limaçon of Pascal (of the father, not of Blaise) of the figure (for the curve with polar equation D 1 C 2 cos ) has a curvature strictly increasing from 5=9 to 3 (more than a half) and is surely closed and convex. In Sect. V.8 we will encounter, in a more general context, the use of orthogonality to the first harmonics in obtaining functions having multiple minima and/or points where they become zero. Can we float logs when there is weightlessness? The four vertex theorem will answer the question. Here are the details of this anecdote (we will encounter logs again, but then with gravity, in Sect. VII.13.C). Two physicists sought out the author in 1991 with the following elementary problem: let C be a convex compact set in the Euclidean plane and ˛ 2 Œ0; =2 a given angle: to find a line D which cuts C at an angle ˛ at the two points of intersection. The physical motivation is simple: we consider a very long prismatic object in equilibrium in an interface situation between two liquids, everything under weightlessness. (A log floating on water is a simple interpretation; in fact, the physicists were interested in long molecules in problems of condensed matter.) The equilibrium positions correspond to the lines 23 sought above, where ˛ is given by the formula cos ˛ D 13 , the ij denoting the 12 interfacial tensions of the system, 1 and 2 for the two liquids and 3 for the object itself. This is a very simple problem; typically it, like many analogous problems, must be solved by the intermediate value theorem. The idea here is to consider the area function of a triangle formed by two tangents to C for which the angle at the vertex



Fig. V.8.3.

Fig. V.8.4. In the figure on the right, the two shaded surfaces must be equal; this shows, in the limit, the desired condition on the angles. V.8.4. Raphaël, di Meglio, Berger, c EDP Science Calabi (1992)

equals 2˛ (area A represented in black in Fig. V.8.4). The derivative of the area will be zero if and only if the angles at the base are equal, and thus equal to ˛. We have thus found two positions of equilibrium, the maximum and the minimum of this function. Moreover, one at least is thus stable. Unfortunately (but ultimately fortunately) for the author, these physicists knew the four vertex theorem, and thus in fact conjectured the existence of at least four equilibrium positions. This is true, but the proof is more difficult; we owe to Calabi at least four different solutions. Here is the one he prefers. We give it because, beyond its elegance, it uses the Euler equation introduced above for representing strictly convex plane curves. The curve is parameterized by the angle of the tangent with a fixed axis, but the curve is defined as the envelope of its tangent at the point of angle ; the function sought is thus the distance u. /. We consider two tangents to the curve at angles C ˛ and ˛: the triangle formed will be that sought if the two lines with angle passing through the points of contact with the two tangents coincide. Calabi calculates their distance . / and finds Z ˛ . / D uO 000 . / C uO 0 . / .cos ˆ cos ˛/ dˆ: ˛



Fig. V.8.5.

Here the function uO is the natural mean value R˛ .cos ˆ cos ˛/u. ˆ/ dˆ u. / O D ˛ R ˛ : ˛ .cos ˆ cos ˛/ dˆ Let us caste a glance at the correspondence u 7! uO between functions, which in fact yields a correspondence between curves given by their Euler equation. We

Fig. V.8.6. Certainly four positions (ellipse), but in general more. Raphaël, di Meglio, c EDP Science Berger, Calabi (1992)



have seen that the curvature equals u C u00 ; thus the zeros of correspond exactly to the vertices of the curve of equation uO and there are at least four of them, Q.E.D. More details can be found about this personal account in Berger (1993b), Raphaël, di Meglio, Berger, and Calabi (1992) and Raphaël (1992). In dimension 3 the problem is of a totally different order of difficulty, because in general an arbitrary convex set never admits a plane section making a constant angle with the tangent plane; it is thus necessary that the separation surface is somewhat deformed about a plane (to be determined); see Raphaël and Williams (1993) for this problem in the physics of condensed matter, which remains essentially completely open. We must now return to the problem posed right at the beginning of this section: to realize an arbitrary (continuous) function on the circle as the curvature of some simple closed curve. The four vertex theorem shows that this is not possible without

Fig. V.8.7.



some supplementary condition. But it turns out that: each function that has at least two maxima and two minima can be effectively realized in this way. This result awaited (Gluck, 1971), it still being required there that the function in question is never zero, i.e. that we work only in the realm of curves that are strictly convex. But we now know that the result is valid in all generality; see the very complete and historical exposition of De Turck, Gluck, Pomerleano, and Shea Vick (2007). Note that the analogous result in higher dimensions, for example for surfaces, is valid for every function; see Gluck (1972). For example there exist surfaces where the Gauss curvature (see Sect. VI.5) has only a single minimum and a single maximum. Returning to the case of plane curves, the proof of this converse theorem is difficult. The idea concerns sliding masses along the circle in such a way that the center of gravity is the origin. We begin with a distribution as in the figure; we effect two types of sliding and obtain thus four points as in the figure; they “encircle” the origin. Thus a hom*otopy in two parameters will give a ruled surface that will cover the origin at one point at least. For a discrete presentation of the four vertex theorem with the aid of polygons, and more, see Ovsienko and Tabachnikov (2001). V.9. Generalizations of the four vertex theorem: Arnold I The four vertex theorem haunted many mathematicians seeking to know whether there are concepts hidden behind it. We will soon see such concepts appearing, but we first look at some intermediate results. The four vertex theorem is a Euclidean theorem; we can say it asserts that, among the osculating circles to the curve, at least four have contact with it of order higher than 3. But, subsequent to Chaps. I and II, we like purely affine geometry and also projective geometry, which are simultaneously somewhat richer than the Euclidean version, as we shall see. We first give the results and subsequently see how it is with the proofs. We first see what replaces the four vertex theory in the purely affine case. In the affine plane, the conics replace circles. Just as the osculating circle at a point of a kinematic curve can be obtained as the limit of the circle passing through three points that tend toward the given point, so each (affine) kinematic curve a conic being determined by five points admits at each point a well-determined conic having a contact of order at least 5 with the given curve. What corresponds to the vertices in the Euclidean case are then the conics that have a contact of order 6 at least; they are called sextactic. The corresponding points are called affine vertices. We thus have Mukhopadhyaya’s theorem (Mukhopadhyaya, 1909): Each simple convex curve in an affine plane possesses at least six affine vertices. This theorem remains valid in the projective context, conics there always having the same citizenship rights as does the notion of convex curve; see Sect. IV.4 as needed. What is peculiar to the projective context is precisely the case of non convex curves, precisely the simple closed curves that aren’t contractible and thus



Fig. V.9.1.

are hom*otopic to a projective line. We could define an inflection point of a smooth curve as a point where the curve crosses the tangent, but the right definition consists of stipulating that the two first vector derivatives are collinear. The curve crosses its tangent in general, but not necessarily; see the graph of x 7! x 4 . Then an old theorem, attributed to Möbius, is: Each simple closed curve in the projective plane that is hom*otopic to a projective line possesses at least three points of inflection. For this theorem of Möbius and its generalizations, see p. 4 and Sect. 3.7 of Haupt and Künneth (1967). It should be mentioned that if a cubic in the affine plane has only two points of inflection, it is because the third is at infinity. We will encounter this result in Sect. V.14. Notice that a projective line has no inflection point, but in a more precise sense (consider the first two vector derivatives) all its points are inflection points. In contrast to the other results mentioned, Möbius’s is by nature purely topological. Mukhopadhyaya’s is a generalization of what he had used in the Euclidean case of four vertices and remains ultimately rather mysterious. The idea is to see, in the Euclidean case, how some osculating circles are situated with respect to others (think of Fig. V.6.6). In the projective case, he likewise studied the various osculating conics to the curve: as in the Euclidean case, at a point of contact of order 5, but not more, the conic crosses the curve at the point considered. In Bol (1950) a more conceptual proof can be found, in part analogous to the proof of Blaschke given above. For a curve in the projective context we define a projective arc length and a projective curvature h (here these invariants depend on the third derivatives): Bol shows that for the reparameterized curve, and in arbitrary projecR tive coordinates fxi g (i D 0; 1; 2), we have hxi :xj D 0. This leads to many conditions, from which we infer that h must become zero at least six times; this provides six projective (and affine) vertices. It is here that we must mention that there exists an entire affine differential geometry, with notions of affine length and affine curvature. For example the sextactic points are those where the affine curvature is zero. The inflection points are, in contrast, the points where we can’t really even define affine elements. For this geometry, amidst a broad literature, see Guggenheimer (1963), Buchin (1983) and Chap. 9 of the synthesis (Dillen and Verstraelen, 2000). This affine differential geometry indeed exists in all dimensions, for example for surfaces, but is then much more



complicated. The essential reason, which may be inferred from what we will say just a little further on, is that the invariants associated with surfaces and arising from third derivatives are not the quadratic forms of the Euclidean theory of surfaces (see Sect. VI.6), but rather cubic forms. These are algebraic objects that are much more intricate to manipulate than quadratic forms. We are indebted to Arnold for placing the six vertex theorem in a more general context, which permitted him to obtain a whole series of generalizations. All this has to do with the study of points called extactic of order d.n/ C 1 D n.nC3/ C 1. 2 Here we consider the algebraic curves of degree n which have a contact of order d.n/ D n.nC3/ with a projective curve (typically nonalgebraic). At each point there 2 exists one of them, because we will see in Sect. V.13 that the space of all (plane) algebraic curves of degree n is of dimension n.nC3/ . We then find 2 for the lines 2 (n D 1), 5 for the conics and 9 for the cubics. The points where the contact is of order greater than d.n/ D n.nC3/ are called n-extactic or, n being understood, 2 simply extactic. Involved is an attempt to show that there are, with increasing n, more than the preceding would lead us to expect. On the other hand, we shall see that Arnold’s context only works, at least presently, for curves that are sufficiently close to standard curves. Here very briefly is the scheme. Just as the inflection points correspond to a zero second derivative trivially parallel to the first derivative, the sextactic points of zero projective curvature correspond to third derivatives, linear combinations of the first two derivatives. The n-extactic points correspond to n-th derivatives that are linear combinations of the preceding derivatives. Arnold interprets these as flattening points of the Veronese curve with associated with a plane curve C (we recall the Veronese surface from Sects. I.7 and II.1). We are working in the projective context; the Veronese mapping V W RP2 ! RPd.n/ is, in hom*ogeneous coordinates, .x W y W z/ 7! .x n W y n W z n W x n1 y W x n1 z W : : :/ and we define the n-th Veronese curve of C to be the curve V .C/ of RPd.n/ . We see rather easily that the n-extactic points of C correspond to flattening points of V .C/, which by definition are the points where the osculating hyperplane has contact of multiplicity at least d C 1 with V .C/. A small technical annoyance: this correspondence isn’t valid without a supplementary convexity condition, which helps to explain the statement of the theorem to follow. This flattening condition is thus the vanishing of det Y; Y0 ; : : : ; Y.d.n/C1/ , an (ordinary) differential equation of order d.n/ C 1, where Y W t 7! Y.t / is a parametric representation of V .C/. It can be shown by long and difficult work that we are in a situation generalizing that of the second proof of the four vertex theorem, specifically that we have a periodic function for which the Fourier series is lacking a certain number of initial harmonics . We then know (from a theorem with the



names Hurwitz-Kellogg-Sturm-Tabachnikov) that it has sufficiently many zeros: P .a k>n k cos k C bk sin k / has at least as many zeros as its first term, i.e. 2n. And so Arnold’s theorem is: Under an appropriate nearness condition for an algebraic curve, as technically required at present: for each n every plane curve admits at least d.n/ C 1 extactic points. For example consider a plane cubic having an oval. Then each curve sufficiently close to this oval admits at least 10 points where the osculating cubic at this point will have contact of order higher than 9. We have been inspired by presentations at L. Guieu’s seminar at Montpelier; a text by Arnold (among others, treating somewhat related subjects) is Arnold (1996). A recent synthesis is given in the Thorbergsson and Umehara (1999), which in particular treats space curves; and see also the very recent Thorbergsson and Umehara (2002, 2004). V.10. Toward a classification of closed curves: Whitney and Arnold II We begin now to take into account results concerning general, i.e. not necessarily simple, closed curves. Here we deal with global results, the final goal being, if possible, a complete classification of these curves. We first remark that such a classification has only a topological sense; plane curves are considered equivalent when they have the “same shape”, the “same appearance”. We suppose everything to be C1 for simplicity. Precisely, two curves are considered equivalent when they are obtainable one from the other by a diffeomorphism of the plane R2 . The Jordan theorem implies that all simple closed curves are equivalent among themselves, and to a circle in the Euclidean setting. Note that there is an orientation problem, both for curves and for diffeomorphisms of the plane. We will not always specify the context in which we work, both for simplicity and because it isn’t at all essential. The description below of more or less recent results has to be brief; we leave it to readers to consult the works cited for most of the definitions required. In the spirit of this book the idea is to show that, in order to progress in the theory of curves, we need to ascend several rungs up the ladder. In all areas of mathematics, the ideal classifications are those where we dispose over a series of invariants whose equality yields the equivalence of the objects considered. Section V.6 provides us with a first direction for investigation: with each closed curve C is associated an invariant, its winding number wind.C/ (defined only within ˙1 if there aren’t two orientations, one for the curve and one for the plane). By construction, and since it is an integer, it is invariant under diffeomorphisms of the plane. But it is also invariant under hom*otopies between curves (see the definition right at the end of Sect. V.XYZ). This invariant isn’t sufficient for classifying curves modulo a plane diffeomorphism, as is shown by the figures:



Fig. V.10.1.

However, Whitney rushed in immediately behind Hopf’s breakthrough proof of the turning tangent theorem by proving, in Whitney (1937), that: two plane closed curves are hom*otopic (in the class of closed curves) if and only if they have the same winding number). A hom*otopy between two curves (here it’s a particular case of a very general and completely analogous definition) is a continuous deformation of the one into the other, with the condition that all the intermediate curves are again really curves of the type considered, in this instance immersions (and not just any, in which case all curves would be trivially hom*otopic among themselves). The proof consists of constructing a hom*otopy by integrating at the vector velocity level the hom*otopy between the two mappings of the given circle by the tangents (the vector derivatives). The name Grauenstein is sometimes attached to this result because Whitney had said that the theorem and its proof were due to him. In fact, the theorem was already present without proof in Boy (1903). We remark, for the sequel, that such deformation in general necessitates passing through intermediate curves possessing triple points and/or points of self-tangency. Unfortunately and inevitably in view of the above figure numerous curves with the same winding number don’t have the same shape, aren’t so to speak situated the same way in the plane, the precise definition being that there don’t exist diffeomorphisms of the (whole) plane sending the one onto the other. This is equivalent to saying that there doesn’t exist a hom*otopy between the two where all intermediate curves are generic in the sense of Sect. V.10.1. In order to progress and be able to hope for a reasonable classification, we appeal to the notion of robustness (that we will see in Sect. VI.10 and also in Sect. VII.13.D). The idea is that the really interesting geometric entities are those that we see; we can neglect those that have probability zero of materializing. If there is a measure, which isn’t always possible, we can replace probability zero by measure zero. A more precise way of broaching the problem is to consider only those objects as interesting whose form is stable under small deformations, which is the origin of the word robustness. This idea is due to Whitney in the context of curves; but it is René Thom to whom we are indebted for finding the general context for the notion of genericity, of robustness, as well as the tool that is necessary and sufficient for treating it, that of transversality. For the planar closed curves, the right notion is that of generic (closed) curve:



(V.10.1) We say that C is generic (normal for Whitney) if the only multiple points of the curve (points where the parameterization is no longer injective) are the double points with distinct tangents.

Fig. V.10.2. Forbidden for a generic curve

This definition is obviously robust; but it is also optimal, for Whitney showed in the article mentioned that each closed curve can be made generic by a small deformation. This is visually evident for eliminating self-tangencies and triple points, but for the general case it is necessary to use Whitney’s structural results on differentiable functions. Throughout the article he succeeds in calculating the winding number of a generic oriented curve C, with the aid of its double points, from the formula that he proves: wind.C/ D ˙1 C NC .C/ N .C/; where NC .C/ (resp. N .C/ ) is the number of positive (resp. negative) double points. A double point is called positive if, when we arrive there for the first time, we see the second branch of the curve cross before itself from left to right (mutatis mutandis for the negative case). The ˙1 can be determined by the orientations. Readers will have the courage to verify Whitney’s formula in the two cases of Fig. V.10.1.

Fig. V.10.3.

The proof consists of applying the turning tangent theorem to each loop furnished by the double points and then forming an appropriate sum. A nice corollary is: (V.10.2) We always have wind.C/ 6 n C 1, where n denotes the number of double points.



Whatever the beauty of Whitney’s formula, it scarcely does more than scratch the surface of the desired classification for the shapes of curves. The two curves of Fig. V.10.1 have the same winding number and the same numbers NC and N , specifically 5; 4; 0. If we exclude the result, rather isolated in its spirit, of FabriciusBjerre and Halpern mentioned right at the end of this section, no work on the classification problem appeared before 1994, the date of the revolution brought about by Arnold (1994b); see also Arnold (1994a), (1994c)). Numerous novel concepts and results are to be found there. We mention some of the most striking. First a complete classification of minimal generic curves: a generic curve is said to be minimal when its total number of double points n.C/ is minimum for a given winding number, say n.C/ D wind.C/ 1, the case of zero winding number being excluded.

Fig. V.10.4. The standard (minimal) curves

To classify the minimal generic curves, we must introduce the combinatorial figure called the Gauss diagram of a curve. It amounts to marking values on a circle (representing a parameterization of the curve) that yield, in increasing order of their appearance on the curve, the double points. Two values are joined by a chord in the disk when they give the same double point:

Fig. V.10.5.

It is easy to see that these chords never cross if the curve is minimal; they thus divide the disk into a number of connected components equal to wind.C/ and we make a tree, for which the vertices are precisely the components and whose edges correspond to the chords separating two of these components. Finally we root the tree, marking in black the connected component whose boundary chords correspond to the points located exterior to the curve. An entire book Fiedler (2001) is dedicated to Gauss diagrams and the study of space curves.



For curves a bit simpler than those of Fig. V.10.21, we obtain:

Fig. V.10.6. Gauss diagrams

Arnold’s classification theorem states that, for each integer n, the equivalence classes (by diffeomorphisms of the plane, i.e. the forms of curves) of generic minimal curves with n double points are in bijection with the rooted trees with n vertices. The proof uses nothing conceptual beyond Whitney’s formula. Shown below are the 9 classes of curves associated with the 9 possible trees for the minimal curves with four double points, as well as the case of two double points; readers should treat the case of three double points: Readers can look at larger numbers of double points to see the type of growth of the number of classes. It is important to remark that Arnold’s theorem is definitely in the spirit of introducing combinatorics into geometry; see for example Sects. II.1, then VIII.4 and VIII.10. In Biggs, Lloyd et al. (1995) can be found the exact number of rooted trees as a function of the integer n; in any case the growth is exponential in n. Before continuing, we mention that the complete classification of shapes not necessarily minimal curves has not terminated; it is not even certain that this classification is at all inspiring. It can be found up to five double points in Arnold (1994b). The corresponding table might suggest to some readers that the notion of minimal curve is the right one; but from an aesthetic point of view the table might seem discouraging. One of the obstacles to the classification is this: a Gauss diagram can always be associated with a generic curve, but in general the chords can now intersect and, furthermore, each diagram doesn’t necessarily originate from just one curve.



c American Mathematical Society Fig. V.10.7. Arnold (1994a)

We now look at some motivations for the sequel. Arnold was inspired by the desire to place the four vertex theorem in a more general context; he tried to compare it with results of what are called symplectic geometry and contact geometry We can only explain in a few words what the latter is about, referring to Arnold (1994a, 1994b) and their bibliographies, for it concerns a very general subject allowing treatment of numerous problems in mechanics; see below the knot associated with a plane curve. Moreover, Gromov’s prediction is that “in the twenty-first century a good part of geometry will be symplectic”. See also a whole series of references below. What now follows issues very largely from this philosophy. Arnold’s idea for finding new invariants, finer than the three introduced by Whitney, consists of putting ourselves high on the ladder and looking at the space of all planar closed curves: without seeking to give a very precise formulation which would be quite ponderous since this space C is of infinite dimension we consider the hypersurface D in C , called the discriminant, which is formed by the curves which are not generic. The above figure is of course oversimplified. Classifying the


Fig. V.10.8. Classification of closed curves with 4 double points. The table also c gives the Gauss diagrams and the values of the Arnold invariants. Arnold (1994a) American Mathematical Society

Fig. V.10.9. The discriminant and its intersections with a hom*otopy of curves




forms of curves amounts to classifying the connected components of C n D D G , which is the set of generic curves. An invariant is thus a function defined on G that is constant on the connected components. The idea of Arnold is to construct invariants while requiring them to jump by 1 or by 2 when a path connecting two generic curves of different classes cross the discriminant. We must also fix their initial value for the curve types, called standard by Arnold, which are shown in Fig. V.10.4 and which clearly are the most natural. One of the important steps in proving that invariants can be so defined consists of showing that the discriminant is naturally co-oriented, which is to say that we can canonically choose a side of this hypersurface. In particular when a path goes through it, we will know if this occurs positively or negatively. For the passage through a triple point, this co-orientation results from the fact that there are two types of passage through a triple point, for the passage through a point of self-tangency, matters are simpler still:

c American Mathematical Society Fig. V.10.10. Arnold (1994a)

This finally leads directly to three invariants, for we easily see that a path between two generic curves can be deformed into a path for which the nongeneric curves have only at most three types of singularities: the triple point, points of common tangents having the same sense and points of common tangents having the opposite sense. By examining what happens upon crossing the discriminant, it is shown that these three invariants in fact exist; they are denoted St, JC , J . The invariant St jumps by 1 upon crossing D, whereas JC jumps by 2 and J by 2.



The construction of these invariants itself indicates the minimum number of triple points and self-tangencies that are necessarily encountered when two generic curves are connected by a hom*otopy, whence the fundamental theorem: In a hom*otopy between two generic curves C and C0 , at least j St.C/ St.C0 /j triple points, 12 jJC .C/ JC .C0 /j points of positive self-tangency and 1 jJ .C/ J .C0 /j points of negative self-tangency must be encountered 2 among the intermediate curves. We see in the examples that these three invariants, whatever their interest, are not sufficient for classifying generic curves. There also remains the fact that Arnold doesn’t have a general formula for calculating them, apart from the case of St for minimal curves. In fact he shows that read on the associated rooted tree the value of St is nothing other than the sum of the distances of all the vertices of the tree to its root. Thus we see that, for our two favorite curves (Fig. V.10.1), we have the values 4 and 10 indicated below. Consequently, passing from the one to the other requires at least six points of each of the three types:

Fig. V.10.11.

This explains the notation St, which signifies strangeness. The curve on the left is the most natural curve with winding number 5, whereas that on the right is the least natural, the most strange. More generally, for minimal curves, the standard curve is the least strange (Fig. V.10.4 or V.10.11 on the left); the curve on the right in Fig. V.10.10 is the most strange. Readers can confirm that the latter curve is the most difficult to draw. For minimal curves we indeed always have St D 2JC . For not necessarily minimal generic curves, we find in Chmutov and Duzhin (1997) a series of formulas for calculating the three invariants of Arnold. For JC and J the formula involves the following: the curve separates the plane into connected components; with each of these we can associate an index. Take any interior point and see how many multiples of 2 the vector that joins it to a point of the curve turns when the entire curve is traversed; see Sect. V.4. At each double point four connected components are thus associated in this way; the formulas in Chmutov and Duzhin (1997) furnish our three invariants as a function of all these indices.



Here now are various complementary ideas. We may be interested in closed curves traced on the sphere. The classification is in fact simpler, but even so not complete. We merely remark that the winding number only makes sense modulo 2, which means it only takes on the values 0 or 1. To see this, “make a portion of the curve pass through the north pole”: the two curves below are in the same class on the sphere:

Fig. V.10.12. How to pass from winding number 2 to winding number 0

A spectacular theorem involves the tennis ball: If a simple closed curve on the sphere separates it into two regions of equal area, then the curve possesses at least four inflection points.

Fig. V.10.13. The tennis ball

The stated result can be found at the end of Arnold (1994a); it fits into the context of wave fronts as we will see right away; for curves on the sphere see also Arnold (1995). But in fact, as can be seen in Thorbergsson and Umehara (1999), this theorem was practically already in Segre (1968). A new approach to curves in space is to be found in Tabachnikov (2002) and the very recent Uribe-Vargas 2004a, 2004b). The notion of wave front comes up in geometry when we seek to study curves parallel to a given curve: we transport points a constant distance t along the normals to the curve C, which yields the curves Ct . We can see in the drawings that these parallel curves are in general going to develop singularities, typically cusps; we want to know how many. In Arnold (1994a) there is a theorem on four cusps for



sufficiently small deformations of a circle (attention: they can be counted twice). Contrary to Arnold’s theory of three invariants, which only uses simple tools for the proofs, here we need to climb the ladder and place ourselves in an abstract threedimensional space. Note also the use of a cancellation theorem in at least four values as for the four vertex theorem. We may note that the general context of wave allows us to effectively treat the problem much more subtle than it might appear of envelopes of lines. Here they are the normals to the initial curve.

Fig. V.10.14. Evolutions of wave fronts: (a) wave fronts: parallel to x C y 3 D 0 in the neighborhood of the inflection point; (b) wave fronts: parallel to a parabola and (c) c Cambridge University Press normal to a parabola. Porteous (1994)

We now treat in more detail a doubly interesting construction, for it is also connected to the study of vision, i.e. making a knot in space from a plane curve. To do this, we consider the three-dimensional space formed by lines in the plane which are marked at a point. This space E is thus the product of the plane with the circle of lines passing through a point. It has the topology of the interior of a torus: here the core of the torus and the parallel curves represent the lines passing through a given point; the plane is represented by the open sectional disks of the torus formed by planes perpendicular to the core. With a curve C in the plane we associate the subset C of E composed of directed lines that are the tangents to the curve and their point of contact. If the curve is generic, it will not have double tangents and thus the closed curve C will not



Fig. V.10.15. At left, a (relatively) realistic drawing where above each point of the plane a topological circle (called the fiber above this point) is represented, composed of a set of lines passing through this point (a projective space of dimension 1!). At right, the view is, in contrast, somehow completely changed, another representation of the total space E has been drawn in which the plane, although infinite in reality, is represented by an open disk (which has the same topology), which varies along a circle S1

have double points. This will be a simple closed curve of the space E; but what is interesting is that in general it is knotted. To go further, it is necessary to put a supplementary structure on E called the contact structure: the lifted curves of type C are not just any curves of E. At each of their points they are tangent to a distinguished plane of E; typically the parallels of the torus are not appropriate. This contact structure is precisely that given, at each point of E, by a tangent plane that depends on the point when the point of E considered traverses the circular inverse image of a point of the plane; these circles are called the fibers. This plane always contains the tangent to the fiber, but turns somehow. This abstract space E is one whose consideration is indispensable in the study of wave fronts, a notion that was very briefly introduced above (Fig. V.10.14).

Fig. V.10.16.

A curve of E is a lifting of a plane curve if and only if it is tangent at each of its points to this family of tangent planes of E. The verticals correspond to the point curves, the horizontals to the lines.



Fig. V.10.17. It is shown in the two figures above why there may be double points for C if C has points of self-tangency, and why no double points of C if the tangents to C at each one of its double points are different. The figure below illustrates clearly that C is in general knotted

To return to the notion of contact structure, which is fundamental in mechanics, there is Appendix 4 of the great classic Arnold (1976), for which there is an English translation Arnold (1978) and which has seen numerous editions. There is a very close relationship between contact structures and symplectic structures, which are also essential in mechanics. Apart from Chap. VIII of Arnold, references are: McDuff (2000), McDuff and Salamon (1998), Hofer and Zehnder (1994). To come back to our knot, it is thus special and is called Legendrian. A small miracle: such a knot possesses an invariant, introduced by Bennequin in 1983: it has been found that this invariant for C coincides with the invariant St.C/. However, in the final reckoning, the theory of plane curves is finer than that of knots, which is too bad, for we would have hoped that this new theory of plane curves would come to the support of knot theory. We can consult first and foremost the very popular Sossinski (1999), then for example Kaufman (1994) and Turaev (1994), the connections with mathematical physics being very strong and profound. For another very recent and very geometric approach, see A’Campo (2000b, 2000a). The contact structure, defined above, that automatically (canonically) defines the plane, has appeared very recently at least with precise mathematical formulation as a part of the structure of the human brain associated with vision; a reference is Petitot and Tondut (1998). It seems that the cortex includes a stack of planes, each of which is a discretized representation of the contact structure of the plane that is realized by a particular neuronal connectivity (that is called a functional architecture). The stack of these implantations isn’t three dimensional just because



E is, but as is conjectured only for redundancy of verification and improving the resolution. Furthermore, it has been shown by very sophisticated experimentation that, when one cell is activated, the neighboring small cells are pre-activated, but not all, only those that are close in both position and direction, i.e. that don’t turn too much.



Fig. V.10.18.

That is to say, the pre-activation translates the integral curves of the contact structure. It is thus that the brain can, in receiving signals, distinguish curves at least certain curves, typically the contours of objects that play an essential role in human vision, also in the technology of image compression, presently much studied. We will encounter contours again, this time of surfaces, in Sect. VI.10. An atypical result in the theory of closed curves is that of Fabricius-Bjerre (1962), which is rather similar to the Plücker formulas toward the end of Sect. V.14. It says this for generic curves, in a sense to be made precise: for a closed curve, four integers are introduced: the number D of double points, the number I of points of inflection, the number NC ) (resp. N ) of double tangents where the normals have the same orientation (resp. opposite orientation): then NC D N C D C 12 I. The initial proof used “finite geometry”; see Sect. V.16. A more powerful proof, due to Halpern, uses the notion of index of vector fields; see V.9.8 of [BG].



c Presses Universitaires de France Fig. V.10.19. [BG] Berger, Gostiaux (1987)

V.11. Isoperimetric inequality: Steiner’s attempts The history of the isoperimetric inequality announced in Sect. V.1 is extraordinary and has continued to the present; it’s a perfect example for illustrating Jacob’s ladder, and not merely one of its rungs. Here is a statement that is both precise and complete: (V.11.1) For each simple closed curve, if the enclosed area is denoted by A and the 2 length of the curve by L, then LA > 4, equality holding only if the curve is a circle. We can formulate its content equivalently by saying: (V.11.2) For all the compact domains of the Euclidean plane, if A is the area and L 2 the length of the boundary, then LA > 4, equality holding only for circles. The current basic reference for practically all the isoperimetric inequalities is the remarkable (Burago and Zalgaller, 1988); we can also look at what there is in Berger (1990b) and Berger and Gostiaux (1987) for the proof with Stokes’ formula. We also need to add Osserman (1978) and Osserman (1979) and we will add more recent references in the sequel. However, only Porter (1933) treats the historical aspect in much detail. The assertions Sect. V.11.1 were part of statements admitted without discussion, without requirement of proof, until Steiner devoted a portion of his energy toward attempting to prove it. Steiner is, with respect to “elementary” geometry, one of the greatest geometers of the nineteenth century, both for the variety of his work and for its depth; we will encounter his name several times. Yet he failed with regard to the isoperimetric inequality. Here are two of his attempts.



The first is ultra elementary and based on the property of diameters of the circle: they are viewed from some point of the circle at a right angle (equal to =2). We comment on the key drawings for this proof (Fig. V.11.1). We take a curve which 2 realizes the minimum of LA and goes on to show by reductio ad absurdum that it is a circle. Take any point x on and its antipode y which divides the length into two equal parts. By symmetry and the minimum property, we see that the area is also divided into two equal parts by the line xy. We then replace the domain by the union of the upper part and its symmetrical part. For each p of the upper arc (and the same for the lower) there must be a right angle between px and py, otherwise we can construct a domain with strictly greater area in rotating the two shaded pieces, end of argument.

Fig. V.11.1. (a) If the area of C1 is > than that of C2 , the domain obtained by replacing C2 by C02 (symmetric to C1 with respect to xy ) has the same perimeter as the given domain, and a strictly larger area. The significance of (b) and (c) is left to the reader

Weierstrass had great difficulty convincing Steiner that the above argument isn’t 2 a proof, for it lacks the existence of a curve minimizing the isoperimetric ratio LA . To meet his objections, Steiner then proposed the now famous Steiner symmetrization, one of the most beautiful and useful inventions in all geometry. It is also much used in analysis, along with its generalizations; see for example Bérard, Besson, and Gallot (1985). We encounter symmetrization operations several times: in Sect. I.2 for Steiner’s problem, in Sect. II.1 for isodiametric inequalities in Borsuk’s problem and for the MacBeath number in Sect. VII.9. The basic idea is that these symmetrizations diminish (or increase) various numerical invariants attached to the object that is symmetrized. We describe it in the plane, but we will amply see in the sequel, in particular in Sect. VII.5.A, that not only can it be defined in all dimensions but also in hyperbolic spaces and on spheres. It was used, for example, at the end of Sect. II.1. This symmetrization is most often used for domains which are a priori convex, but that isn’t required here. We work with a plane domain D and we consider some line H



c Nathan Fig. V.11.2. I [B] Géométrie. Nathan (1977, 1990) réimp. Cassini (2009) Édition

(letter chosen to depict a hyperplane!): the symmetrization of D with respect to H, denoted H .D/, is obtained as in the figure by intersecting D with lines orthogonal to H. The segments (or unions of segments) of these intersections are subsequently set symmetrically astride H, keeping the same total length. The set of these new segments is the H .D/ in the figure; it has three essential properties: it is symmetric with respect to H, it has the same area as D and for the lengths of the boundaries we have L.D/ > L.H .D//; moreover there is equality for these lengths only if D was initially symmetric with respect to H. Thus Steiner constructed an operation that strictly decreases the isoperimetric ratio (choose a good direction!) as soon as we aren’t dealing with a circle. Again Weierstrass had to convince him that something 2 essential was lacking in his proof: the fact that LA had in addition to satisfy a uni2

versal inequality LA > k > 0 for a certain k, or something like that what analysts call an a priori inequality. Readers can try to prove such an inequality if they think they can do better than Steiner and Weierstrass. It is even asserted that Weierstrass said to Steiner: “take the sequence nl : for each nl there is one that is strictly smaller, and nonetheless there isn’t a positive lower bound”. This well illustrates the point at 2 which physical intuition makes it impossible to conceive that LA could be arbitrarily small. This anecdote can be found on p. 4 of Blaschke (1949), but above all in Porter (1933). It wasn’t until (Schwartz, 1884) that the first correct proof was published. In fact, Weierstrass was the first, in the 1870s, to give such a proof, but that was in his course at the University of Berlin. Schwarz thanks Weierstrass explicitly in his text. The “worst” thing was that this text had as its goal a proof of the isoperimetric inequality for surfaces in space, which is obviously much more difficult; in fact his text also proved the plane case. His proof also introduced for the first time the “circular” symmetrization, transforming an arbitrary surface into a surface of revolution. The proof was carried out in two steps: the first shows that the symmetrization improves the isoperimetric ratio, with characterization of the case of equality; then things



are proved for surfaces of revolution. In both cases inequalities are proved for a suitable type of integral, for the product of two functions among others, which is the celebrated Schwarz inequality in current terminology. We will see in Chap. VII what is necessary for finding a complete proof “a la Steiner”, that is to say completely geometrically. Here we say only that this was done by Blaschke, who truly answered the objection of Weierstrass by showing with a compactness argument that when symmetrization is carried out with respect to more and more lines in varied directions, then a limit object is actually obtained. That it is a disk is the trivial part, for it is clear that only the circle is symmetric with respect to all directions. 2 The ratio LA is thus finally bounded, in an ideal fashion, from below. A really 2

trivial drawing shows that there does not exist an upper bound: LA can be arbitrarily large. However, in the realm of affine geometry, there has been recent interest in the 2 upper bound LA when renormalization by an affine transformation is allowed; what has been discovered will be treated in detail in VII.10.E.3. We must also mention the very beautiful formula obtained in Hurwitz (1902) for the strictly convex case, which gives an estimate of the deficit: L2 1 A6 (area of the developable): 4 4 The developable of a plane curve is the envelope curve of its normals. The figures below motivate it, but this isn’t a proof! For ellipses that are more and more flattened the area of the developable is larger and larger:

Fig. V.11.3.

V.12. The isoperimetric inequality: proofs on all rungs Unlike the person in the street, readers armed with a certain mathematical culture will very quickly propose at least four possible attacks for the isoperimetric inequality. (1) A simple closed curve being essentially a periodic object in one parameter (even if there are two coordinates), we must be able to get away with an analysis a la Fourier.



(2) The inequality in question is a relation between the area of a domain thus its interior and its boundary. But there exists a very general formula relating the two: the Stokes formula, which is an equality between an integral over the boundary and an interior integral, this for two objects derivable from one another in a universal way, the first being of our choice. We would thus certainly have to be able to prove the isoperimetric inequality with the aid of Stokes’ formula by cleverly choosing the function to be integrated on the boundary. (3) Since we know that the circle is optimum, shouldn’t we be able to deform systematically (and in a continuous manner, as opposed symmetrization, which is a discrete operation) a curve by letting it become more and more a circle, in a natural way, which surely (?) decreases the isoperimetric ratio. We can even hope that the limit exists and is a circle. (4) The final approach is the simplest: first show that there exist curves such that the isoperimetric ratio is minimum, then show that such a curve is necessarily a circle. These four approaches have been successfully pursued but, apart from the first, are rather recent or even very recent; here briefly is what has transpired. The proof by Fourier analysis appeared in Hurwitz (1902). But it is necessary to be adroit and parameterize the two coordinates fx. /; y. /g of the curve C by the angle that the tangent makes with a fixed axis and not, as would be more natural, by the arc length. Subsequently, letting K. / be the radius of curvature, the calculation yields the equality: L2 X ak2 C bk2 AD 4 2 k2 1 k>2

P where the Fourier series is that of K D a0 C k .ak cos k C bk sin k /. But a1 D b1 D 0 because the curve is closed. Part of the interest in this proof is that the deficit L2 4A in circularity, is measured by a sequence of geometric invariants. Here is some information on this 2 natural problem: to go further than LA > 4 and interpret the deficit L2 4A geometrically. This is an important characteristic of mathematicians: We always want to “go further”. The invariants introduced by Hurwitz have not really been exploited, but in any case they form a discrete infinite sequence of invariants characterizing the curve within an isometry of the plane. This sequence is the series of ratios between the .ak2 C bk2 /. Hurwitz obtained related results: the one on the area of the developable mentioned at the end of the preceding section and some that could lead to the four vertex theorem (Sect. V.8). In Santalo (1976) can be found a proof that is not “a la Fourier”, but uses integral geometry, i.e. geometric probabilities in disguise; see Sect. I.2. It also provides an explicit estimate of the deficit. For what can be done better presently regarding the deficit, see Osserman (1979) and 1.3.1. of Burago and Zalgaller (1988),



especially the very beautiful Bonnesen inequality: L2 4A 6 2 .R r/2 , in which R (resp. r) denotes the smaller (resp. greater) radius of a circle containing C (resp. contained in C).

Fig. V.12.1.

We mention an open problem from this realm: in analogous fashion, prove the isoperimetric inequality for surfaces (or better for manifolds in all dimensions) with the aid of spherical harmonics a very natural desire, since Fourier analysis on the sphere is given by the spherical harmonics. We will speak of this again in Sect. VI.11. It wasn’t until (Schmidt, 1939) that we had a proof using Stokes’ formula, but the proof is extremely difficult, although it was the first that worked in the not necessarily convex case. It would be desirable that this proof of Schmidt be cast in modern language. In Knöthe (1957) there is a very simple proof using Stokes, but it is concealed among much more general results in arbitrary dimension. It was explained to the author by Gromov in the train between Paris and Bonn and was featured for the first time in 9.3 of [BG], in [B], and 12.11.7 of Berger and Gostiaux (1987). We will encounter Knöthe’s mapping in Sect. VII.8; see also 2.2 de (Giannopoulos and 2 Milman, 2001). Stokes’ R R formula for a compact domain D of R and its boundary C, says that D d˛ D C ˛ ds, where ˛ is an arbitrary differential 1-form and d˛ the 2-form which is the exterior derivative of ˛. The form ˛ is not obvious to find; it is obtained by constructing a mapping of D into the unit disk D which is obtained thus: we normalizes the area of D to equal , we then slice D and D by parallels in a fixed direction, so that the areas are in the same ratio, then divide the sectional intervals so that they have the same ratio. This mapping is interpreted as a vector field f on D and we write Stokes’ formula Z Z divergence.f / dx dy D hf unit normali ds D


for the scalar product with the unit normal of the curve C, which is the flux of f . By construction the Jacobian of f , i.e. the determinant of the differential of f , is equal to 1, whereas the divergence of f is its trace. We complete the proof with Newton’s inequality (VII.2.1). In return, this proof is subtle in the non convex case.



Fig. V.12.2. Knöthe’s mapping. II [B] Géométrie. Nathan (1977, 1990) réimp. Cassini c Nathan Édition (2009)

This proof calls for several remarks. First there are problems of differentiability, then of convexity, problems that are circumvented by suitable approximations; these presently have not been detailed anywhere. But above all the proof applies without any modification in arbitrary dimension; it’s this that is given in the references above. Finally the proof yields as a corollary the case of equality: only the circles (spheres) attain the minimum; but see Sect. VII.14 or Burago and Zalgaller (1988) for details concerning this uniqueness: the idea is that the domain D, beginning with dimension 3, must be sufficiently regular, or convex, to eliminate objects such as a sphere with hairs, since hairs change neither the area of the sphere nor the enclosed volume. Typically such hairs don’t exist either on convex sets or on submanifolds. The Steiner symmetrization, used for making an arbitrary curve circular, is discrete, harsh. Can the same result be obtained in a continuous fashion? We must then think of curvature: circles are the curves of constant curvature. The idea is then to deform a curve C by transporting along the normal at a point a distance proportional to the curvature at this point, thus (see the figure) tending to equalize differences in curvature. If the curve is not convex, we will need to carry things out with the appropriate sign; to have a continuous deformation, we define a one-parameter (the parameter can be thought of as the time t ) family of curves C.t / such that, for each t , the deformation is that of the figure. We then need to write the C.t / precisely in the form C.sI t /, where s is a suitable parameter for each C.t /, and thus C.sI t / is a point of the plane dependent on two real parameters. We thus obtain a partial differential equation for the deformation that is of parabolic type in t . This equation thus admits, thanks to classical theorems, solutions for sufficiently small values of t : we start on the curve C.0/ D C. In contrast, the existence for all t is a difficult study, begun in Gage and Hamilton (1986). For interested readers, the parabolic partial differential equation is: .V:12:1/

@C D .t; /N.t; / @t

where t is the time, the curve parameter C.t; / which evolves with time, the curvature and N the unit normal vector of C pointing toward the interior.



Fig. V.12.3.

Readers will surmise, correctly, that as t tends toward infinity, C.t / converges toward a limit object, typically a point. This is not a circle as we had hoped. To gain satisfaction, it remains to show two things: the first is that, appropriately renormalized, the curves C.t / really do converge to a circle; the second, for the goal we 2 .C.t// are pursuing, is to prove that the isoperimetric ratio L A.t/ is decreasing (nonincreasing, more precisely) in t. For such a renormalization, a sort of absolute zoom, there are several choices. For example, we might require that the area be constant, but holding the length constant is also a possibility. The area is the simplest to treat. Since we then have a circle in the limit, we will have proved the isoperimetric inequality as well; and we will not have failed to notice here the promised intrusion into modern geometry of the dynamics of evolutionary systems. All that precedes is true; the final point for the exact problem studied is in Gage (1986). However, interest in the preceding does not come down to a very conceptual and dynamic proof of the isoperimetric inequality for plane curves. The interesting point is that this technique may be used in more general contexts, for example for obtaining periodic geodesics on surfaces, a difficult problem that we will examine amply in Chap. XII. We mention also that this deformation of curves into circles comes up in physics in what happens when metal in fusion is poured on a plate at very high temperature. A generalization in the case of surfaces will be seen in Sect. VII.13.C. We mention this “descent” method again in Sect. VII.14 along with its very flamboyant recent application in proving the Poincaré conjecture. There is also a mechanical context where curves are deformed, not proportional to the curvature at a point, but in an entirely different spirit. With each closed (here R not necessarily simple) curve C is associated a sort of energy integral C k 2 .s/ ds of the square of the curvature.R We place ourselves in the space of curves and consider the functional F W C 7! C k 2 .s/ ds. The curves which render this functional critical, i.e. of derivative zero with respect to all deformations, are called elastica, which are the forms taken on by metal rope. To find them we take any rope and decrease the functional by following, in the space of closed curves, the gradient directions (those of steepest descent) of the functional F. For recent progress, see e.g. Angenent (1991). Finally, in Sect. VII.13.C we will discuss at some length surface deformations analogous to those of Gage-Hamilton above for curves, for they bear on a problem of industrial physics and metrology; see also Angenent (1992).



For curves, a proof like the one here is nowhere else to be found, for the result is almost obvious once we have acquired the profound results of geometric measure theory (GMT) which we now discuss. This is because the technique in question is applicable in the very general context of Riemannian manifolds and in other contexts too; see for example TOP.1.B and TOP.10.C of Berger (1999), or 7.1.2 and 14.7.2 of Berger (2003). But the basic idea is in Gromov (1980). It’s the most natural possible idea for studying all minimization problems, in particular to prove “Steiner’s postulate” and what made it so deficient: the need to show the existence of curves 2 minimizing the ratio LA . If we were able to make a suitable compact set out the set of all curves, the ratio in question being obviously continuous, there remains nothing more to say than that a continuous function attains a minimum somewhere on the compact set. In fact, let’s for the moment assume existence and suppose that the minimizing curve is differentiable. Then a little differential calculus shows that if 2 .t/ the derivative of the ratio LA.t/ at t D 0 is zero for every variation C.t / of curves about C.0/ D C, then C has constant curvature: it is thus a circle, Q.E.D. The computation is made as follows: we obtain the curves by transporting a variation f .s/ along the normal n.s/ for a parameterization of the curve by arc length; then (see the computation of the periphery in Sect. V.6): Z L Z L @.A.C.t /// @.L.C.t /// f .s/ ds and k.s/ f .s/ ds: D D @t @t 0 0 It’s a classic trick of the calculus of variations that, if we want the second integral to be zero each time the first is, it is necessary that the curvature k.s/ be constant, and thus the minimizing curve is clearly a circle.

Fig. V.12.4. Variations of area and length

Compactness theorems for sets of curves were obtained in the wake of work on geometric measure theory (GMT) begun in 1960, then more generally for manifolds in a broad sense to be defined precisely and in such a way that numerous geometric minimization problems have reasonable objects as solutions. We will encounter several examples in Sect. VI.11 and, for the case of convex sets, in Chap. VII. In every case here, GMT assures that the minimizing curve is differentiable. We thus very



rapidly obtain a solution, modulo the complex and lengthy foundations of GMT itself, of the isoperimetric inequality for surfaces not only in space (see Sect. VI.11), but also for all Riemannian manifolds not to mention the very popular theme of minimal surfaces and soap bubbles. We may resolutely wish to ignore that the curves of constant curvature are circles, for both in higher dimensions and in the more general context of Riemannian geometry the analogous problem can be resolved by the following packing method, which is due to Gromov. We give this proof in detail, for it doesn’t to our knowledge appear anywhere else explicitly (in the Euclidean case, strictly speaking) except for I.5.G of Berger (2003). Furthermore, we will see in Sect. VI.11 that it extends to all dimensions. In contrast this technique is used in depth, and is unavoidable, in the case of Riemannian manifolds with a Ricci curvature that is bounded below; see Chapt. TOP. 1.A of Berger (1999) or Berger (2003). The idea is to fill the minimizing domain D by normals to the boundary‘curve C. In fact, we proceed in the reverse direction: for each point m of D we look for a point of C which is at minimum distance; at least one such point q (called a foot) = foot.m/ exists by compactness. The fact that the distance is minimum assures that the segment mq is normal to C. The whole domain is thus filled by the normals to C, and the property of the minimum then proves that the distance mq is less than of equal to the radius of curvature k 1 (which is, we recall, constant) of C at q. In the figure, k 1 is also denoted foc.s/, denoting the focal value and representing the distance at which the normal at c.s/ meets the envelope of normals (and thus the normals that are “infinitely close”). We stop on the normal at the value of the cut cut.s/, a distance beyond which the foot of m is no longer c.s/. We have cut.s/ 6 foc.s/ and computation of the double integral shows that Z L Z cut.s/ Z L Z k 1 1 A D Area.D/ D .1 k t / dt ds 6 .1 k t / dt ds D L; 2k 0 0 0 0 whence the inequality k 1 L > 2kA. But we don’t know what k is! The astute2 ness of Gromov was to notice that the quotient LA is also minimum, by writing that its derivative is zero (looking at the above formulas and setting f D 1), we find k 1 L D 2A. There is thus equality throughout in the above inequality, which im-

Fig. V.12.5.



plies that foc.s/ D the constant k 1 , which is tantamount to saying that our packing is that of a circle. Just because we have proposed four natural methods doesn’t mean that there aren’t others. We find no less than ten at the very beginning of Burago and Zalgaller (1988); or in this book look a bit in Sect. VII.14. See alsoRitore and Ros (2002). V.13. Plane algebraic curves: generalities Chapter IV gave us almost complete satisfaction for the simplest curves after the circles, i.e. the conics. The books on the subject of plane algebraic curves are numerous, we mention only Coolidge (1959), a very precious little bible, valid up to the date of its first publication in 1931; then a classic work Walker (1950) from the intervening period; finally the completely modern Fulton (1969) and the very recent Chenciner (2006); and a separate mention for the very pedagogical Brieskorn and Knörrer (1986). But the more general algebraic curves are in particular very well treated in the bible Griffiths and Harris (1978), which covers a good part of algebraic geometry. But given the importance, the naturalness and indeed the beauty of the subject, there exist a whole array of other references; for the history, see Dieudonné (1985). We will broach but very few matters; our choice is dictated by two requirements: to be brief and to stay within the spirit of this book, elementary at least at the level of the problems that are posed. Moreover, we will above all else insist upon problems encountered elsewhere in the book, such as the 3264 conics of Chasles, the inflection points of cubics (Serre’s conjecture), the Poncelet polygons. Which is to say: we decline to be interested, at least at the outset, in objects that are already very high on Jacob’s ladder . We want to stay well below at the beginning of a problem, of a concept. Another reason is that, contrary to curves that are only differentiable, algebraic curves were very quickly and intensely studied (Descartes and Newton, for a start) and by the end of the nineteenth century numerous results, both local and global, were already known. The problems that have long remained open, that are indeed in some instances still unsettled, are rather those where algebraic curves encroach upon the theory of numbers, the exemplary case being that of Fermat’s theorem and the use of elliptic curves. In comparison to those that are complex, the real algebraic curves have enjoyed a resurgence of interest, as have the real algebraic varieties of higher dimension, this while the natural objects of robotics are real algebraic. In Sect. V.15 we will see some interesting problems concerning real algebraic curves. For us an algebraic curve will almost always be, to begin with, a plane algebraic curve, i.e. a set of points of R2 (or of C 2 ) defined by an equation P.x; y/ D 0, where P is a polynomial in the two variables, real or complex, x and y. We consider



the complex case right away, since the study of conics has shown us that this is almost an absolute necessity. But then we know that we also need to extend things to the projective context. Then an algebraic curve will be the set of points of CP2 (or RP2 ) defined by P.x; y; z/ D 0, where P is a hom*ogeneous in x, y, z. We won’t forego passing from the affine situation to the projective, for we have known since Chap. I how this is done. It’s an important fact that the projective algebraic curves are always compact sets. In all that precedes, the degree of an algebraic curve is the degree of the polynomial (the smallest degree; we need to avoid powers of the same polynomial) which defines it. In the complex case we have precisely a Nullstellensatz, which states that two irreducible polynomials that yield the same curve are proportional. We can thus consider, as for the conics, the space of all curves of a given degree n, a projective space of dimension n.nC3/ . We thus find 2 for lines, 5 for conics, 9 2 for cubics, etc. To characterize the Nullstellensatz a bit more, we need pay attention to the case where the polynomial P of the definition can be written P D QR, in which case we say that the curve is decomposable. Otherwise it is called irreducible. In the sequel we always assume irreducibility without mentioning it explicitly. We can and should as well study algebraic curves over any field, but especially over commutative fields. For intuitively clear reasons those over finite fields are connected to combinatorics, and also in an obvious way to the theory of numbers; see for example Chap. V of Silverman (1986) and all of Silverman and Tate (1992). But there are also the fields that are fundamental in arithmetic, specifically the field Q of rational numbers. In all these cases, we work either in the affine or in the projective context. The case of cubics is exemplary; see below. Bézout and singularities. We treat essentially the complex case. The problem here is that of the intersection of two algebraic curves C and C0 of respective degrees n and m, not to forget the trivial case of intersection with a line. Each line intersects a curve of degree n in at most n points, exactly n if we are in the complex case and count multiplicities (defined as the order of a root of a polynomial), since we obtain an ordinary equation of degree n. Two merged points of intersection correspond to a tangent. For the natural question of knowing whether only algebraic curves have this property of always being intersected by a line in a bounded finite number of points, see Sect. V.16. We return to the general case: by eliminating y between the equations P and Q of C and C0 by appropriate algebraic manipulations, we obtain an equation in x of degree equal to nm and we see that there are at most nm common points. But this is the trivial part of the matter; the case of conics in Sect. IV.6 already has shown us that the subtle problem is that of multiple roots, the common points “merged a certain number of times”. It is necessary to do two things in the general case: first to define for a common point how many merged points must be counted; then to show that the sum of all these multiplicities is in fact equal to nm. The figure below is exemplary;



the two curves are cubics, n D m D 3. They have three common points, the one obvious point counts for 1; for the other two we can try to guess “by deformation”. The drawings thus suggest counting 3 for the point of tangency but with crossing (think for example of a curve and its osculating circle, see Sect. V.6). For the more complicated point, we must deform the cusp and observe at how many points it intersects each branch of the double point of the first curve. The small sketches thus suggest counting this point for 3 C 2 D 5. In total we thus find 1 C 3 C 5 D 9.

Fig. V.13.1.

For implementing the preceding, the deformation method is too complicated; besides, it would only be valid in the case of fields such as R or C, not in the case of finite fields whereas all that follows is completely general. In brief, we do this: at a point of a curve, we define the branches that pass through it, and with each of these branches we associate a development of a local parameterization (for example, for a curve with a simple cusp it is .t 2 ; t 3 /); the technical name is place. At a common point, the order of the one curve with respect to the other is calculated by replacing the expansion of places of the one in the equation of the other. The local parameterizations can all be made with the help of development in fractional powers of t , called Puiseux developments. It remains to show: first, that at a common point the sum of these operations is symmetric in the two curves, in spite of an asymmetric definition; then to show that the total sum is indeed nm. These matters had been expedited with complete correctness by the end of the nineteenth century. The final result is called Bézout’s theorem. Complete proofs are to be found in the references; the book by Walker is the most accessible for a first approach. For the study of singularities and of global formulas, see below “Duality and Plücker’s formulas” at the end of Sect. V.14. We have emphasized Bézout in order to indicate the formidable difficulties to be dealt with in order to create an intersection theory for higher dimensions and to



obtain, for N hypersurfaces of CPN , a number of “points” equal to the product of their degrees, of which we had need of in Sect. IV.9.

V.14. The cubics, their addition law and abstract elliptic curves We are going spend some time on cubics, because it turns out that these curves play an essential role, not only for the Poncelet polygons encountered in Sect. IV.8, but in a large part of mathematics and are of first importance in number theory (recall that we say either number theory or arithmetic indiscriminately; these two names cover essentially the same areas of mathematics). Here are some reasons why we treat cubics in preference over other curves: in the first place, these are the simplest curves after the conics; but then in fact the cubics are richer than the conics, because (at least for those without singularities) they are the only ones that possess a group structure, which is Abelian. This wasn’t the case for the (real or complex) conics: they admit a group that acts on them, but it is of but one parameter. The case of the Euclidean circle, seen as a group, is deceptive: it is not an intrinsically algebraic object. Besides, just as the conics and more generally, the unicursal algebraic curves, admit a parametric representation, the cubics admit parametric representation (of course not rational) with the aid of the so-called elliptic functions, which are the simplest functions after the sine and cosine the term “elliptic” has its origin in their use for calculating the arc length of an ellipse. Like the sine and cosine functions, which are of dazzling importance in all mathematics, these elliptic functions turn up in many places; we will see this for the geodesics of ellipsoids of revolution in the following chapter and for the movement of a solid about its center of gravity in Sect. VII.13.C. Next, we can describe this group law geometrically. The cubics are a common thread between Abelian integrals (integrals of algebraic functions), geometry and number theory. In summary, one of the riches of the elliptic cubic curves is that they are mathematical objects that are susceptible to three equivalent definitions, but definitions completely different in their nature; moreover, the connections between them are very subtle, not to mention the concepts that are needed for interposing them. Typically, in the case of the complex field, these three versions of the same object are: (i) the singularity-free cubics of CP2 ; (ii) the toruses of dimension 2 viewed as the quotient of C by a lattice (and the accompanying elliptic functions); (iii) the abstract algebraic curves of genus 1. The Abelian group law of plane cubics. The cubics (we also say elliptic curves and we will see why) are by definition algebraic curves defined by a polynomial of degree 3, whatever the base field: R, C, Q, a finite field, etc. Let us first look at this



famous group structure, for which the interest is double, first while it is situated geometrically in the concrete case (curves of KP2 ), but also abstractly when we regard abstract cubics as toruses (see below). In what follows, claims are made without justification; readers should refer to the works cited for the proofs. We dispose at the outset of the cubics with a singularity. Once there is a double point or a cusp, we can parametrize the cubic by two rational functions of degree 3, the parameter being for example the slope of a line passing through the singular point. The algebraic curves thus parameterizable are called rational or unicursal, and are of much less interest that the fortunate ones that remain. We begin with almost any commutative field. A cubic C always admits an inflection point p. This point is going to be the identity element of the group that we seek. If m and n are two points of C, the line mn joining them intersects C at a third point q. Then the line pq intersects C at a third point f .m; n/. We show (we need to do a little algebra and with some care, thus climbing at least one more rung of the ladder) that the law .m; n/ 7! f .m; n/ is an Abelian group that we denote simply m C n, the point p which is the origin and identity element being thus the 0. The only non obvious axiom is the associativity of this law: we can prove it geometrically with the help of the lemma (which requires the careful algebra of which we spoke): if three lines intersect a cubic in the points ai , bi , ci (i D 1; 2; 3) and if simultaneously the ai and the bi are collinear, then the ci too are collinear. Here is the diagram: In other words, three points x, y, z of C are collinear if and only if x Cy Cz D 0. All this has a precise sense, even with the points that get counted twice (as in the case of the conics, mm will be tangent to C at m). In particular the tangent to C at x intersects C at the point 2x, etc. There is also no question of reality here since we are dealing with equations of third degree which already have two roots, so that the third always exists. One of the most utilized cases for graphics, for number theory and for error correcting codes is where the chosen point of inflection is at infinity and the curve is symmetric with respect to the x axis, because it can be shown that a singularity-free real cubic can always be written, after a suitable hom*ography, in the form y 2 D P.x/ D x 3 C ax C b. There are but two possible forms according as P has three real roots or only one. With this group law we can now resolve practically all geometric problems posed, in the complex case at least, for it can be shown (again a bit of algebra, to be done carefully) that the 3n points of intersection of fxiP g with an algebraic curve of degree n obey the necessary and sufficient condition i xi D 0 which calls for a small celebration! The points of inflection p are the triple points: 3p D 0. There is one or three (at most) in the real case, but always 9 in the complex case; we saw in Sect. I.8 that they have very simple coordinates via the hom*ogeneous equation x 3Cy 3Cz 33axyz D 0. And we can now explain the counterexample to Sylvester’s situation in Sect. I.1: for if p and q are two points of inflection, then 3p D 0 and 3q D 0, and thus the third point r where pq intersects C is .p C q/, and thus is again an inflection point because 3r D 3p 3q D 0.



Fig. V.14.1.

Fig. V.14.2. Group law on a plane cubic. Case where the inflection point chosen as an identity element is at infinity

We now intersect C with a conic to obtain six points (with their orders of multiplicity). For example, at each point m of C there exists an osculatory conic, i.e. one which has with C at m five merged points (a conic is defined by five points). It intersects C at a sixth point n, which is the point 5m. If we think of the four vertex theorem (for the superosculating circles ), or of the sextactic points encountered in Sect. V.9 for not necessarily algebraic curves, we then look for the points m such that 6m D 0. But we know them, for to say that 6m D 0 is to say that 2.3m/ D 0,



thus that the tangent to m at C intersects it at p D 2m and thus p is an inflection point since then 3p D 6m D 0. The sextactic points are thus the points of contact of the curve with the tangents emanating from its points of inflection. We will see in the duality (at the end of the present section) that through each point of the plane, in the complex case, there pass six tangents to C. If the point is on the curve, there will be only four such tangents apart from the tangent to this point, which counts for two, and three only if it is a point of inflection. The sextactic points are thus 9 3 D 27 in number. In the real case with two connected components we find 9 of them (other than the inflection points themselves), 6 on the oval and 3 on the infinite branch (a projective line topologically). The sextactic conics corresponding to points of inflection are the tangents at these points, but considered as conics degenerated into double lines.

Fig. V.14.3.

We mention that all the points m for which we have km D 0 for some integer k play an important role in the cubics that come up in arithmetic; they are called torsion points. Over Q their existence depends on the type (see below) of the cubic.

Fig. V.14.4.



Projective classification of planar cubics. The projective-minded geometer will take the viewpoint ii) still further: it can be shown that the cross ratio of four tangents (when there are four, which occurs according to the type of cubic in the real case) issuing from a point of C is independent of this point and is thus an invariant of the curve (take inspiration from the figure below and see a proof in Walker 1950). But above all it can be shown that this invariant is characteristic, i.e. two complex cubics (without singularities) are projectively equivalent if and only if the cross ratio is the same for both. The proofs can be very geometric and rather simple; see the end of Walker (1950), whence a projective classification of cubics. In the case of y 2 D P.x/ at three points on the x axis, the four tangents issuing from the point at infinity on the y axis are thus the vertical tangents at the three points P.x/ D 0 of the x axis and the point at infinity. Thus the cross ratio is immediately calculable with the third degree polynomial P.

Fig. V.14.5.

To be more precise in the classification, we need to recall the effect of permutations on the cross ratio, for in our cubics the four tangents issuing from a point are not ordered. The correct invariant is thus not just one of the cross ratios , but rather 2 3 the function .2C1/ ; see Sect. I.6. We will soon see how to calculate it explicitly .1/2 from the equation of the curve. We show that the cubics (without singularity) are never parameterizable by rational functions, as are the conics or the cubics with singularity. But we would like to parameterize them using reasonable functions, generalizing the parameterization of the circle by sine and cosine. Said circle can also, by the way, be written in rational 2t 1t 2 functions . 1Ct 2 , 1Ct 2 /. These functions exist, they are the elliptic functions; see R above. Just as sine and cosine originate from the integral p dx 2 , the elliptic func1x tions originate from first integrals that can’t be calculated with rational functions, R dx or with sine and cosine, i.e. the integrals px.x1/.x/ . We also see clearly that



we can parameterize the cubic with equation y 2 D x.x 1/.x /. These elliptic functions surely exist, but we need to study them in the complex domain, where they belong to the class of doubly periodic holomorphic functions (with points where they become infinite, called poles referred to as “meromorphic functions”). Digression: plane lattices (alias the flat torus). The connection with the group law in the complex case is going to fall naturally under the point of view (ii) (see the beginning of the present section) with a question we should have asked a long time back, as we did for the conics in Sects. IV.4 and IV.7: what is the topology of a singularity-free conic? We stay in the projective setting. For the conics we have a circle in the real case and a sphere in the complex case. Here we have a torus in the complex case; in the real case we can have one circle or two (see the above figures). But in the real case, if there are two circles, in RP2 the one is convex (has an interior) while the other is not, for it is hom*otopic to a projective line. In the complex case, we need to visualize the torus T2 D S1 S1 in CP2 . If the topology is always that of S1 S1 and if addition modulo the periods is always the same as the naive one on S1 S1 , it turns out quite differently as soon as we compare cubics as objects of type (i), (ii) or (iii), i.e. as curves of the projective plane, complex tori or abstract algebraic curves. In each case the classifications obtained in these three cases coincide. In addition to some of the references already given, Serre (1970) is very dense and very precise. Here we need to introduce some objects that we will encounter amply in the sequel (Sect. IX.4): the lattices known as the flat tori. It is necessary to understand thoroughly that the geometry of a torus that is embedded in R3 (see the geometry of surfaces in the following chapter) is never compatible with a group law; it only holds at most for one period for tori of revolution and rotations along the meridians; but the rotations along the parallels don’t respect the geometry. Whether we want to or not, for these tori we have to climb a bit up the ladder. A lattice in a Euclidean plane is obtained by specifying two independent (thus necessarily nonzero) vectors .u; v/ of R2 , the lattice ƒ strictly speaking being the subset of R2 composed of the points ku C hv for all integers k, h. We thus define an associated torus as the object obtained by the equivalence relation on R2 D C, obtained by identifying two points x and y when y x 2 ƒ. The topology shows itself by identifying the opposite sides of the full parallelogram fxu C yv W x 2 Œ0; 1; y 2 Œ0; 1g (Fig. V.14.6). When we parameterize a cubic with elliptic functions whose two periods are precisely u and v, topologically we obtain the torus C=ƒ; but above all the Abelian group law is the law of addition of vectors in C induced by the equivalence relation. The torsion points are then seen very clearly, the nine points of inflection, etc. Note here that the geometry is affine rather than a group geometry: contrary to the embedded case, where it is necessary that the identity element of the group is an inflection point, here we can take any point whatsoever as the identity element. We will look for collinearities in the figure, thanks to periodicity.



Fig. V.14.6.

Fig. V.14.7.

It remains to connect the classification of cubics by the cross ratio invariant encountered above with that of pairs fu; vg of associated periods. What matters is just the ratio uv , where u and v are regarded as complex numbers. Note that, for the complex cubics that are simply the complexifications of real cubics, the ratio uv is real (and conversely). But in fact we can change the basis for a lattice, the set of these changes (conserving orientation) is the modular group SL.2IZ/, i.e. the group of ma ab W ad bc D 1 . This trices with integer coefficients and determinant one: cd group joins R and C as being one of the most important groups in all of mathematics. Thus the good invariant for the complex cubics is the complex number uv , but mod au C bv u , 7! ulo SL.2I Z/, that is to say, in taking the quotient by the action cu C dv v ab belongs to SL.2I Z/. In general this quotient of C by where the matrix cd SL.2I Z/ is most frequently represented by the famous modular domain:



Fig. V.14.8. The modular domain

To have the true quotient, we need to identify the two vertical half-lines on the one hand, and the two half arcs of the circle on the other. The points on the vertical axis represent the rectangular lattices, those on the boundary of the lattice cells. The square lattice corresponds to the point .0; 1/ D i and the hexagonal regular lattice to the two lowest points. This gives the torus that is richest in symmetries. As for the right geometric structure to place on the modular domain, it’s not the Euclidean (which is deceptive) that we see so clearly, but the locally hyperbolic metric induced by the Poincaré half-plane (see Sect. II.4), so that we have invariance under the natural group SL.2I Z/; see also Sect. IX.4. We are thus within our rights to ask for the connection between this uv and the invariant coming out of the cross ratio, given the two classes of cubics. The connection is (unfortunately, but inevitably) transcendental, as we will see. But we will do more and will find the “explicit” link between (i) and (ii) (see the beginning of the present section), i.e. between the equation of a planar cubic and the lattice that it determines. In fact the connection is made from (ii) to (i) in the following way, which R dx is an operation inverse to the calculation of the integrals px.x1/.x/ introduced above. Passage from (ii) to (i). We prove all that follows; see the references given, to which it is interesting to add that of Serre. We P begin 1with a fixed lattice ƒ. For each integer k, we define the series Gk .ƒ/ D 2ƒ0 2k , where the (double) sum is 0 over all nonzero elements of ƒ (notation ƒ ). Then the abstract cubic defined by ƒ has the equation: y 2 D 4x 3 g2 .ƒ/x g3 .ƒ/; where we have put g2 .ƒ/ D 60 G2 .ƒ/ and g3 .ƒ/ D 140 G3 .ƒ/. The two so-called Weierstrass canonical elliptic functions fpƒ .z/; p 0 ƒ.z/g parameterizing this curve are given by the series: X 1 1 1 @pƒ .z/ 0 ; and the derivative pƒ pƒ .z/ D 2 C .z/ D : 2 2 z .z / @z 0 2ƒ



0 The parameterization sought is fx D pƒ .z/; y D pƒ .z/g. Contrary to the case of the circle with sine and cosine, the Weierstrass functions cannot be everywhere defined in the usual sense, for they must have singularities (called poles) where they become infinite, but in a reasonable way, i.e. as rational functions, because a bounded holomorphic function (in particular one that is periodic) must reduce to a constant. And here finally is the relation with the cross ratio of the four tangents issuing from four points that classifies the cubics. For the equation y 2 D 4x 3 g2 .ƒ/x g3 .ƒ/, we will forget its lattice ƒ from time to time; we introduce its discriminant , for which being nonzero guarantees three distinct roots, and being zero at least a double root, i.e. D g23 27g32 . Thus the modular invariant j.ƒ/: g3

j D 1728 2 . The terminology comes from the fact that j.ƒ/ is invariant under SL.2I Z/ when we let SL.2I Z/ act on an expression z D uv , where fu; vg is a basis of ƒ. With the associated notation j.z/ we thus have j.z/ D j

az C b cz C d

ab in SL.2I Z/. This posed, for our cross ratio we thus have: j D cd .2 C 1/3 28 . Moreover, the function j.z/ is holomorphic in the entire half2 . 1/2 plane Im.z/ > 0 and defines, by passage to the quotient by SL.2I Z/, a bijection from the modular domain (Fig. V.14.8 above) onto C. To go back from the equation y 2 D 4x 3 g2 .ƒ/x g3 .ƒ/ to ƒ it is necessary to use the inverse function of j , which does not have a simple expression. As information useful for the sequel: the invariants Gk .ƒ/ introduced above, written in Gk .z/ (for z D uv ), are no longer invariant under SL.2I Z/, but still behave az C b very well because they satisfy Gk .z/ D .cz C d /2k Gk . These are called cz C d weighted modular forms 2k and are essential in arithmetic, on the one hand because it can shown that there aren’t any others, and better yet because they are generated by G2 and G3 . Precisely, the algebra M of all modular forms is the algebra CŒG2 ; G3 of the polynomials in G2 and G3 ; the set Mk of those with weight 2k has for a basis the family of the G˛2 Gˇ3 with 2˛ C 3ˇ D k. On the other hand they serve for computing the functions that count the number of points of given norm in a lattice of arbitrary dimension; we encounter this in Sect. X.8. Yet once again a succinct, but perfect, exposition can be found in Serre (1970).

for each

Abstract elliptic curves (iii). Let us now recall the “higher” notion of a 1-dimensional manifold as compared to that of a plane curve, see Sect. V.3. In the setting of algebraic geometry (here complex for simplicity), we can also define in an abstract manner, independent of any embedding, the notion of algebraic curve (alge-



braic variety of dimension 1). The definition copies that of differentiable manifolds see Sect. XYZ but here we require that the changes of chart be given by rational functions. Such a curve possesses an invariant, its genus. The abstract algebraic curves of genus 1 are none other than the elliptic curves which are obtained as quotients C=ƒ. They can be embedded in CP2 , but also in the CPn of higher dimension. Return to the Poncelet polygons. It’s this abstract presentation that provides everything readers want in connection with Poncelet’s theorem on polygons and the cubics in Sect. IV.8, and we now have practically everything that is needed. We still need to show that the abstract curve defined by a point of the first conic and a tangent issuing from this point to the second is really an elliptic curve. Now the object obtained covers the first conic two times algebraically, with the exception of four points of intersection with the second. But in these four points the covering singularity is of the form z 7! z 2 in C. Now there exists a very general formula called Riemann-Hurwitz for calculating the genus of such an algebraic curve; see 2.1 of Griffiths and Harris (1978). It says this: consider a holomorphic mapping f W S ! S0 between two algebraic curves. Apart from a finite number of ramification points, the covering is throughout a constant number n of sheets. At the ramification points q the mapping is written z 7! z v.q/ and we have: genus.S/ D 1 C n .genus.S0 / 1/ C

1X .v.q/ 1/: 2 q

Here we indeed find the genus 1, Q.E.D. Then in Sect. IV.8 we used the classification of the involutions of an elliptic curve, which states that all are of the form x 7! x C a. This result is a particular case of the theorem that states that the only holomorphic mappings of an elliptic curve into itself are the group automorphisms, followed by a translation; see 2.6, p. 326 of Griffiths and Harris (1978). An involution whose square is the identity is thus necessarily clearly of the form x 7! x C a. Finally we use the expression of a cubic as y 2 D P.x/ for finding the explicit Cayley conditions via the pencil of conics determined by our two conics. The classification of elliptic curves over different fields isn’t yet complete, but it has known steady even spectacular progress, which among other things has led to resolution of Fermat’s theorem. Even though this is but a crude caricature, the idea of showing that the equation an C b n D c n has no nontrivial solution in integers for n > 3 has consisted of studying the elliptic curve, nowadays called Frey’s curve (but for what it will be in the future, see Hellgouarch, 2000): y 2 D x.x an /.x C b n / and to show that the Fermat relation is too strong: the elliptic curve in question must finally have too numerous properties, ultimately contradictory.



Duality and the Plücker formulas. What can we do for curves of degree > 3? We stay with the complex numbers for the better part of this subsection. There is no longer any group law by which we could completely describe, for the cubics, the intersections with the other algebraic curves. There is no longer anything so simple starting with degree four (at least for general curves the unicursals are a trivial case to be treated separately). There exists a theorem, called Riemann-Roch, that best describes these various intersections, but it cannot be stated in just a few words; we refer the reader to Walker (1950) and Coolidge (1959) for “elementary”, i.e. visual, expositions, at least to begin with, not requiring us to climb many rungs. However, we can’t get to the heart of things by staying just with those works; we must ascend higher in order to progress. We have already encountered the genus of algebraic curves. For a singularity.n 1/.n 2/ free curve of degree n the genus equals . For the visual person, and 2 2 for the complex projective curves of CP , which are thus compact oriented surfaces, it is absolutely necessary to know that our genus coincides with the topological genus that we will encounter in Sect. VI.1, specifically the number of holes. We should find zero holes for curves of degree one and two lines and conics which we have seen to be topological spheres; and even a torus, for the cubic. But for higher degrees and for singularity-free curves, we don’t get all the integers, but only 0; 1; 3; 6; 10; : : : . The topological appearance of all the compact surfaces of three-dimensional space is shown above. These figures represent the topology of complex algebraic curves as situated in the complex projective plane CP2 , which is real fourdimensional! The spherical case is that of the (complex) conics; the toroidal case (one hole) is that of the cubics without double point. In contrast, we can find all the genuses if singularities are permitted. The genus will soon be defined for abstract algebraic curves, but its connection with the surface with singularities of CP2 is more delicate; see Griffiths and Harris (1978). The curves of genus 0 are called unicursal or rational, which means that they are parameterizable by rational functions. The lines and the conics are such, as are cubics having one singularity; more generally, we can apply Plücker’s formulas, which will bring out the fact that many singular points imply unicursality. The notion of duality was essential in studying the conics; see Sect. IV.4. Here too an algebraic curve C (over any field) has a dual curve C , i.e. the set of its tangents taken in .KP2 / , a set of lines of KP2 identified with KP2 itself is the

Fig. V.14.9.



set of points of an algebraic curve; but, if C is of degree n, then C is of degree n.n 1/. This degree is, by its construction, the number of tangents that can be placed on the curve starting with a point of the plane and is called the class of the curve. Here is how we show that it equals n.n 1/: the line xX C yY C zZ D 0 will be tangent to the curve P.x; y; z/ D 0 if, when we write the equation (of degree n in X, Y, replacing x by its value) which gives the points of intersection, easily seen to be of degree n 1, has a double root. Moreover, we have a proper duality: .C / D C. But then this number n.n 1/ opens up an abyss under us, for when n is greater than 2 we get a weird degree for .C / . The explanation is that the class in fact depends on singularities. For example a singularity-free cubic will have a dual of order 6, but that dual will have an enormous number of singularities. Following a first attempt by Poncelet to remove this “apparent” contradiction, the complete answer would lie in the Plücker formulas, which had been rigorously proved by the end of the nineteenth century. Here are these formulas, which apply to the complex case, for curves having only double points, cusps of the type .t 2 ; t 3 / and inflection points of the type .tI t 3 /. We have two direct formulas: m D n.n 1/ 2ı 3


D 3n.n 2/ 6ı 8;

where n is the degree, m is the class, the number of points of inflection, ı the number of double points and the number of inflection points. In the duality, the double points correspond to double tangents, the inflection points to cusps, whence we get two other Plücker formulas by applying the first formulas to the dual curve: n D m.m 1/ 2 3i


D 3m.m 2/ 6 8;

where is the number of double tangents of the curve. Readers will not have forgotten to verify that this time there is no contradiction in the degree of .C / ! The value 3n.n 2/ for the “general” number of points of inflection is explained thus: with each curve P.x; y; z/ D 0 is associated its Hessian, which is the determinant @2 P of the 6 second derivatives @x 2 , etc. and is of degree 3n.n 2/. Now the points of inflection are easily seen to be the points of intersection of the curve with its Hessian. The first formula shows the elementary result: an algebraic curve of degree n cannot have more than n.n1/ singular points. 2 Thus, the correct invariant for algebraic curves, in the duality, is neither the degree nor the class, but exactly a sort of mixture of the two and is nothing other than the genus already encountered several times above, for its value is effectively both .n 1/.n 2/ . C /. .ı C / and .m1/.m2/ 2 2 Just because we have Plücker’s formulas doesn’t mean that we know all the actual possibilities for the numbers that arise; not all combinations of integers satisfying the four Plücker formulas are possible; see p. 109 of Coolidge (1959). For the current state of the problem, see Laumon (1976); for the real case, see Sect. V.15.



V.15. Real and Euclidean algebraic curves The topology of real algebraic curves. It’s the first of all questions and a problem that is typical for what we pursue in this book, i.e. requiring an ascent of the ladder, being of extreme difficulty and in the end not yet being settled. We consider a singularity-free algebraic curve C of degree n in RP2 : what is its topology? The complex case would be trivial: it would be a surface with .n1/.n2/ Dg 2 holes, where g is its genus. In the real case, then, we have seen that it was a circle for all the conics, and one or two circles for the cubics. Moreover, one of the circles is, in RP2 , hom*otopic to a projective line, while the other is contractible. We say that a topological circle of RP2 is an oval or a line according as to whether it is hom*otopic to zero or to a projective line, respectively; there are no other possibilities. In each degree there never exists more than one line: in fact a little algebraic topology shows that two lines must intersect in at least one point (as do two true projective lines), which will thus be a double point, but these have been excluded. And, in odd degree, there always exists one line, because each real equation of odd degree possesses at least one real root: we ignore the possibility of this line and observe the ovals that remain. Two questions arise: first that of their number, then that of knowing how they are placed with respect to one another: they can be mutually exclusive or one might enclose another. If at first it’s just a question of the number of these ovals, we have had the perfect answer since Harnack (1876): the maximum number is given by 12 .n 1/.n 2/ C 1 in degree n. Moreover, it is attainable; see the elementary (but dated) exposition A’Campo (1980). We indeed find 1 for the conics, 2 for the cubics. The proof of the bound is by contradiction and amounts only to finding a curve of some degree that intersects the curve under consideration in too many points vis-a-vis Bezout. In the other direction, the idea works by recurrence and applies the small parameter method: if C D 0 is the equation of a curve of degree n (with already the maximum number of ovals and well chosen besides) and D D 0 is the equation of a line intersecting only one of the ovals in n points, we consider the curves of the equation CD D ". For " sufficiently small, we obtain a curve with at most n 1 ovals. The method can be seen better for 4 and 6 in the figures below. For the first we take two conics with equations C1 and C2 and the curve with the equation C1 C2 D " with " sufficiently small and with the appropriate sign; we then have four connected components. For the case of degree 6, we again takes two conics C1 and C2 and, in addition, four lines Li (i D 1; 2; 3; 4) and proceed in two stages: first with the curve K4 with equation C1 C2 C"L1 L2 L3 L4 , then with the curve with the equation K4 C1 D . We find eleven connected components: The question of knowing how these different ovals are positioned is of a different order of difficulty, and in fact has not yet been answered. For curves of degree 6 it was already a part of the 16th Hilbert problem. Apart from some rather partial intermediate results, it is only recently that the problem has made spectacular



c Springer and N. A’ Campo Fig. V.15.1. A’ Campo (1980)

progress. We give briefly a selection of results; for complementary results see the two non-technical expository texts Risler (1993) and Itenberg and Viro (1996). As in Sect. V.10, the most interesting curves are those called extremals (M-curves in the current literature), i.e. those that have the maximal number of ovals. As soon as three ovals are nested, the maximum number is decreased (this can’t happen for n D 4). There remains for the extremal curves but a single integer p for which the values are unknown, specifically that of the ovals situated in the interior of an oval, the others all being exterior, as in the “symbolic” figure: We should remark that these drawings are oversimplified; the reader will find circles intersecting the symbolic curves drawn in many more points than permitted by Bézout; in contrast, Fig. V.15.1 is realistic.

c Springer and N. A’ Campo Fig. V.15.2. A’ Campo (1980)



Fig. V.15.3.

We now discuss the most important case, that of even dimension n D 2k. ovals on the exterior and thus The curves constructed by Harnack gave 3k.k1/ 2 .k1/.k2/ on the interior. On the interior it’s difficult to set a value. In dimension 2 6 the first really prickly case Hilbert succeeded in getting 8 on the interior, thinking that there were but the two possibilities, 1 and 8, for p. But in 1974 Gudkov found a curve with p D 4 and showed furthermore that 8, 4, 1 are the only possibilities in degree 6. All these results use the method of small parameters, together with a fine analysis of Hilbert of the singularities that can be obtained upon deformation of nonsingular curves. The true revolution is in Viro (1990a), based on his completely new method, discovered in 1983 and valid in arbitrary dimension; the essential idea is well explained in Itenberg and Viro (1996). We can start with a drawing of the triangulation type T, called patchwork , for which the vertices are the points with integer coordinates .i; j / of the square f.x; y/ W jxj C jyj 6 m2 g, where m is any integer. P With T we associate the polynomial i;j i;j x i y j t .i;j / , where the summation is taken over the vertices of the triangulation, the i;j are signs, t is a real parameter and .i; j / an appropriate convex function on the triangulation T. It is shown that for t > 0 sufficiently small this yields a singularity-free curve in x, y of degree equal to m and whose connected components are directly connected with T. With appropriate T’s we can construct curves with p sufficiently varied with respect to the degree n D 2k. Here is the patchwork yielding a curve of degree 10 for which p equals 32: this curve was the first to invalidate an old conjecture going back to Ragsdale in 1906. To this day the realizable pairs .n; p/ are not known. For example, still for the case n D 2k and where q D 12 .n 1/.n 2/ p, do we always have for the Mcurves the inequalities p 6 3k.k1/ C1 and q 6 3k.k1/ C1? Or, what is equivalent, 2 2 2 is jp qj 6 k ? Another problem, that of the maximum number of points of inflection, was broached without rigor by Klein: he thought this number was a third of that for the complex case, i.e. n.n 2/ for the degree n. The complete proof, in (Ronga, 1998), uses other than algebraic geometry the theory of singularities (transversality, etc.) already encountered in the 3264 conics of Chasles in Sect. IV.9.



Fig. V.15.4. A patchwork that yields a curve of degree 10 and with p D 32. Itenberg, c Springer and I. Itenberg Viro (1996)

Real Euclidean curves: evolutes, involutes and caustics. For each plane Euclidean curve (not necessarily algebraic) of class C1 , we can define two notions: that of evolute and that of involute. The evolute C˘ of the curve C is the curve that is the envelope of the normal lines to C: for a circle it reduces to a point, a completely degenerate case; for an ellipse we already find a curve of degree 6 with four cusps, etc. The four vertex theorem shows that the evolute of a simple closed curve always has at least four cusps, for our envelope always has a cusp when the curvature is maximum or minimum at the corresponding point. Pay attention see Fig. V.15.4 to what is obtainable with some double counting: in the case of the second figure we’re dealing with a curve of constant width, whose evolute is completely traversed twice. We have seen in Fig. V.10.14 how this is a particular case of the more general notion of wave front. Some curves are interesting to mention: the hypocycloids and epicycloids ; these are curves for which the evolutes are the “the same”, which means precisely that they are obtainable from the original curve by a similitude; see 9.14.34 of [B] or p. 345 of [BG]. We define these curves (whence their name) as those obtained by letting one circle roll on another on the exterior or the interior tracing the path of a point fixed on the rolling circle. The case of a circle rolling on a line is well known we then speak of a cycloid and of multiple importance in applications. The evolute is obtained by translation; it is thus essentially identical. We first show that it is a brachistochrone and the only one; see Sect. 1.5.4 of Dombrowski (1999), a delightful work both for its pedagogy and for the history of this theorem. That is, a ball placed somewhere along the curve and initially at rest slides to the bottom in a time that is independent of its starting point. This is perhaps difficult to believe, but it is true and explains why the pendulums of old clocks were hung by flexible metallic leaf and not about



Fig. V.15.5.

an axis of rotation. In this way the lowest point of the pendulum would describe approximately, to be sure a cycloid, and thus the duration of the swing was much more independent of amplitude than was the case for the classical pendulum. Also, the form of the cycloid explains why a car that is driving on a gravel road throws gravel forward, and not backward, since the gravel which adheres to the tire will describe a portion of a cycloid. But now it is not difficult to go in the reverse direction, from C˘ to C. It suffices to know the formula for the first variation; see any book on Riemannian geometry, e.g. I.1.C of Berger (2001b). In particular, the following is shown: let a, b be points of C such that, when we pass from a to b, we don’t encounter any cusp on C˘ between the associated points a˘ , b ˘ . Then the arc length of C˘ between a˘ and b ˘ is equal to the difference in the lengths aa˘ and bb ˘ . We can thus reconstruct C with some degree of caution starting with C˘ , unrolling a string fixed to C˘ . The involute of a curve is the curve obtained, starting at some point of the given curve, by the unrolling procedure. We obtain in fact a family of curves, said to be parallel. For the distance between the two curves, measured along their common normals, is constant. If we take the involute of a curve having an odd number of cusps, we then find curves of constant width. What more can we ask regarding the case of algebraic curves? We mention two results from the end of the nineteenth century that seem little known to contemporary geometers. The first deals with the singularities of the evolute, which is again clearly an algebraic curve; formulas may be found in Coolidge (1959). The second problem is to know when the involutes are algebraic in this case, which isn’t true generally. The involute of a circle is a spiral and absolutely not algebraic, nor incidentally is an ordinary cycloid. The answer has been known since Humbert (1888) and is not at all elementary. It is directly tied to the problem of knowing when the arc length is an algebraic function of the endpoints, which isn’t



Fig. V.15.6.

the case for circles or ellipses. The answer is that the desired curves are the caustics of algebraic curves: the caustic of C for one point source of light (or one direction if the point source is infinitely remote) is the envelope curve of the lines obtained by optical reflection from the light source considered. For example, one of the caustics of a circle is the epicycloid with two cusps. Humbert’s result is not so



Fig. V.15.7.

surprising given what has been said about the reproductive properties of the epi- and hypocycloids. But the proof requires the whole arsenal of the so-called Abelian integrals and an important theorem of Abel respecting them. This problem is important in physics, in research on the totality of nonlinear electromagnetism theories; see Gibbons and Rasheed (1995). Metric properties. Finally and as always in the Euclidean setting, it is correct to ask whether there exist metric definitions of algebraic curves, analogous to those of conics. This subject, not much in fashion nowadays, is well detailed in Chap. X of Coolidge (1959), but we don’t know of a nice result that is really pleasant to state and leave it to readers who consult the book to make up their own minds about the various results. Here we present just one: if we intersect a planar algebraic curve whose complexification does not contain cyclic points see Sect. II.XYZ with a circle and the number of points obtained is equal to the degree of the curve, then the center of gravity of the points obtained depends only on the center, but not on the radius, of the circle. Already for conics, other than circles, the result isn’t clear. Let us say that we would like definitions of algebraic curves by means of distance relations, and not solely by metric properties relating certain distances or angles; we can if necessary be content with the articulated polygons below. Here is a nice result that was pointed out to us by Chern. It uses Abelian integrals and is differential in nature: if we intersect an algebraic plane curve C by lines at the points fxi g, and if the number of lines equals the degree of the curve, P k.xi / then i sin 3 D 0, where k.xi / denotes the algebraic curvature of C and i is the i angle the curve makes with the line (modulo ). This result is taken from p. 84 of Segre (1957), where it serves as a very special example. It seems to us that the entire book abounds with results that are interesting but presently little in fashion. Articulated polygons. In Sect. II.3 (see also Sect. II.XYZ and Sect. V.7) we encountered the inverter problem, in particular for describing a linear segment with a



plane articulated system. Each point of an articulated system with a single degree of freedom describes a piece of an algebraic curve, since everything there is polynomial. The converse is true; it’s the Kempe-Koenigs theorem: every algebraic curve of R2 can be realized by an articulated system. This realization can generally only be local, for an articulated system can get stuck after a certain time. Readers may refer to the completely elementary but partial exposition in Chap. V of Hilbert and Cohn-Vossen (1952) and the much more profound version in Lebesgue (1950). The case of articulated quadrilaterals, even parallelograms, has unexpected depth. The profound reason lies in the mathematical structure defined as the set of quadrilaterals whose side lengths are given. One of the components can be fixed, leaving a degree of freedom. To study this structure, it is deceptive to be interested as above in curves described by points bound to the system. For example, for a trapezoidal rectangle of given dimension, the larger side being fixed, the middle of the smaller side describes a complete curve of degree 6, called Watt’s long inflection curve, which was used for constructing a machine that imitates the human gait. But the correct description is seen by examining the relation between the points of the two circles whose centers are the extremities of the base whose distance apart is given. But we see that this is of the exact same algebraic nature of correspondence (2,2) as that of the Poncelet polygons. It is thus once more an elliptic curve that gives the sought-after structure. This was discovered a long time ago see Darboux (1879) but hugely ignored. The interesting dynamic geometry text (Benoist and Hulin, 2004) deals with quadrilaterals of given side lengths. This reveals in depth the results of Emch (see Sect. IV.8), which explain the Poncelet polygons, and more, with the aid of articulated systems consisting of juxtaposed parallelograms. The theory of articulated systems, this time with several degrees of freedom and thus providing algebraic objects of higher dimensions, has recently been the object of several works that are astonishing for non-platonists. It is thus that the structure of a plane polygon with five sides (two real parameters) yields a surface, part of a complex surface discovered in the 1970s: the K3 manifolds of Calabi-Yau. Starting with six sides (three real parameters), we get so-called mirror manifolds, which didn’t appear in mathematics until 1980 and play an essential role in mathematical physics; see Voisin (1996). All of this is contained in unpublished work of M. Kontsevitch of IHES. The case of quadrilaterals was already treated from this point of view in Darboux (1879). We obtain the space of these quadrilaterals within isometry by fixing two points as explained above. In a way analogous to what was explained in Sect. IV.12, for our quadrilateral there exist two canonical involutions described in Fig. V.15.9 below: For this point of view, see Fig. IV.8.7 and the references Emch (1900), Emch (1901) and Barth and Bauer (1996). Being involutions of an elliptic curve, their composition is a translation. This translation is always of order 6, i.e at the end of six repetitions we get the initial object, which can be shown by drawing a cube



c Elsevier Fig. V.15.8. (a, b) Lucas (1960)

Fig. V.15.9. To be repeated six times to test drawing abilities

and observing the hexagons formed by the edges transversed appropriately. We already found a transformation of order 6 in Fig. II.2.10, and readers may investigate whether or not there is a link between the two cases. For an elementary exposition of articulated systems with numerous drawings, see Rideau (1989). For articulated polygons in R3 made conceptual, see Kapovich and Millson (1996). The recent text (Benoist and Hulin, 2004) studies the dynamics of foldings of quadrilaterals when they are iterated indefinitely. V.16. Finite order geometry Here we study the converse, if it exists, of the fact that each line intersects an algebraic curve of given degree n in at most n points in the real domain, and in exactly n points (with multiplicities) in the complex domain. This type of question, which consists of studying curves and more generally submanifolds of affine or projective space which are intersected by each line in a finite given number of points (called the order of the object in question), is called finite order geometry, and was the



object of numerous studies over the decade preceding 1900. The entire book Haupt and Künneth (1967) is dedicated to it; references can be found there as desired. We take from it some typical results that are not too technical, that are of two types: first, locally, it is shown that certain conditions of finite order imply that the object is more or less differentiable, that its singular points aren’t too menacing; then come the global results, which use the preceding as needed. We treat the case of arbitrary dimension, so as not to disseminate relatively isolated results in this book. At first we stay with the truly real domain. We need to realize right off, beginning with order 2, that we are at loose ends: every strictly convex curve is of order 2, valid locally or globally. We thus seem lost, but it is almost impossible to comprehend the stubbornness of mathematicians. First, we have already stated that the four vertex and the six point sextactic theorems had been proved initially by methods of finite order geometry (Sects. V.8 and V.9). There is also Möbius’s theorem, which states that each noncontractible closed curve in the real projective plane possesses at least three inflection points. A theorem of Juel from 1914 states that each connected simple plane curve of order 3 can only have the topological forms shown below:

Fig. V.16.1.

By curve we understand exclusively a continuous curve. In Marchaud (1965) it was proved that when a surface of order 3 (cubic from R3 or RP3 ) possesses lines in sufficient number (i.e. at least 8), then all these lines belong to a cubic algebraic surface; see p. 393 of Haupt and Künneth (1967) and the references mentioned there. For a generic surface (not a cone, plane or ruled surface) the number is 15 or 27. This configuration of 27 lines was encountered in Sect. I.9. Also Marchaud (1936) seems at first very spectacular: each surface of R3 of order 2 is either a piece of a convex surface or a piece of a (ruled) quadric. It is, however, necessary to realize what is being called a surface (otherwise finite point sets, etc., will be solutions). For Marchaud a surface of order n is simply a set for which each plane section is a curve of order n. For those interested in circles, we find this in Juel: a surface of E3 that is intersected by each circle in at least four points is a cyclide (surface obtainable from a torus by an inversion mentioned at the end of Sect. II.7). We also find some strictly finite order geometry in Sedykh (2000); and there is in Darboux (1880) a study of the maximum number of families of circles (more generally of conics, for it is basically an affine problem) which can belong to a surface.



Fig. V.16.2. As soon as a surface is neither convex nor a part of a (ruled) quadric, there exists an abundance of lines that intersect it in at least three distinct points

But there doesn’t seem to be any connecting thread when we peruse the thick book Haupt and Künneth (1967) and we strongly sense a lack of unity. How can we climb a ladder when we can’t even find the ladder? It is in Thom (1969) that at last things are placed in a proper setting the one he himself invented of singularities and of transversality. Based on this ascent, three things from this pioneering book should be pointed out that throw an entirely new and renewed light on finite order geometry. First, the notion of finite order for each subobject of dimension N is generalized at the outset; typically it is a submanifold W of dimension N of RM : the condition of finite order for this object W is that each affine subspace of RM of dimension M N intersects it in at most n points. Thus Thom proved this “almost everything” type of result: If V is a compact differential manifold of dimension N, then in the space of all differentiable mappings from V into RM (with a Cr topology), there exists an everywhere dense open set U such that for each mapping g 2 U the image g.V/ is of finite degree. Thus the objects of finite order aren’t rarities, but instead the general rule, but it is important to realize that the degree of g for all possible choices of g is not bounded on U. To quote Thom’s second point, it’s necessary to define the notion of local order of a point v of a VN RM : it’s the limit superior of the number of points of intersection of V with an .M N/-dimensional subspace for which the distance to v tends to zero. Next we must speak of the local order of a generic submanifold, including for example those of the space U above. This local order grows with the codimension M N. The first theorem implies that a compact embedding of minimum order has an order that does not exceed the generic local order. For example it is known (Thom) that the generic local order for the surfaces of R3 equals four. It is effectively attained for each surface; but we can, furthermore, for each number of holes, find a surface of order 4 exactly. This is due to Calabi; take a suitable tubular neighborhood of the graph drawn below: But in return the preceding brings out that:



Fig. V.16.3.

The objects whose total order is less than the generic local order are likely very rigid. That is exactly what the two results of Marchaud and Juel given above say, results that so to speak return now to the regular order. The third result was announced exclusively by Thom, and clarified and completely proved in (Pohl, 1975); it is also Pohl who answered the most essential question that we used for motivation: Let V2k be a compact submanifold of real dimension 2k of the complex projective space CPkCn such that each element of an everywhere dense set of complex projective subspaces of dimension n intersect it in exactly m points, where m is a fixed integer. Then V2k is a submanifold, either a (complex) algebraic variety of dimension k, or an image under a projective transformation of CPkCn of the canonic embedding RP2k CPkCn . In fact this last type of subset is actually intersected in exactly one point by each complex projective subspace of dimension n. To our knowledge Pohl’s work excepted this extraordinary breakthrough of Thom has not been followed by new discoveries. For example, Pohl’s result requires that the submanifold be of class C4 . A recent text on this difficult subject is Meyer (2006). V. XYZ Higher dimensions. Algebraic geometry can of course be done in all dimensions. We have pursued it in degree 2 for the quadrics, but we had to remain reasonable. However, we have done a little algebraic geometry in dimension 5 while studying the 3264 conics of Chasles. We merely mention too, in the spirit of the resurgence of algebraic geometry in the real case, problems of topology for example: the number of connected components, but above all the topology of these components by themselves since, as soon as we go beyond surfaces, the topology of compact manifolds is an immense realm, not yet fully explored. The bulk of the progress is due to Viro; see Viro (1990b). It is incidentally from his general theory that he subsequently deduced refinements for the case of plane curves. We are not able to resist the very geometric theorem on the algebraic surfaces of order 3, called cubic here, already encountered in Sect. I.9: a cubic surface of CP3 which isn’t



ruled (doesn’t consist of a one parameter family of lines) always possesses (contains) 27 lines. Their combinatorial configuration is imposed; see Schläfli’s double six in Sect. I.9. Here, in the real case of cubic surfaces of RP3 in contrast to the inflection points of, of which there can only be 3 in the real case as opposed to 9 in the complex case there exist real cubic surfaces possessing 27 lines; see Fig. I.9.5. The twin books Fischer (1986a, 1986b) from which the figure is taken are fascinating and the photos superb. Differentiability of functions, of mappings. The class Cp is that of functions having derivatives of order up to (and including) p and such that these derivatives are moreover continuous; the merely continuous case is expressed as C0 . If derivatives of all orders exist, we speak of the class C1 . It can involve numerical functions, but more often we are in the general setting of mappings U ! Rn where U is an open subset of Rm . The fundamental property of these classes is that of being closed under the composition of mappings. It is this property that makes what follows possible. But before all else the notion of diffeomorphism: two open subsets U, V of Rm will be called diffeomorphic if there exists a bijection f W U ! V such that both f and f 1 are of class C1 (or any class other than C0 and mutatis mutandis). We are restricted to m D n, for this is imposed by the differentiability. In all these definitions that are going to come from differential objects, we omit specification of the class. As for references, many works on differential geometry are good. Only rarely will we allude to the classes Ck;˛ of functions having derivatives of order through k and such that these derivatives are Lipschitz of ratio ˛. A function is .x;y/ Lipschitz of ratio ˛ when it satisfies fkxyk 6 ˛ for all x, y with x 6D y. The notion of a Lipschitz function extends immediately to the case where the source space is an arbitrary metric space; we can also speak of Lipschitz mappings between two given metric spaces. Of course the functions of class C1 whose differential is of norm bounded by ˛ are Lipschitz with ratio ˛. But the converse is false. We mention however, even though we won’t use it, the fundamental discovery of Rademacher in the setting of functions on Rn : a Lipschitz function is almost everywhere differentiable. This theorem has been the constant object of extensions; see for example Cheeger (1999). There is finally the class of functions called real analytic, denoted C! . These are the functions that are infinitely differentiable and for which, moreover, the Taylor series at each point converges and coincides with the function itself over an open interval containing the point. The analytic functions are the only ones that are capable of being “predictive”, in contrast to the general C1 functions, the classic example 2 being the function that is zero on 1; 0 and which equals e 1=x on 0; 1Œ. From what is on the left nothing can be predicted for what is on the right! General topology. We have already used classic notions from general topology and will use them extensively in the sequel, without specifying each time what is in-



volved: topological space, open sets, closed sets, compact set, complete spaces (typically when we deal with metric spaces), etc. Readers will be able to find references to their liking. Measures. The notion of measure is a subtle one, even in the new millennium. It belongs on Jacob’s ladder a bit above metric spaces, topology, simple connectivity, etc. The book Oxtoby (1980) is interesting, for it also treats the notion of “almost everywhere” in an exclusively topological setting, often called Gı -dense, the only such notion possible in those situations where there doesn’t exist a measure that is adequate for the geometry considered (especially in those cases where one can’t exist by the profound nature of things). Manifolds. It is impossible to define a differentiable manifold of dimension n quickly and well, especially since in practice it is always necessary to restrict ourselves to those that are in a sense denumerably infinite, as explained below. This is likely why Elie Cartan used this notion without ever defining it. The quote Cartan (1946–1951) is famous: “the notion of manifold is rather difficult to define with precision”. We won’t do much better than Cartan. Let us say first that a topological manifold of dimension n is a separable topological space (satisfying Hausdorff’s axiom ) for which each point is contained in an open set that is homeomorphic to an open subset of Rn . A chart of a topological manifold M (of dimension n) is a pair .U; ˆ/ consisting of an open subset U of M and a hom*omorphism ˆ of U onto an open subset of Rn ; a differentiable manifold is a topological manifold that can be covered by a set of charts (an atlas ) such that, for each pair of charts .U; ˆ/, .V; ‰/ for which U \ V is nonempty, the mapping ‰ ı ˆ1 W ˆ.U \ V/ ! ‰.U \ V/ is a diffeomorphism.

Fig. V.XYZ.1.



The coordinates associated with a chart ˆ are the xi (i D 1; : : : ; n): U ! R defined by transforming by ˆ the canonical coordinates fui g (i D 1; : : : ; n) of Rn : xi D ui ı ˆ.

Fig. V.XYZ.2.

We consider here only differentiable manifolds that admit denumerable atlases, for which we define tangent space at each point and the tangent bundle, which is the collection of all the tangent spaces Tm M for the various points m of M. This definition is consistent with the various charts thanks to the rule that gives the differential of the composition of two mappings. For geometry, the tangent vectors are the velocities of curves traced in the manifold. In a chart with coordinates fxi g the tangent spaces also have automatically associated coordinates; they are such that if c is a curve of M, then the coordinates of the velocity c 0 .t / will be simply the .xi .c.t //0 D d.xidt.c.t// . We also have the notion of differentiable mapping f W M ! N between two manifolds (understood to be differentiable from now on), thus also of diffeomorphism. Such a mapping f W M ! N admits a differential df : this object df .m/ at m 2 M is a linear mapping df .m/ W Tm M ! Tf .m/ N. For “pure” geometry, the nicest definition entails the effect of f on the velocities of the curves.

Fig. V.XYZ.3.

It is following Whitney that we know the form of this definition, but we owe to him also the essential theorem that every differentiable manifold of dimension n is diffeomorphic to a differentiable submanifold of an RN of dimension N sufficiently large as a function of n. A submanifold of an RN is defined as for the geometric



curves, by equivalent conditions: being everywhere locally a graph, knowing that things are diffeomorphic to the situation of a linear subspace. We can also define the notion of submanifold of a manifold. We say that f W M ! N is an immersion if df is everywhere injective. We say that f W M ! N is an embedding if it is an injective immersion.

Fig. V.XYZ.4.

We see that what was said in Sect. V.3 with respect to curves can now be expressed thus: there are but two manifolds of dimension 1, the line R and the circle S1 . A geometric curve is a submanifold of dimension 1 of R2 ; it is simple closed if it is an embedding of S1 in R2 ; it is a kinematic curve if it is an immersion of R in R2 (it will have been noted that each open interval of R is diffeomorphic to R itself). A closed kinematic curve is an immersion of S1 in R2 . Here we are dealing only with plane curves, but by replacing R2 by RN we obtain analogous objects in space (of three dimensions for R3 , etc). For the question of the classification of manifolds of dimension greater than 1, see Sect. III.1.D of Berger (2001b). The notion of hom*otopy is very general and plays a role in numerous contexts. The simplest case is that of loops. A loop in a topological space T (which is not prohibited from being a differentiable manifold) is a continuous mapping f W Œ0; 1 ! T such that f .0/ D f .1/; the origin of this loop is this point f .0/ D f .1/. Two loops with the same origin are called hom*otopic if we can pass continuously from one to the other; which is to say precisely that there exist a continuous mapping F W Œ0; 1Œ0; 1 ! T such that the restriction of F to Œ0; 1f0g (resp. Œ0; 1 f1g) is the first (resp. the second) loop. But we can do the same thing with two curves without a marked origin (called a free hom*otopy); we will encounter all this in particular in Chap. XII. But when we deal with more definite objects than



just continuous mappings (topological curves, for example) we usually require that supplementary conditions be preserved along the deformation. We then speak rather of isotopy; we saw it in Sect. V.10 above. A topological space is called simply connected if each loop (or, what amounts to the same thing, each closed curve) is contractible, i.e. is hom*otopic to a point (a degenerate curve); see also Sect. III.5. An essential fact is that every reasonable space, manifolds in particular, admit a universal covering that is simply connected (and essentially unique). In particular a covering of a simply connected space can only be trivial, that is, a bijection; we will see a typical application in Sect. VI.7. The sphere is simply connected (of dimension two), the circle (of dimension one) isn’t; that was essential in Sect. V.3. For a formulation of all these notions, see in part [BG], but the complete details are always a bit long; see also works on algebraic topology. Bibliography [B] Berger, M. (1987, 2009).Geometry I, II. Berlin/Heidelberg/New York: Springer [BG] Berger, M., & Gostiaux, B. (1987). Differential geometry: Manifolds, curves and surfaces. Berlin/Heidelberg/New York: Springer A’Campo, N. (1980). Sur la première partie du seizième problème de Hilbert. In Séminaire Bourbaki 1978–79: Vol. 770. Springer lecture notes in mathematics (pp. 208–227). Berlin/Heidelberg/New York: Springer A’Campo, N. (2000a). Generic immersion of curves, knots, monodromy and gordian number. Publications mathm´ atiques de líInstitut des hautes études scientifiques, 770, 208–227 A’Campo, N. (2000b). Planar trees, slalom curves and hyperbolic knots. Publications mathm´ atiques de líInstitut des hautes études scientifiques Alias, J. (1984). La voie ferrée. Paris: Eyrolles Angenent, S. (1991). On formation of singularities in the curve shortening flow. Journal of Differential Geometry, 33, 601–633 Angenent, S. (1992). Shrinking doughnuts. In N.G. Lloyd, L.A. Peletier, & J. Serrin (Eds.), Nonlinear diffusion equations and their equilibrium states, 3. Boston: Birkhäuser Arnold, V. (1978). Mathematical methods of classical mechanics. Berlin/Heidelberg/New York: Springer Arnold, V. (1994a). Topological invariants of plane curves and caustics. University Lecture Series. Providence, RI: American Mathematical Society Arnold, V. (1994b). Plane curves, their invariants, perestroikas and classifications. Advances in Soviet Mathematics, 21, 33–91 Arnold, V. (1995). The geometry of spherical curves and the algebra of quaternions. Russian Mathematical Surveys, 50, 1–68 Arnold, V. (1996). Remarks on the extactic points of plane curves. In The Gelfand mathematical seminars (pp. 11–22). Boston: Birkhäuser Barth, W., & Bauer, T. (1996). Poncelet theorems. Expositiones Mathematicae, 14, 125–144 Benoist, Y. & Hulin, D. (2004). Itération de pliages de quadrilatères. Inventiones Mathematicae, 157, 147–194 Bérard, P., Besson, G., & Gallot, S. (1985). Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy. Inventiones Mathematicae, 80, 295–308 Berger, M. (1972). Enveloppes de droites. Bulletin de l’Association des Professeurs de Mathématiques de l’Enseignement Public, 283, 311–314 Berger, M. (1993). Encounter with a geometer: Eugenio Calabi. In P. de Bartolomeis, F. Tricerri, & E. Vesentini (Eds.), Conference in honour of Eugenio Calabi, manifolds and geometry (pp. 20–60). Pisa: Cambridge University Press



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Chapter VI

Smooth surfaces VI.1. Which objects are involved and why? Classification of compact surfaces We will study contemplate the next simplest objects after curves, i.e. surfaces. We studied curves essentially in the plane, whereas surfaces appear in the Euclidean three-dimensional space E3 . However, we will see soon enough the necessity of considering abstract surfaces; see Sect. V.XYZ. We didn’t encounter this problem for curves, for the only abstract curves are the line and the circle, and we can always visualize them, with their internal geometry, as situated in the plane. This impossibility of visualizing certain surfaces has already been encountered in Sect. I.7 with regard to the projective plane and in Sect. V.14 with regard to elliptic curves. We will encounter it once more in Sect. VI.4 below with regard to hyperbolic geometry. The word smooth is usual for saying differentiable, having a differential, requiring the existence of a tangent plane at the very least. In another direction there are the polyhedra, that will be treated amply in Chap. VIII. Using Sect. V.XYZ, a surface in E3 (more briefly just surface) is a submanifold of E3 of dimension 2, in contrast to an abstract surface, i.e. a differentiable manifold of dimension 2. A surface in E3 can thus also be an embedding in E3 of an abstract surface (abstract manifold of dimension 2), but we will exceptionally also consider immersions of surfaces in E3 . Here we are not going to try to present everything that is known some of which is quite recent but rather concentrate on what isn’t known about surfaces. In particular, we will mention some rather spectacular instances of problems that are open but nonetheless simple to state, some recent. On the other hand, we won’t be climbing very high on Jacob’s ladder. A rather complete exposition of the basic definitions for surfaces, of synthetic nature without proofs, can be found in the last chapters of [BG], which can be supplemented for definitions and basic results by parts of Berger (2003). Readers can also consult the textbooks do Carmo (1976), Stoker (1969), Klingenberg (1973) and Montiel and Ros (1997). The rather old reference Struik (1950) remains much appreciated, Gray (1993) is written systematically using the computer algebra system Mathematica and we should add the recent synthetic work Burago and Zalgaller (1992). Finally Porteous (1994), in its title a bit audacious, presents an approach to curves and surfaces which is rather “orthogonal” to classic approaches, holding to the singularities and transversality of Thom, a book that is very interesting, because it uses an approach which is not very theoretical, but rather systematically geared toward practical applications and the real world. M. Berger, Geometry Revealed, DOI 10.1007/978-3-540-70997-8_6, c Springer-Verlag Berlin Heidelberg 2010




We will borrow from it numerous times in the sequel. Note too that our Chap. XII will be entirely devoted to large scale behavior, i.e. geodesic flow on a surface, an idea from Hamiltonian dynamics. As motivations for the study of surfaces, there are first those that are, as with curves, facets of everyday life thus de facto simple models for doing physics or for testing certain hypotheses. The most striking case is that of our planet, which is rather well represented by a flattened ellipsoid of revolution. The spherical model is very useful, but was treated amply in Chap. III. As in Sect. III.1, the essential thing is to introduce the so-called intrinsic (or internal) metric of a surface, given by the lower bound of the lengths of curves traced on the surface, joining two points considered. Contrary to the case of the sphere, the search for these shortest paths doesn’t in general have more than a theoretical answer. Specifically, they belong among the geodesics but, on the one hand, it isn’t possible to calculate these explicitly except in very special cases and, on the other hand, it is hard to know in practice when a geodesic stops being a shortest path; this is the problem of the cut locus. For complete surfaces, these shortest paths always exist between two arbitrary points but aren’t necessarily unique; see the definition of cut locus in Sect. VI.3. The problems about surfaces are twofold. There are those that concern the intrinsic metric for its own sake, then those concerning the connections between the intrinsic metric and the way in which the surface appears in space, which reveals relationships between the two fundamental forms. Here is a question that may occur to readers, above all if they peruse the table of contents or do some browsing. Why do we bother with curves and surfaces when we have done geometry in arbitrary or high dimensions in preceding chapters? It is because they are essential for all of Riemannian geometry, a subject too immense for us to give more than a brief summary. Readers can refer to the synthesis Berger (1998), duplicated in Berger (1999), but above all Berger (2003). However, we should also mention that another reason concerns hypersurfaces of En : as soon as we have n > 3, then generically the intrinsic metric determines the way in which they are embedded, an old remark that can be found in Killing (1885); thus there isn’t the subtle mixture of two fundamental forms that we observe when n D 3. This mixture is only provisional and still under scrutiny; see for example MartinezMaure (2000). Here now are some examples of surfaces. We have devoted all of Chap. III to the sphere, so-called Euclidean spheres (sometimes called round spheres to distinguish them from surfaces that have only the topology of a sphere. We have also encountered the quadric surfaces (ellipsoids, etc.) at the end of Chap. IV. Very important for both theory and practice are the surfaces of revolution, which are invariant under the group of rotations about a fixed line (and are smooth). These are determined once we know a meridian (or the meridian, since all meridian curves are essentially the same). The parallels are sections by planes that are perpendicular to the axis of revolution. According to the topological classification of plane curves (see Sect. V.3)



there are thus but four (connected) topological types for surfaces of revolution: the sphere and the torus for the compact case, the plane and the cylinder for the noncompact case.

Fig. VI.1.1.

We have encountered the ellipsoids and, more generally, the quadrics, in Sect. IV.10. The ruled surfaces are those composed of a one-parameter family of lines in space. The developable surfaces, for example the cylinders and the (pointed!) cones, are a particular case that will be seen in Sect. VI.7. Although beyond the main scope of our goals, we point out for interested readers that the tangent plane of a ruled surface, when the contact point describes one of the lines of the family (called generators), varies in hom*ographic fashion, so that the cross ratio (see Sect. I.6) of four points of a generator is equal to the cross ratio of four planes tangent at these points (the set of planes passing through a given line has, by duality, a structure of the projective line). This hom*ography evidently depends on the generator considered; moreover it is singular precisely for the developable surfaces, for the tangent plane remains constant as we move along a generator.

Fig. VI.1.2. For a general ruled surface, the tangent plane turns when we move along a generator, and in a hom*ographic manner. For a developable surface the tangent plane remains constant along a generator



The hyperboloids of one sheet are ruled and can also be surfaces of revolution; see Fig. IV.10.2. Although not an essential point for this chapter, it is impossible to talk about surfaces without mentioning their topological classification. We saw in Sect. V.3 that the only abstract curves are the circle and the line. An essential and natural result from algebraic topology is that the only compact orientable surfaces are those of the figure below; they are classified by the number of holes (called the genus for reasons that we have seen for algebraic curves in Sect. V.13): 0 for the sphere, 1 for the torus. The compact orientable surface with g holes is called the torus with g holes; it can be regarded as the connected sum of g ordinary tori.

Fig. VI.1.3. Connected sum of two tori

What has then happened with our real projective plane RP2 ? Its problem is that it is not orientable; but with it the classification of abstract compact nonorientable surfaces also complete: every compact nonorientable surface is the connected sum of g projective planes RP2 (g > 1), which defines the genus of the surface in question. It amounts to the same as saying that a compact nonorientable surface of genus g is a sphere with g holes, in which each hole is filled with a Möbius strip. We can give another, more visual, classification by introducing the Klein bottle:

Fig. VI.1.4. Construction of the Klein bottle: a nonorientable and immersed surface, c Springer with only one circle of self-intersection. Hilbert, Cohn-Vossen (1996)

In Sect. I.7 we saw sketches of the real projective plane RP2 , but the Klein bottle seems much easier to draw. Here is the announced classification: every compact nonorientable surface is either the connected sum of a torus with n holes and RP2 , or the connected sum of a torus with n holes and a Klein bottle. The connection with the preceding classification is easily made if we know that RP2 C RP2 D Klein, 3RP2 D torus C Klein (here C denotes the connected sum operation).



projective plane (cross-cap)

Klein bottle

Fig. VI.1.5. Classification of orientable and nonorientable compact surfaces

These abstract compact nonorientable surfaces, like the real projective plane, are not embeddable in R3 . They are only immersible, an immersion being by definition a differentiable mapping of the given surface into the space, where the tangent mapping is of maximum rank everywhere, here of rank 2; we don’t require that such a mapping be injective, which is to say that we don’t prohibit self-intersections otherwise we would have an embedding. As intuitive as it may be, this classification of surfaces is never easy to prove completely, above all in the embedded case, in spite of the pedagogic efforts of many mathematicians. Readers can consult one of the references below, but should pay attention to whether Morse theory facilitates matters as Gramain (1971) claims, but the classic proofs by slicing provide more information on the fundamental group. Proofs can be found in Gauld (1982), Hirsch (1976), Moise (1977), SeifertThrelfall (1980), Wallace (1968), Massey (1991) and Stillwell (1980). One of the difficulties is in showing that each surface is triangulizable, and triangulizability is often assumed by the classifiers. A recent reference for triangulating a surface quickly and well is Colin de Verdière (1990). For a triangulation with S vertices, A edges and F faces, the Euler-Poincaré characteristic of the surface is provided by the formula S A C F D . In the orientable case the number g of holes is related to by D 2.1 g/. A portion of a triangulation can be seen in Fig. VI.7.7. We encounter an amusing version of the classification of surfaces with the regular star polyhedra in Sect. VIII.5: two of these don’t have the topology of the sphere, but instead that of a surface with four holes. To see this, it suffices to use the preceding formula. VI.2. The intrinsic metric and the problem of the shortest path We now pose for general surfaces the same problem as was posed for the sphere in Sect. III.1: what is the shortest path from one point to the other? We will see that there is no perfect answer, but only local results.



The intrinsic or “internal” metric d of a surface S of E3 is defined by dS .p; q/ D d.p; q/ D inf flength.c/ W c a curve traced on S joining p to qgI i.e. the distance between two points is the lower bound of the lengths of curves connecting the two points. Observe that this distance isn’t in general the one that is induced by the embedding S E3 , a distance which would be dinduced .p; q/ D kq pk, but which scarcely has any practical interest. It is however clear that dS > dinduced , which allows us to show subsequently that d really is a metric on S: readers will have the courage to verify the three axioms. Now we watch for two essential difficulties that will occupy us: perhaps there is no curve that joins p to q or perhaps the distance isn’t realized by any curve.

c Presses Universitaires de France Fig. VI.2.1. [BG] Berger, Gostiaux (1987)

If a curve realizing the distance exists, it is a shortest path and is given this name, or that of segment, in analogy to the Euclidean case. The term minimizing geodesic is long out of use, even colloquially; see below. The first difficulty is simple: for obvious reasons it suffices (and is also necessary) that the surface be connected. Secondly, the problem of existence of a shortest path is of a whole other difficulty and requires ideas from general topology and study of the calculus of variations, which furnish the geodesics that we will see in the next section. In fact, this apparently simple question wasn’t correctly posed and resolved until (Hopf and Rinow, 1931), one reason for this tardiness being that general topology only dates from the start of the twentieth century. We give here the (perfect) answer of Hopf and Rinow: the surface S may or not be intrinsically complete, i.e. complete with respect to its intrinsic metric, where completeness means that every Cauchy sequence of points of S has a limit in S. The awaited answer is: if S is intrinsically complete, then two arbitrary point of S are joined by at least one segment. For proofs see the references mentioned, but once this is established we must pay attention to two things. The first is that completeness isn’t necessary; e.g. an open disk possesses all desired shortest paths between its points. The second is more subtle: a surface in E3 can be complete for its intrinsic metric without being complete for the induced metric: roll a plane into an (infinite) cylinder; the rolled-up plane



is intrinsically complete, but not complete for the induced metric; see Fig. VI.2.2. Readers can construct a surface rolled-up on the interior of a torus of revolution, thus bounded as a subset, but unbounded for the intrinsic metric; in particular this surface isn’t compact.

Fig. VI.2.2.

VI.3. The geodesics, the cut locus and the recalcitrant ellipsoids In order to investigate segments of a general surface, even of revolution, we use methods of the calculus of variations, which makes use of the tools of differential calculus as a first step. Let us suppose the problem is solved, that is, let us specify a segment c W Œ0; 1 ! S. We make small variations in it and express that the curve c has minimal length. In particular the derivative of the length must be zero (this is only necessary and certainly not sufficient in general). It seems that the first person to have made this calculation was a Bernoulli for the Bernoullis were a whole dynasty specifically Johann (1667–1748), who taught it to his student Euler; see Hauchecorne and Suratteau (1996). This is perhaps the moment to say that this reference is excellent for historical data on the mathematicians whom we mention, and can be complemented by Berger (2005) with its annotated bibliography on the history of mathematics, in France in large part. The proposition that follows appears for the first time in the writings of Euler (as in Chap. V we have parameterized the curve c by its arc length): In order that the curve c of S, supposed traversed at constant speed, satisfy the condition of zero derivative for each variation, it is necessary and sufficient that the second derivative c 00 .t / (the acceleration) is normal to the surface (i.e. orthogonal to its tangent plane). A geodesic of a surface is any curve having this property. In a chart of the surface the equation of the geodesics is a second order (vector) differential equation, but with terms quadratic in the first derivatives. Two things follow: the first is that, for a sufficiently small parameter (which may be considered as time or as distance),



Fig. VI.3.1. Geodesics. The acceleration vector is always normal to the surface. Berger c Università di Parma and A. Concari (1994)

there exists a unique geodesic emanating from a given point with a given velocity vector. The second is that in general we can’t be sure that geodesics exist for “large” times; we see this clearly on any (incomplete) surface with punctures or holes. A surface being given, we want to know graphically if there exist curves for which the acceleration is constantly perpendicular to the surface and, if so, we want to describe them as best possible. Intuitively, we are already convinced: for example, if a portion of the surface is convex, we can stretch an elastic string or rubber band between two points, which will then realize the shortest path. Physically, a ball set in motion on the surface will have a well determined trajectory if we know the starting point and starting velocity (in magnitude and direction).

Fig. VI.3.2. Nonexistence of shortest paths. The geodesics are not defined for “all times”

Another “practical” way of finding these curves, these paths, is to have a small rudimentary vehicle roll (and only roll) along the surface, a “vehicle” composed of two parallel and identical wheels fixed rigidly to an axis. The two wheels mark two curves on the surface that enclose a curve of the type sought, the better the approximation when the wheels are close together. On a plane, it is a way of going straight ahead.



Fig. VI.3.3. A small vehicle, with two parallel wheels very close together, rolling on a surface, gives an approximation to geodesics (better when axis length is small)

Note. As the acceleration is normal to the surface, the speed (a scalar) along a geodesic is constant. From now on we normalize the speed by taking it constantly equal to 1. To return to the problem of extending geodesics indefinitely, we recall that in Hopf and Rinow (1931) it was proved that: There is equivalence between the property that a surface be complete (always for the intrinsic metric) and the property that all geodesics be defined for all time (from 1 to C1). From now on we will only consider complete surfaces, except where the contrary is explicitly stated. There remain at least two things to be done right away. The first is to calculate, when possible, the geodesics for certain surfaces. The second is to say when a geodesic c, restricted to an interval Œa; b, is a segment joining c.a/ to c.b/. It is already clear for the sphere that a geodesic is not in general a segment on all intervals for which it is defined. It is the same for all other compact surfaces and for all their geodesics. We can say quite a bit about the geodesics for all the surfaces of revolution ; in fact, they can be calculated with the aid of integrals. For if we project a geodesic c onto p ı c on a fixed plane P perpendicular to the axis of revolution since the normals to a surface of revolution intersect its axis the plane curve p ı c of P (now no longer parameterized by arc length) will have a vector second derivative .p ı c/00 passing constantly through a fixed point. We thus know that, in polar coordinates, this curve .; / satisfies 2 0 D constant (where 0 denotes the derivative with respect to time). This is the “area law” for motions with central acceleration. This is not sufficient for calculating the geodesics, but already permits us to say lots of things about them. For example, when the meridian is strictly convex, the geodesics oscillate in the band enclosed between two parallels that have the same projection on P. We will encounter this in Chap. XII, but remark right away that, in such a band, there is a dichotomy: either all geodesics close up on themselves (and we can



say that they are periodic), or else they are all everywhere dense in the band, but never close. We will come back to this question in Sect. XII.3.B.

Fig. VI.3.4. At left, what happens on a sphere. At right, what happens on a surface of c revolution (m0 is derived from m by a rotation through an angle ˛ ). Berger (1994) Università di Parma and A. Concari

A particular case is that of quadrics of revolution, where we will stay with ellipsoids. We have mentioned that an ellipsoid provides a good approximation of our planet. The integrals that furnish the geodesics are elliptic integrals; see Sect. V.14, and for a recent exposition see Chap. 3 of Klingenberg (1995), but pay particular attention to some of the fine details. Since elliptic functions are well known, tables were made that allowed for numerical calculations for finding shortest paths prior to the availability of modern techniques (e.g. satellites and computers), and that was fundamental in geodesy. We can consult the classic works on elliptic functions, e.g. Appell and Lacour (1922), but that does not permit us to answer the question: when does a geodesic cease to be minimizing? See below for more on this theme which we will take up again in Sect. XII.3.B. Readers can study the different types of geodesics for a torus of revolution; this was the object of Bliss (1902–1903). It is interesting, for assessing the evolution of mathematics, to observe that what we propose as an exercise was, in 1902, an item of cutting-edge research. Since Jacobi it has also been known how to calculate geodesics for all the quadrics. Let us again stay with ellipsoids E (not of revolution, sometimes called triaxial ellipsoids , where the axes can have different lengths); we use hyperelliptic integrals . More than the calculation, what is interesting is the global behavior of these geodesics. To describe it we need (and this suffices) to use the family of quadrics hom*ofocal to the ellipsoid considered: see Sect. IV.10. These hom*ofocal quadrics slice E in two families of curves which are the proper curves of the second fundamental form that we will encounter in Sect. V.6, which means that their tangent is



always a proper direction of this second fundamental form; these curves turn out to be (it’s a theorem) the lines of curvature. In Fig. VI.3.5 we note four special points of E, called the umbilics of E (just two are shown). We will encounter this also in Sect. XII.3. We once again show Fig. IV.10.6 because of the quotation below.

Fig. VI.3.5. At left: ellipsoid E intersected by hyperboloids of families hom*ofocal to E, that of hyperboloids of one sheet and that of hyperboloids of two sheets. The traces of the hyperboloids of two sheets are the lines of maximum curvature, those of the hyperboloids of one sheet are the lines of minimum curvature. At right: an ellipsoid with three unequal axes has four umbilics, but we see only two of them in the figure, ! and ! 0

We take from Porteous (1994, p. 260), these phrases of Monge, who conveyed his enthusiasm for the above figure and for what a room would be like that had an ellipsoidal vault whose tiles followed the lines of curvature: Finally, two chandeliers suspended from the umbilics of the vault, with whose suspension the entire vault seemed to compete, served to illuminate the room during the night. We won’t enter into much greater detail in this regard; for us it suffices to have indicated to artists a simple object whose decoration, although very opulent, could have nothing arbitrary, since it consisted principally in unveiling for all eyes a very gracious arrangement which is in the very nature of this object. Just as the geodesics of surfaces of revolution oscillate between two suitable parallels, so the geodesics here oscillate between pairs of associated lines of curvature when they are intersected by the same hom*ofocal quadric ; the latter are of two types, according as they correspond to hyperboloids of one sheet or of two sheets . The case of geodesics that emanate from the umbilics is exceptional: they all pass through the “antipodal” umbilic , but we should pay attention to the fact that when they return to the point of departure, it is along a different tangent, and they are never periodic, with the exception of the plane section that contains the four umbilics. Thus the umbilics aren’t like the poles of a surface of revolution. But, just as for the surfaces of revolution, a band of geodesics between associated lines of



curvature depending on the band are either all periodic or else never close and are then everywhere dense in the band. A final property of the geodesics of the ellipsoid: the lines tangent to a chosen geodesic are all tangent to the same hom*ofocal quadric; we will return to this in Sect. XII.3.C.

Fig. VI.3.6. At left: a line of curvature realized as the envelope of a family of geodesics (the rest of the envelope is symmetric and hidden). At right: geodesics emanating from an umbilic

We can now attack the “global” problem of the shortest path, more precisely the problem of possible non uniqueness. To the best of our knowledge its study goes back to Poincaré (1905). It is not very difficult to establish what follows. Let c W Œ0; 1Œ! S be a geodesic emanating from a point c.0/ D p and with initial velocity vector c 0 .0/ D v: either it is minimizing up to infinity, which means that for each T > 0 the path c W Œ0; T ! S is a shortest path between c.0/ D p and c.T/, or else there exists a positive value beyond which c is no longer minimizing. This value is denoted cut.c/ (it can also be denoted cut.v/ since v and c determine one another) and is called the cut value.

Fig. VI.3.7. The two possibilities for a cut point: two shortest paths or/and conjugate points

What makes the subject difficult is that at cut.c/ two situations can arise, possibly simultaneously: either a second geodesic d arrives at q D c.cut.v//,



minimizing and distinct from c, or else the point q is conjugate to p on c; by definition, this means that the geodesics c˛ in a neighborhood of c D c0 and emanating from the chosen point p envelop a curve tangent to c at q. In other words, the curve ˛ 7! c˛ .cut.v// has velocity zero at ˛ D 0. In general the figure may show a cusp, and things may even get bad in the C1 case and even in the case of the sphere where all the geodesics pass through the same antipodal point. We also need to know that if c.t / is conjugate to p on c, then c isn’t minimizing after t , it ceases to be minimizing before t or no later than at t. Whence our interest in knowing the envelope of the geodesics that emanate from p. Finally we call cut locus of a point p and denote by cutloc.p/ the set of points c.cut.c//, where c runs through all geodesics emanating from p. For standard spheres, the cut locus of each point reduces to a single antipodal point; the situation is thus very special: there is always a unique shortest path except for antipodal points, at which we may depart in any direction and always arrive at the antipode, with same distance for all starting directions. The structure of the cut locus was studied by Poincaré in the real analytic case, denoted C! ; see Sect. V.XYZ for these notations. This study was completed only in Myers (1935a, 1935b) and Myers (1936). In the C! case the cut locus of a point is always a graph whose free extremities correspond to conjugate points; at each terminal point of this graph there occur exactly two minimizing geodesics and, at a vertex of the graph of order k, there occur exactly k.

Fig. VI.3.8. At left: cut locus with a triple point; At right: the cut locus of a point of the exterior meridian of a torus of revolution; the number of minimizing geodesics reaching a point of the cut locus is indicated

In cases that are just C1 the cut locus can be pathological, even fractal. Its structure has recently been clarified in Hebda (1994). In contrast, it is easy to see that the cut locus is always the closure of the set of points reached by at least two distinct minimizing geodesics.



What about the cut locus for our favorite surfaces? One motivation is the following: even though the geodesics contain the segments (the minimizing curves), and even if we know their equation, we are told nothing about the cut locus values. Now it is important to know the cut loci if we want to completely understand the intrinsic metric structure of a surface, in particular for the ellipsoid of revolution of our planet. It is in fact disconcerting to have two shortest paths for traveling from one point to another: which should we choose? It is even a philosophical problem: Aristotle, Buridan and Spinoza were interested, from this point of view, in the difference between humans and animals. It is astonishing to confirm that the only two cases, to our knowledge, where the cut locus of all points is known are very particular surfaces of revolution (e.g. the sphere and the torus; see Fig. VI.3.8) and the Zoll surfaces that we will encounter in Chap. XII, because they have the property that their geodesics are all periodic, although they are not round spheres. For the sphere, the cut locus of each point is reduced to its antipodal point. For certain Zoll surfaces (see Sect. XII.3.B), there is a description in Besse (1978) of the structure of the cut locus, also a proof although rather dense. Except for the two cases mentioned, the cut locus is completely unknown for all other surfaces. We can’t silently pass over the antipodal property of the sphere. It’s an old problem of Blaschke from 1927, when he conjectured that only the sphere has this property. This was proved only in 1963, by Leon Green. We find his proof, among others and the study of what happens in higher dimensions, in Besse (1978), which is entirely devoted to this problem and its generalizations. The proof requires at least two rungs up the ladder: the geometry of phase space, specifically the unitary fiber, i.e. the set formed of all the unit vectors tangent to the surface in question. The fiber of the geodesic flow of the surface considered, a manifold of dimension 3 D 2 C 1, will be the object of Chap. XII. There we will need to apply Liouville’s theorem to it; see Sect. XII.6.A. Returning to ellipsoids: for ellipsoids of revolution, the cut locus of a pole consists just of its antipode; if there are three axes, the cut locus of an umbilic is reduced to the antipodal umbilic. And for the other points? It is in fact asserted in Braunmühl (1882) that the cut locus of ellipsoids, at each point except at umbilics, has the topology of an interval. This doesn’t tell us where it is, but at least we would know that there aren’t any triple points or worse. Unfortunately, Braunmühl’s text relies on an assertion of Jacobi relating to ellipsoids: the envelope of the geodesics emanating from a non umbilical point possesses exactly four cusps (see the reference to Arnold just below). If that is true, then we have essentially intervals, for the results quoted earlier show that two of the cusps belong to the cut locus; the two others, which correspond to local maxima for the distance along the geodesic to



the point of contact of the envelope, don’t intervene. There thus remains an interval whose two extremities are the two cusps, that are themselves local minima. For the claim of Jacobi from 1864, already mentioned at the end of Sect. V.10, and for its modern context, see Arnold (1994, §13, p. 39).

Fig. VI.3.9. An old sketch of Braunmühl (1882). A conjecture due in part to Jacobi. c Springer Braunmühl (1882)

Fig. VI.3.10. A figure which seems evident, but remains conjectural. Braunmühl c Springer (1882)

But it isn’t known how to prove Jacobi’s conjecture even for ellipses of revolution, even for those that are close to being a sphere, and again even for the points of their equator. The fact of knowing the equations of the geodesics isn’t sufficient in the present state of our knowledge. To realize the difficulty of the problem, which is in fact a problem of algebraic geometry, think of an ellipsoid of revolution that is very elongated vertically, and fix a point on the equator; see Fig. VI.3.11. As the curvature (see below) along the equator is close to zero, the geodesics emanating from this point and near the equator will have an envelope that turns a large number of times, which spirals about the equator. It is surely reasonable to conjecture in this case precisely that the cut locus is the part of the meridian composed of points that are between the two conjugate points which are above. More generally, it is conjectured that the cut locus is always contained in a line of curvature, and this is supported by computer studies. See Itoh and Kiyohara (2004) and Sinclair (2002).



Fig. VI.3.11. It is difficult to know what happens for geodesics that turn very slowly

We can also realize the difficulty as follows. Take a sphere and deform it only very little. At the outset the geodesics emanating from a given point all pass through the antipode; topologically, we need to know what happens when we deform the family of lines passing through a point of a plane. We find in Arnold (1994) the following result: the families of nearby lines have an envelope with at least four cusps. But this doesn’t say that there are only four; in fact it is easy to construct families of lines possessing an arbitrary number > 4 of cusps.

Fig. VI.3.12. Cayley’s astroid, envelope of the normals to a triaxial ellipsoid of R3 . An octant has been cut out and translated for a better comprehension of the surface. c A. K. Peters Ltd. Joets, Ribotta (1999)

Although not directly related to our problem, below is a picture of Cayley’s astroid (the envelope of the normals to an ellipsoid), a surface studied by Cayley in the middle of the nineteenth century, taken from Joets and Ribotta (1999), where it is studied for the case of dimension four; see also Banchoff (1984). On this topic, we don’t know any reference that treats the problem of intrinsic geometry of ellipsoids of dimension greater than two. For example: what are the properties that generalize what was encountered above with the umbilics in dimension 2? We will return to envelopes of normals in Sect. VI.10.



VI.4. An indispensable abstract concept: Riemannian surfaces The following section studies the connection between abstract surfaces and surfaces of R3 . It is scarcely possible to understand it well without the fundamental idea introduced by Riemann in 1852 of Riemannian surface (in fact, Riemann placed his definition in arbitrary dimension). The starting remark is that, to define the intrinsic metric of a surface S E3 , we use the Euclidean norm of E3 only for the tangent vectors to S. Whence the idea of Riemann: to define a geometry on an abstract manifold M of dimension two (differential surface, see Sect. V.XYZ ) by placing the differential .dx1 ; :::; dxn / of an arbitrary Euclidean structure P at each point i j g dx dx , requiring only that this structure depends continuously, or differi;j i;j entiably, on the point describing the surface. Such an object is called a Riemannian surface; if we want to specify the Riemannian metric, we denote this by .M; g/. We will encounter constructions of such abstract surfaces almost never realizable in E3 for the objects called “space forms” that we will see in Sect. VI.XYZ.

Fig. VI.4.1. An abstract ds 2 on the sphere S2 ; a field of ellipses in the plane

The above sketches are given in E3 , but this is only to remedy our difficulty in seeing “abstractly”. We need to exert ourselves to think, to “see”, abstractly. In the case of a surface that is “already” in E3 , this “abstract” structure is simply on each tangent space for this surface the Euclidean structure induced by the embedding of the tangent plane in the space. In the abstract case, the embedded case occurs by taking for the definition of length of a curve c W Œa; b ! M the Rb quantity length.c/ D a kc 0 .t /k dt . In a chart fx; yg of M (see Sect. V.XYZ) we define the Riemannian metric by g D a dx 2 C 2b dxdy C c dy 2 , this expression signifying that for a tangent vector v of associated coordinates fu; vg in this chart (see Sect. V.XYZ) we have kvk2 D au2 C2buv Ccv 2 . Many books use the notation ds 2 D g to indicate the fact that g provides the infinitesimal arc lengths of curves (see Sect. V.6). An essential historical example of an abstract Riemannian surface is the hyperbolic plane (see Sects. II.6 and II.XYZ). It is defined on the open disk x 2 C y 2 < 1 by: 4

ds 2 D


.x 2

2 2 2 .dx C dy / :

C y2/



And it is thus that in 1854 Riemann defined hyperbolic geometry for the first time with absolute correctness and, moreover, in all dimensions, where the formula is essentially the same. The previous partial results used Beltrami’s surface, which has the major disadvantage of being doubly incomplete: on the one hand we get only part of the hyperbolic plane; on the other it is necessary to truncate the surface or to cover it while turning about, which is an abstract representation. We will see right away in Sect. VI.5 that a theorem of Hilbert shows that we can never realize the whole hyperbolic plane as a surface of E3 .

Fig. VI.4.2. The Beltrami surface, a doubly incomplete representation of the hyperbolic plane. There is a singularity at the boundary that can’t be extended; see also Fig. VI.4.4, but above all the general result of Hilbert on p. 413

In this abstract setting the shortest path problem is distinctly more difficult. We proceed as above by the calculus of variations and thus define the notion of geodesic, but here we find only a differential equation whose solutions we can’t interpret as curves with acceleration normal to the surface. It wasn’t until the turn of the XXth Century that matters were first well understood, by Ricci and Levi-Civita; see for example Berger (2001) or Berger (2003). But if we want to be global we have to wait for the result of Hopf-Rinow, equally valid in the abstract setting of Riemannian surfaces. Thus finally we have geodesics: we have a proper metric space structure and the geodesics extend indefinitely if this metric is complete. Note what is not well explained in most works: the fact that the axiom d.p; q/ D 0 ) p D q requires, in the definition of an abstract surface, that the topology is separated. We could talk about cut locus again, but we won’t have need of it. An isometry between two Riemannian surfaces .M; g/ and .N; h/ is a diffeomorphism f W M ! N that preserves their metrics, which is equivalent to saying that the differential df W TM ! TN preserves the corresponding norms of the tangent vectors. The Gaussian curvature K. There’s hardly any mathematics without invariants! We’re indebted to Gauss for having found the fundamental invariant of Riemannian surfaces (in Sect. VI.6 we will return to the history and its context). We call this invariant Gaussian curvature or simply the curvature and denote it by K. We will



also say occasionally total curvature, but that can lead to confusion with the integral formulas such as (VI.6.3) and (VI.7.1). This function K W M ! R hasn’t a simple direct expression for abstract surfaces; if it is calculated in a chart in terms of the gij , we find a really complex expression involving the first and second derivatives of the gij . In contrast, its geometric interpretation is simple: the curvature measures the infinitesimal rate of divergence (or convergence) of the geodesics. That will become very clear in Sect. VI.6, but we need to see it here in an abstract way. Precisely, here is an equation due to Gauss: let c˛ .t / be a one-parameter family ˛ of geodesics c˛ defined in the neighborhood of an initial geodesic c0 such that the curves ˛ 7! c˛ .t / are orthogonal to c˛ . The field of vectors tangent along the extent of c0 is the field @.c˛ .t// ˛ .t// . Then if we put Y.t / D k @.c@˛ k, this norm satisfies the second order @˛ 00 differential equation Y .t / C K.c.t // Y.t / D 0.

Fig. VI.4.3. Finite diagram of a one-parameter family of geodesics

A particular case consists of taking the one-parameter family ˛ of geodesics c˛ emanating from a given point m 2 M, where ˛ traverses the unit circle of Tm M. For each t , the curve C.mI t / W ˛ 7! c˛ .t / can be called the circle of center m and ˛ .t// of radius t . Then the equation of Gauss yields for the norm k˛ k the partial development in t :

d.c .t // t3


D t K.m/ C o.t 3 /: d˛ 3 First, we observe that this development does not contain a t 2 term (a Riemannian surface is Euclidean up to order two); next it does not depend on ˛ but only on m. Thus the total length of circles of small radius equals: length.C.mI t // D 3 2 t t3 K.m/ C o.t 3 /. We can say that the metrics with K < 0 are super-Euclidean and those with K > 0 sub-Euclidean. Whatever the importance of the curvature, it is not in general a characteristic invariant: a diffeomorphism f W M ! N can preserve the curvatures KM , KN without



being an isometry (see the definition at the beginning of the next section); it suffices to take any two small open subsets of the surfaces M and N and take for f any mapping that preserves the level curves of K. Few works mention this important point; see however Spivak (1970, pp. 310, 311 of Vol. II). Still less do they treat the problem of knowing whether it is possible to find characteristic invariants for a Riemann surface. This problem was attacked very early by Darboux (1887, 1889, 1894, 1896) (or Darboux (1972)). We extract what follows from Chap. XIII of Cartan (1946–1951): in order that two surfaces for which the functions K and kd Kk2 are independent be isometric by a mapping f , it suffices that the mapping preserve the four functions K, kd Kk2 , hK; kd Kk2 i, k.kd Kk2 /k2 . If there is a relation of the type F.K; kd Kk2 / D 0, then it suffices that f preserve kd Kk2 and the Laplacian K. Moreover, the surfaces of this type are diffeomorphic to surfaces of revolution. The notation df for a function on a manifold designates its differential, which means the linear form on the tangent spaces which associates df .x/ D x.f / with each tangent vector x, i.e. it is the derivative of f for x. For the geometer this is the ordinary derivative d.f .c.t/// .0/, where c is any curve whose velocity at the origin is dt @f @f the vector x. In coordinates, df D @x ; . For a function f on a Riemann sur1 @x2 face, by duality the scalar product allows transformation of the differential df into a vector (in fact a field of vectors) called the gradient/ of f and denoted grad.f /. The gradient is thus defined by the relation: df .x/ D hgrad.f /; xi for every vector x. We thus set kdf k D k grad.f /k. Finally, the Laplacian f of a function on a (Riemannian) surface is the operator that generalizes the Laplacian in the plane defined in Sect. IX.4.

Fig. VI.4.4. The different types of meridian defining a surface of revolution of constant curvature; only one the six types yields a true surface (i.e. without singularities). [BG] c Presses Universitaires de France Berger, Gostiaux (1987)

If an .M; g/ has constant curvature K, say K D k, then its metric is in contrast to the variable curvature case completely known and completely



determined by k, at least locally. This can be seen from the Gauss differential equation Y00 .t / C K.c.t // Y.t / D 0: if this constant k is zero, the metric is everywhere locally Euclidean, and we can thus say once again that the curvature measures the deficiency of being Euclidean. If the constant k is positive, then the metric is everywhere locally isometric to that of the sphere of radius p1 . If k < 0 then the k metric is everywhere locally isometric to a geometry of the hyperbolic plane (see Sects. II.5 and II.XYZ) appropriately normalized, the case K D 1 being just that of the hyperbolic geometry exhibited in Sects. II.5 and II.XYZ. Readers must then ask themselves: what about the typical global case if M is compact? It’s a fundamental question of geometry called the space form problem which is not on our agenda; see ideas and references in Sect. VI.XYZ. It is in fact a particular case of the problem that follows. Before continuing, we mention that the determination of the surfaces of revolution of constant curvature is easy; see Sect. 3.9 of Klingenberg (1973) or Exercise 7 of Chap. 3 of do Carmo (1976). In fact, the burning question is now this: We possess on the one hand the surfaces of E3 which are automatically Riemannian, and the abstract Riemannian surfaces on the other. What is the relationship between these two categories of objects? VI.5. Problems of isometries: abstract surfaces versus surfaces of E3 We now consider two surfaces S and S0 , abstract or in E3 : an isometry between S and S0 is a bijective mapping f W S ! S0 that preserves their intrinsic metrics, that is, dS .p; q/ D dS0 .f .p/; f .q// for all p and q in S. For a long time it was not realized (or was considered obvious) that this condition implies the differentiability of the mapping f , which is at least of class C1 ; see Calabi and Hartman (1970) for a very precise position on these problems, with their pitfalls. So it is good to note that the same definition of the intrinsic metric permits us to give the equivalent definition: an isometry f W S ! S0 is a bijective mapping, of class C1 at least, that preserves the norms of tangent vectors. For a surface S, we let Tm S denote the vector space (a plane) of vectors tangent to S at the point m 2 S. Then the condition for isometry is kf 0 .v/k D kvk for each vector v tangent to S. To see this more clearly, we note the evidence: when we are in E3 each tangent plane to a surface inherits a planar Euclidean structure, i.e. a positive definite quadratic form. We need thus to see that the intrinsic metric of a surface S is constructed starting with the set of quadratic forms on the set of Tm S; we have seen this at the beginning of Sect. VI.4. It’s the set of these quadratic forms that are called the first fundamental form of the surface. An isometry is a differentiable mapping that preserves the first fundamental form. As general references on the problems of this section, consult: Gromov (1986), Vol. V of Spivak (1970), Burago and Zalgaller (1992), and various parts of Burago and Zalgaller (1992), which attaches great importance typical of the Russian school for this area to low order differentiability.



The important question of whether the abstract surfaces are more general than those of E3 is not yet settled. There is the local problem and the global problem. In the local case we only want to know if, given a point m of the Riemannian surface .M; g/, we can find a neighborhood U (perhaps very small) of m and an isometry between .U; gU / (the restriction of g to U) and a surface of E3 , i.e. a mapping f W U ! E3 such that f .U/ is a surface of E3 and f an isometry between .U; gU / and f .U/ endowed with the induced metric. In the global case we want an isometry between all M and f .M/. We say, in brief, that .M; g/ is realizable in E3 locally (resp. globally). The question has not been settled locally or globally. Locally we know this: if the metric g of M is real analytic (see Sect. V.XYZ) then it is locally realizable: for each point and for an appropriate neighborhood we have a local isometry (Cartan-Janet, 1926). Beyond this, without analyticity, we have a local realization at each point where K is nonzero; the solution uses standard existence theorems for partial differential equations which are obviated once K vanishes at a point. Subsequently more and more partial refinements have been found, e.g. for the case where the differential d K isn’t zero. In 1971 Pogorelov found an example of a Riemannian surface .U; g/ pointed at m such that no neighborhood of m, however small, can be realized in E3 ; this metric is evidently very strange, e.g. the points where the curvature is zero form a dense subset. Above all it is only of class C2 , and the problem is today totally open whether, starting with C3 , or even with C1 , a local embedding is always possible. So we know today that, even locally, abstract Riemannian surfaces are more general, but a complete result is still awaited. In particular, the regularity of g plays a role that is not well understood, except for the class C1 ; see below. The metric of class C2 of Pogorelov has a curvature K that is Lipschitz (see Sect. V.XYZ). If we want to realize every surface, be it locally or globally, we must use an embedding in EN with at least N > 5; see the references cited. Globally we certainly cannot hope to realize every compact abstract Riemannian surface .M; g/ in E3 , for the simple reason that there exist compact abstract surfaces with negative curvature and that each compact surface of E3 has points at which the curvature is strictly positive. To see this, take any point not on the surface and a sphere about this point that contains the surface but has smallest possible radius. It exists by compactness and is necessarily tangent to the surface at some point; at such a point of contact the curvature of the surface is greater than that of the sphere, thus positive. But in return there exists a very satisfying although very difficult result that bears the names of Weyl, Nirenberg, Pogorelov, Alexandrov: let .S2 ; g/ be a structure of an abstract Riemannian surface on the topological sphere S2 that is



C1 (resp. analytic) and such that the curvature K is everywhere strictly positive. Then it is isometrically embeddable by a C1 (resp. analytic) mapping into E3 . The proof cannot be done “easily”; we will in fact see in Sect. VI.9 that the surface obtained in E3 is unique (within a global isometry of E3 ). We only mention that Pogorelov’s proof from 1969 uses the space of all Riemannian metrics on S2 by showing that, in this space, the subset of metrics with everywhere positive curvature is connected. But we always need very subtle results on partial differential equations; see Sect. VIII.6. The case K > 0 is broached in Guan and Li (1994), but only an embedding of class C1;1 is provided. The proof is achieved by studying the limit of embeddings associated earlier with metrics with strictly positive curvature that approach the given metric. See also the end of Sect. VIII.6 for a schematic key for this type of proof. It wasn’t until Gromov in 1986 and subsequent elaboration by Labourie in 1989 that the existence and uniqueness could be put in the same theoretical setting that goes to the bottom of things; see Sect. VI.7 for the setting and for references. We mention in passing, although it is a bit off subject, a little known result of Gluck (1972) that states, for the surfaces of E3 having the topology of the sphere S2 , that there is no four vertex theorem (cf. Sect. V.8). Gluck proved that, for every function on the abstract sphere S2 , we can always find a sphere S2 embedded in E3 for which the Gaussian curvature is this function. In particular it can very well have but a single maximum and a single minimum. Instead of taking the curvature as a point function, we can take it as a function of the direction of the tangent plane; this is Minkowski’s problem. The direction of the tangent plane is usually given as points of the unit sphere corresponding to unit vectors orthogonal to these tangent planes, and thus a function is specified on the sphere. This problem is today completely understood; see the brief mention in Sect. VI.7. Which embeddings can we expect for abstract Riemannian surfaces, not necessarily compact? A typical case is that of the hyperbolic plane Hyp2 D .R2 ; g D hyp/, a geometry encountered in Sect. II.4. Can we realize this geometry in E3 ? A result of Hilbert in 1901 states that there isn’t any global realization of Hyp2 in E3 ; locally the realization is very easy with pieces of surfaces of revolution (those of Beltrami or others), but this cannot be done globally. The proof consists of following a geodesic and showing that a singularity necessarily develops in the embedding at the end of a certain distance. Hilbert’s result was refined by Efimov in 1964: there is always impossibility for realizing a complete abstract surface .R2 ; g/ as soon as its curvature is bounded above by a negative constant: K 6 constant < 0. The proof is very complicated it uses dimension two so as to make use of functions of a



complex variable; see Sect. VI.XYZ. Moreover, for low order differentiability enthusiasts even though it is not in the spirit of this book the result is even valid in the C2 setting, but then we encounter the difficulty connected with the theorema egregium; see the next section. It is already difficult to show that the area is infinite; but it is evidently of class C3 in Gauss’s equation Y00 .t / C K.c.t // Y.t / D 0. See the excellent introduction in Klotz-Milnor (1972), in all of which readers will find interest. The case C1 . All the preceding extends to embeddings of class at least C2 , if only for defining the curvature. But if we are content with embeddings only of class C1 , then the so-called Nash-Kuiper theorem (see Bleecker, 1997) allows an isometric embedding in E3 of every abstract Riemannian surface, and with lots of choice; the choice is in fact enormous. Once a compact Riemannian surface is embedded in E3 without additional conditions, in a contraction manner, i.e. so that the embedding considered reduces distances, then there exist isometric embeddings of class C1 which are arbitrarily close in the class C0 to the given embedding. For example, there exist lots of surfaces of E3 which aren’t round spheres, but for which the intrinsic metric is nonetheless isometric to that of a round sphere (this in strong contrast with the uniqueness provided by a theorem of Cohn-Vossen that we will study in Sect. VI.9, where the smoothness condition requires that the embedding be C2 ). To see this, take a sphere with radius just a bit smaller; this trivially yields a contraction embedding. The proof of the Nash-Kuiper result Nash, John is hard: it’s a mixture of the theory of underdetermined partial differential equations and geometry. Roughly speaking, the desired metrics are constructed by very intense foldings and crumplings around a contraction embedding. The totally novel theory of Nash from 1956 is in fact very well understood thanks to Gromov; see the h-principle in his book Gromov (1986). VI.6. Local shape of surfaces: the second fundamental form, total curvature and mean curvature, their geometric interpretation, the theorema egregium, the manufacture of precise balls It is finally time to occupy ourselves with the surfaces in E3 , no longer just with their intrinsic metric but also with the way in which they are situated in space. We will say the minimum needed for grasping certain important results and open problems. This situation is described perfectly by the second fundamental form and in particular by the two invariants called total curvature and mean curvature. In brief, we look at how the surface differs from a plane, more precisely how it differs from its tangent plane. For this it suffices to know the shape of the plane curves that are the sections of the surface by planes containing the normal to the surface. Now for us, after Sect. V.6, it is the curvature of these curves that we need study. The result is quite simple and quite old: (VI.6.1) When the plane turns about the normal at a given point m the algebraic curvature of the section curve is a quadratic form.




Fig. VI.6.1. Section by a normal plane

From the point of view of calculus, II involves the first two derivatives of the equation of the surface, for example as the local graph of a function R2 ! R. The first fundamental form by itself requires only first derivatives. In an orthonormal system having as origin a point m of the surface S, where the first two vectors are in the tangent plane at m and where the third is directed toward the normal, in the neighborhood of m the surface has an equation of the form z D f .x; y/ D

1 2 1 ax C bxy C cy 2 C terms of higher order: 2 2

Readers should know that the curvature of the plane curve z D 12 ax 2 is equal to a. For the same reason, the curvature of the section of S by the plane containing the normal and the vector .cos ; sin / equals k. / D a cos2 C 2b cos sin C c sin2 : In other words, it is the value taken by the quadratic form ax 2 C 2bxy C cy 2 for the vector .cos ; sin /. This quadratic form can be defined in an intrinsic fashion as a quadratic form on Tm S, and is called the second fundamental quadratic form, and denoted II. The first fundamental quadratic form, denoted I, is simply the metric on Tm S (x 2 C y 2 ) in the coordinates used above. We note that the sign of the second fundamental form depends on a choice of orientation on S (or on a preferred direction for the exterior normal). As mentioned above, from the point of view of calculus the second fundamental form involves the first and second derivatives of the equation of the surface (for example, in the expression for S above, a, b, c are the second derivatives of f at the origin). The first fundamental form, on the other hand, requires only first derivatives and allows us to pass from the initial coordinate system to a system connected to the tangent plane and the normal.



Now that on the tangent space Tm S we dispose over the two quadratic forms II (the second fundamental form) and I (the Euclidean structure), we can diagonalize II with respect to I, i.e. find an orthonormal system of Tm S where II takes the form k1 x 2 C k2 y 2 . The eigenvalues k1 and k2 are called principal curvatures, the eigendirections the principal directions (they are of course orthogonal). It should be noted that k1 and k2 are the minimum and maximum values of the curvature k. / of the sections of S by the plane turning about the normal. The trace and the determinant of II with respect to I are two essential invariants: (VI.6.2) K.ext / D k1 k2 is called the total curvature despite the danger of confusing it with the integrals of this curvature; H D 12 .k1 C k2 / is called the mean curvature. The notation K.ext / reminds us that the definition uses information exterior to S, as opposed to K, which ultimately uses distances calculated within S. Notice that K.ext / does not depend on the orientation of the normal since it is a product, whereas the sign of H changes with orientation. The points where K.ext / > 0 are called elliptic, those where K.ext / < 0 hyperbolic, those where K.ext / D 0 are called parabolic and those where k1 D k2 D 0 planar; moreover those where k1 D k2 are called umbilics. The principal directions are only well determined away from umbilics, where the principal curvatures are distinct (i.e. where II isn’t a multiple of I). The umbilics are in general isolated points; apart from these points, we thus dispose over two direction fields of tangent lines, those in the direction of maximum curvature and those in the direction of minimum curvature. These direction fields are integrated into two families of curves of S, called lines of curvature, which we will find many times in the sequel. It may seem surprising that the condition k1 D k2 in general defines isolated points, and not a curve on S. The explanation is that in the three dimensional space 2 2 of quadratic forms ax C 2bxy C cy , or in that of the real symmetric matrices a real b , the condition of equality of the two eigenvalues in fact corresponds to two b c independent conditions: a D c, b D 0. This is a remark of which Arnold is fond; see Arnold (1974). The exemplary case where the second fundamental form can be found without any calculation is that of surfaces of revolution. In fact, each normal to such a surface intersects the axis of revolution and, by symmetry, we see that the two principal directions are those of the meridians and parallels, which are in fact the lines of curvature. We obtain immediately the principal curvatures and their reciprocals, the radii of principal curvature, the one being the length of the normal between the point and the axis, the other the radius of curvature of the meridian at that point. Here is a simple point, but of great practical importance, which is in the spirit of industrial fabrication (see Sects. II.2, V.7 and VII.13.C) and for which the proof is left to readers: if all points of a surface are umbilics, then the surface is necessarily a portion of a sphere, a local result; as needed, see any book on differential



Fig. VI.6.2. Finding the radii of principal curvature is trivial for a surface of revolution

geometry, e.g. p. 147 of do Carmo (1976). It follows from this that if we rub two surface portions together (with the classic abrasive emery powder, or any other abrasive, between the two) in all directions, they will finally slide against one another when both are spherical. In fact, we can say for example that they are surfaces of revolution in more than one way, in at least two different directions for the axes of revolution, which can happen only for spheres; or again that all points are umbilics since in the rubbing process we turn in every direction about each point. Group theory enthusiasts can see here also the fact that the only groups of isometries of the Euclidean space R3 having at least two parameters, all finite, are those formed by rotations about a fixed point. The reader should know that there doesn’t exist at present any other way of fabricating spherical surfaces, industrially or even by high precision mechanics, metrology, etc, for example for the lenses of ordinary glasses. The case of astigmatic lenses and so-called progressive lenses are a totally different story. For astigmatic lenses, glass tori of revolution are used. As for progressive lenses, only imperfect approximations to a good optic on a very high field depth can be obtained. Progressive lens technology is improving daily and the story is not yet complete. In high precision work, the surfaces obtained must be studied by interferometry and corrected by appropriate manual abrasion; this is also done for parabolic mirrors for telescopes. The artisans who do this work have “micrometers in their hands”. Nevertheless, computer aided design (CAD) and robots have recently begun to be used for managing the corrections in high precision work, e.g. for mirrors of large telescopes. The manufacture of spherical balls is still a completely different story, simpler in a certain sense. An old method consisted of placing approximately round balls in a receptacle and intermingling them sufficiently long, using abrasives. We will see in Sect. VII.13.C why this intermingling yields true balls (approximate in practice, perfect in theory). But since 1907 a patented process of the SKF corporation has allowed quick and accurate manufacture of balls, with precision within a micron, using the above property with the following device. It begins with two plates in which grooves are cut in the form of a half tori, the difference in diameters corresponding to the diameter required for the balls. These two plates are placed against one another and turned. A suitable abrasive and some approximate balls are injected into each of the toroidal tunnels; at each turn the balls are ejected and reintroduced



c Nathan Édition Fig. VI.6.3. Claude, Devanture (1988)

in random fashion into another “tunnel”. After three machinings in a row of this type with appropriate abrasives we extremely quickly obtain balls that are almost perfect. Professionals themselves are ever astonished by the simplicity of the process and the exceptionally good result. If we want still higher precision, of the order of a micron or better, we sort the balls thus obtained according to their diameter. We will encounter this in Sect. VII.13.C. For small balls, see in Sect. VI.8 below for the mathematical formulation and more details. We are now going to present some classical topics that are given in detail in the various references that have already been mentioned. First, the local form of a surface is completely known when K.ext / 6D 0. If K.ext /.m/ > 0, then in a suitable



neighborhood of the point m the surface is the boundary of a strictly convex subset of E3 (see Chap. VII for notions of convexity). If K.ext /.m/ < 0, always in a suitable neighborhood, the situation is diffeomorphic to that of a hyperbolic paraboloid and its tangent plane. That is to say, matters are as in the figure below: the tangent plane at m crosses the surface when two (differentiable) curves cross at m, two portions of the tangent plane determined by these curves are strictly above the tangent plane and two are strictly below.

Fig. VI.6.4.

On the other hand, once K.ext / is zero at m, even in the domain C1 , we can’t say more in general about the surface; the sketches clearly show that practically everything is possible, in particular crossing its tangent plane infinitely many times in each neighborhood of the point considered. In order to be able to state things, we need to restrict ourselves to the generic case, i.e. to shapes that are stable under small deformations, which must occur in a precise setting. We encountered this notion of genericity in the case of plane curves in Sect. V.10; we will also encounter it for convex sets in Sect. VII.13.D, and for billiards in Sect. XI.10.C. For the genericity of surface shapes, see Porteous (1994). The theorema egregium of Gauss is truly one of the most remarkable theorems of mathematics: (VI.6.3) The Gaussian curvature K of the intrinsic metric of a surface of E3 coincides with the total curvature K.ext / D k1 k2 of its second fundamental form. We are going to comment on this result at some length. A fundamental documentation can be found in the historical book (Dombrowski, 1979), all the more precious in that it is not known how Gauss discovered this result, which was absolutely not “in the air”, in contrast to the majority of great mathematical results that come at the end of a succession of conjectures, etc. It would have been a long time before someone else discovered it; readers will find a hypothesis on this topic below.



Fig. VI.6.5. The surface of revolution generated by the curve below on the right is pathological

The preceding may make readers think that the proof of the theorema egregium is never easy. No conceptual proof of it is known, making us stay at the bottom of the ladder. The simplest proof consists of calculating ds 2 in coordinates. We thus take locally the expression .u; v/ 7! x D x.u; v/; y D y.u; v/; z D z.u; v/ for our surface S. If a curve of the chart is given by t 7! .u.t /; v.t //, then in S it will give the curve c W t 7! .x.t /; y.y/; z.t //, and the norm of the velocity vector will be kc 0 .t /k D .x 02 .t / C y 02 .t / C z 02 .t //1=2 . In th chart we will have x 0 D xu u0 C xv v 0 , y 0 D yu u0 Cyv v 0 , z 0 D zu u0 Czv v 0 , where ab denotes the partial derivative @a . Thus @b 0 2 02 0 0 02 we can write kc k D Eu C 2Fu v C Gv , where E, F, G are functions of .u; v/. Today the standard expression for such a ds 2 is ds 2 D Edu2 C 2Fdudv C Gdv 2 . The calculation is greatly simplified if we take for S an equation of the form z D f .x; y/; the point m being the origin, the x; y plane being the tangent plane and the z axis the normal at m. We thus obtain for ds 2 an expansion restricted to the neighborhood of m D .0; 0/: ds 2 D dx 2 C dy 2 C a.x; y/dx 2 C 2b.x; y/dxdy C c.x; y/dy 2 C o.x 2 C y 2 /; where a, b, c are hom*ogeneous polynomials of degree 2. From this partial expansion we calculate the second fundamental form and we find without difficulty the value 00 00 00 K.m/ D 12 .ayy 2bxy C cxx /. This proves the theorema egregium. This proof did not suffice for Gauss, who wanted to be so sure of his theorem that he launched into the complete calculation of K in an arbitrary chart and for a



general expression ds 2 D g D Edu2 C 2Fdudv C Gdv 2 , an impressive calculation that can be found in extenso in Dombrowski (1979). In Sect. II.2.A of Berger (2001) is found a more intrinsic proof in the spirit of Riemann; it consists of writing the equation of geodesics and of ds 2 in so-called normal coordinates, i.e. where the geodesics emanating from the origin are the lines of the chart in question. For a surface S E3 there exists a natural mapping fS W S ! S2 with values in the unit sphere S2 E3 . To define it, we first remark that at each point of the surface we have two opposite normal unit vectors; to choose between them, it suffices to choose a side of the space that the surface cuts locally in two, or else to have an orientation of the surface in the space. The first way is called a co-orientation ; it is used in an essential way for the curves of Arnold’s theory seen in Sect. V.10. We suppose from now on that we have made such a choice (propagated by continuity). The mapping consists then of associating with each point m of S its chosen vector exterior normal. Practically all the books call it the Gauss mapping ; in fact, it was used a little before by Rodrigues. Its essential property is this: we observe the effect of fS on the areas of S and of S2 . We see intuitively, in observing things infinitesimally along the principal directions, that fS multiplies the lengths in those directions by k1 and k2 respectively, thus the areas are multiplied infinitesimally at m by k1 k2 D K.m/. It will be noted that fS does or does not change orientation accordingly as K is positive or negative.

c Presses Fig. VI.6.6. the Gauss mapping. [BG] Berger, Gostiaux (1987) Universitaires de France



A consequence is: if D is a domain of S then: Z .VI:6:3/; Area.fS .D// D K.m/ d m D

where the element of integration (denoted d m) on the surface is the most natural; readers can assume the task of seeing how a measure on a surface of E3 is defined canonically. If S is a strictly convex surface then the mapping fS is bijective and R thus S K.m/ d m D 4, this formula having been obtained without our having to calculate the curvature K of the surface; and so it was proved by Rodrigues without any calculation that for each ellipsoid we have 4 as the complete integral of the curvature. If we extend the notion of degree, introduced in connection with curves for the mappings S1 ! S1 , to mappings between surfaces, the general formula will be: Z K.m/ d m D 4 degree.S ! S2 /: S

For this notion of degree, see e.g. [BG]. In Hilbert and Cohn-Vossen (1952) we find a calculation of angles for the polyhedrons that can give us some intuition about the theorema egregium, by considering that the deficiency of the angles at a vertex is a “discrete” analogue of the “continuous” curvature. We consider a vertex of the polyhedron where n faces meet and denote the angles of these faces by ˇi . These vertex angles completely determine the metric of the polyhedron.P We call the curvature of the polyhedron at the point considered the quantity 2 i ˇi ; what follows shows that it is a good definition. The Gauss mapping associated with this vertex degenerates into a polygon on the sphere S2 : the vertices of this polygon are the vectors normal to the faces, sides correspond to what happens when we revolve about the corresponding edges. The figure shows that the angles of this spherical polygon are the supplementary angles ˛i D ˇi . But the surface area of this spherical polygon P equals (decompose into P triangles and apply to each piece the formula III.1.2) i ˛i .n 2/ D 2 i ˇi , which is our desired curvature.

N4 β4

β1 α1





β2 N2




Fig. VI.6.7. The polyhedral proof suggested by Hilbert for establishing geometrically c Springer the theorema egregium, when n D 4. Hilbert, Cohn-Vossen (1996)



The proof of the theorema egregium, as much as the result itself, poses a problem for those who (justifiably) prefer low order differentiability: to define the total curvature of a surface S given locally as the graph of a function f , it suffices that f be of class C2 (we derive the normal to the surface), whereas for computing the Gaussian curvature it is necessary to differentiate the ds 2 two times. Now this ds 2 uses essentially the first derivative of f , whence finally the appearance of third derivatives of f . There is a mystery here, invisible somehow to less reflective people who work in C1 without moral concern. This strange phenomenon seems never to be mentioned in textbooks. The dilemma was attacked by Hermann Weyl starting in 1916, to attempt to know how to define curvature with only two derivatives. Since van Kampen (1938) we have a proof of the theorema egregium in the setting where f is only C2 . For more subtle situations, appropriate references can be found in Klotz-Milnor (1972), to which may be added Chern, Hartman, and Wintner (1954). A plane curve is determined within a plane isometry by its curvature as a function of arc length; see Sect. V.7. A surface in space is not determined by its Gaussian curvature, nor even by its two principal curvatures; we will see a celebrated counterexample in Sect. VI.8. On the other hand, it is determined within an isometry of the space by its two fundamental forms. The counterexample shows that it does not suffice to know the principal curvatures, the principal directions also being required; and this in fact suffices. The classic theorem isn’t difficult; we break trail by showing that there are relations between the two fundamental forms. The first is surely the theorema egregium, but there also exist the so-called Codazzi-Mainardi equations, which are a bit tricky to express: we defer to the various references on surfaces, according the taste of readers and their ability to access them. These equations are useful in diverse results; we will see some further on. For a recent approach to the theorem of determination by the two fundamental forms, see Ciarlet and Larsonneur (2000). VI.7. What is known about the total curvature (of Gauss) Surfaces with curvature zero. In Sect. VI.4 we saw that the Riemannian surfaces of (Gaussian or total) curvature zero are locally isometric to the Euclidean plane. But this says nothing about the way that they are embedded in space. For example a cone (with the apex removed, so a not to have a singularity) and an arbitrary cylinder have zero curvature. The local and global classification of (complete) surfaces with (Gaussian or total) curvature K equal to zero is well understood, but this understanding is not so very old. We begin with the local. It is a very old result that if k1 D k2 D 0 in a neighborhood of a point, then over this neighborhood the surface is a piece of a plane in space. If now K D 0 at a point, but k1 6D 0, then in a neighborhood of this point the lines of curvature (corresponding to the zero principal curvature) are inevitably seen as linear segments in space. But we must pay attention to the way in which the



examples of cones and cylinders can be deceptive: a ruled surface, i.e. a one parameter family of lines (or pieces of same, but we can always extend what will follow to complete lines) is not in general a surface of zero curvature. The lines which define a ruled surface are called its generators. The characteristic difference between ruled surfaces and cones and cylinders is that the plane tangent to the ruled surface turns when we move along a generator (see Fig. VI.1.2). To say that it doesn’t turn is tantamount to saying that the curvature is zero, or again that the surface is the envelope of a one parameter family of planes. It can be shown that this is again equivalent to these lines being the tangents to a curve in space. But this can’t be a global figure, since our surface isn’t differentiable at points of the so-called striction curve; see Fig. VI.7.1. In the case of a cylinder, this striction curve is entirely removed to infinity. Such surfaces are called developable, this because they are precisely mappable onto a plane with conservation of the metric.

Fig. VI.7.1. A developable surface and its striction curve. Porteous (1994) c Cambridge University Press

We can think of the surfaces with curvature zero as crumpled sheets of paper. It is an interesting exercise to see that we can fold a piece of paper not only along a line, but along any traced curve whatsoever. Mark off the curve and fold: this works, but don’t use too much space around the folding curve, and the curve must not be too complicated, for otherwise the sheet will intersect itself. More on this subject can be found in Porteous (1994) and in Fuchs and Tabachnikov (1995). Resuming our discussion, locally the surfaces of curvature zero are pieces of developable surfaces, but also locally we can produce pathological objects even of class C1 : take pieces of planes and join pieces of non planar developable surfaces to them. But what is the case globally when we look for surfaces of curvature zero that are complete, i.e. without boundary (see Sect. VI.2)? We can anticipate from the above figure that things don’t work globally. We can guess the answer: the fact that the surface is complete says that it contains entire lines. Let us look at these



Fig. VI.7.2. How to make a developable surface not containing any complete line

generators: they envelope a curve, but this curve is a singularity (see Fig. VI.7.2); the only reasonable case is that where all the points of contact of the generators with their envelope curve are at infinity, i.e. the generators are all parallel to each other. Our surface must finally be a cylinder, but a general cylinder and not necessarily a surface of revolution. As intuitive as the result may seem, it wasn’t until (Hartman and Nirenberg, 1959) (see also 5.8 of do Carmo, 1976) that we had a rigorous proof. This proof however, in spite of being fine and subtle, does not introduce any new concept. Surfaces with positive curvature. What is the general form of a complete surface S of E3 with curvature everywhere positive (resp. nonnegative)? We had a good answer in the case of plane curves in Sect. V.5, this answer requiring the subtle turning tangent theorem or Umlaufsatz. In the spatial case we find that things are simpler; the result goes back to Hadamard (1897). We consider a compact surface S, with K > 0 everywhere, and we want to show that it is the boundary of a convex domain of E3 . The sole idea is to apply the normal Gauss-Rodrigues mapping (defined in the preceding section): S ! S2 . The condition K > 0 implies that this mapping is everywhere locally a diffeomorphism; we thus see easily (since S is compact) that it is a covering of S2 (see as needed the end of Sect. V.XYZ or [BG] for the concept and what follows). But the sphere S2 is simply connected (see also Sect. III.5), which implies that the covering is in fact a bijection. Analogous to the case of curves, we show subsequently that this implies that the surface is entirely on one side of its tangent plane. It is interesting to mention here a difference between curves and surfaces. For (not necessarily simple) closed curves (i.e. immersions of the circle S1 in E2 ) the result was false: the curvature being positive throughout allowed self-intersections. This is not the case with surfaces: if the surface S is only the image of an immersion of the abstract sphere S2 and is of positive curvature, then in fact we inevitably have an embedding (and the surface is the boundary of a convex set). The proof was given by Hadamard. It can be anticipated that the result remains true under the weaker hypothesis K > 0. This says in effect that flat pieces are permitted. Such is the case, but the



Fig. VI.7.3. Why a surface with K > 0 is necessarily convex

proof had to wait for Chern and Lashof (1958). A new tool was needed: the general formulas of Chern and Lashof, which related the integral of the curvature to topological invariants via Morse theory, specifically the Betti numbers. Details and generalization of the result of Hadamard to the noncompact case can be found in Sect. 11.3.2 of [BG], (5,6-B) of do Carmo (1976) and Chap. I of Burago and Zalgaller (1992). In Sect. VI.9 we will review the uniqueness results given in this section for surfaces with positive curvature. Surfaces with negative curvature. Geometrically these are both the surfaces that cross their tangent plane at each point and those for which the intrinsic metric is hyper-Euclidean (see Sect. I.5.D of Berger (2003) for details): the sum of the angles of each sufficiently small geodesic triangle (with sides that are geodesics) is less than or, equivalently, each sufficiently small triangle satisfies a2 > b 2 Cc 2 2bc cos ˛:

Fig. VI.7.4.

“Sufficiently small” can be replaced by the condition that the triangle is the boundary of a simply connected domain; see below. The exact default from will be measured by the Gauss-Bonnet formula below. Readers will have noted that these are the properties of that hyperbolic geometry (Sect. II.4) for which the curvature K constantly equals 1. The surfaces with negative curvature are very important in



dynamics, for then the geodesics diverge, even exponentially; we will encounter them amply in Chap. VII. We will also encounter them in considering the space forms in Sect. VI.XYZ. Finally, the minimal surfaces of the following section always have a K that is negative or zero. The basic text for their study is Hadamard (1898), much in advance of its time and where the theory of chaos was discovered. There are two important things to mention for these surfaces. It may seem at first very intuitive that a complete surface with negative curvature cannot be contained in a bounded region of space. Hadamard was sure of it. This assertion is called Hadamard’s conjecture. It was much studied; see for example Chap. II of Burago and Zalgaller (1992). It was refuted not long ago in Nadirashvili (1996), where the author succeeds in constructing a minimal surface without any planar point (thus with strictly negative curvature) that is contained in a bounded region of E3 . The construction uses Weierstrass’s formulas for minimal surfaces; it “suffices” to find suitable functions which are solutions. These formulas, essential for understanding minimal surfaces, will be given in the next section. The Gauss-Bonnet formula. We consider a triangle of a surface whose sides are geodesic arcs; we suppose furthermore that the triangle determines a domain D, i.e. that it can be “filled in”. The serious formulation is that this triangle is the boundary of a simply connected domain (see if needed the few lines at the end of Sect. V.XYZ). See the figure for a non simply connected counterexample, where readers will show that, the boundary curve being a (closed) periodic geodesic, we have ˛ D ˇ D D R , but that D K.m/ d m D 2. But in the simply connected case we always have the formula relating the curvature to the angles ˛, ˇ, of the triangle: Z .VI:7:1/ ˛CˇC D K.m/ d m: D

It may be noted that this is a generalization of formula III.1.2 of spherical geometry and of formula II.4.1 of hyperbolic geometry. It should be realized that the domain can be enormous, have curves of different signs, but all that is compensated for in the end. Formula VI.7.1 is never easy Rto prove in an elementary way; it does not follow from the formula Area.fS .D// D D K.m/ d m given above for the Gauss mapping. Readers may choose a proof from the classic texts. In any case, the result is concerned only with the intrinsic metric and thus with abstract Riemannian surfaces. For the (approximate) date of the discovery of this formula see Dombrowski (1979). In fact, the above formula is due to Gauss. Bonnet proved (in Bonnet (1855) a more general formula, valid when the sides do not necessarily lie along geodesics. It is then necessary to modify the quantity A C B C C , by adding to it the integral of the geodesic curvatures of the sides. The geodesic curvature of a curve along a surface of E3 measures the deficiency of the curve from being a geodesic. This notion generalizes the algebraic curvature of plane curves, which measures the deficiency of a curve from being a line. The proof is no more difficult than that of Gauss’s formula, but it is nonetheless difficult and non conceptual.



Fig. VI.7.5. The figure lower right shows clearly that simple connectivity is indispensable

Now, we want a conceptual proof, so why not use Stoke’s formula, since it relates an integral on the interior of a domain to an integral along the boundary? This conceptual proof appeared in Chern (1944) and was the start of a revolution in differential geometry, that of “Chern classes”. We use the language of differential geometry; see [BG]. The idea is to introduce the unitary fiber US of the surface S (that we will amply encounter in Sect. XII.6), i.e. the set formed of all unit vectors tangent to S. The canonical projection p W US ! S consists of associating with each unit vector the point of S where it is tangent.

Fig. VI.7.6. The surface is lifted by a vector field into its unitary fiber



The essential thing is that the three dimensional manifold US possesses a canonical differential 1-form and above all that its exterior derivative d is the form lifted onto US by p of the 2-form of the curvature K.m/d m of S (S oriented so that its canonical measure d m becomes a 2-form). Now if the domain D is simply connected, we can define a continuous field of unit vectors on D; thus we can lift D into US. The Stokes formula applied to .D/ is exactly the Bonnet formula, because the 1-form is nothing other than the geodesic curvature. The unitary fiber will not be confused with the direction fiber introduced in Sect. V.10 for plane curves: in the first case, the fiber is the circle formed by the vectors of length 1, in the second it is the circle formed by the linear directions, the quotient by identification of antipodal vectors of the preceding. We note that here we have a typical example of the necessity for introducing objects “above” an object for a good understanding of what happens “below”. In the present case we can say that it has to do with a ladder which has as much a physical as a conceptual nature. The so-called Gauss-Bonnet theorem. The result concerns compact abstract Riemannian surfaces. It is intuitive that they can be triangulated, i.e. divided into a finite number of triangles with simply connected interiors (taking them sufficiently small will accomplish this). If we apply Gauss’s formula to each of these triangles and take the sum, we obtain the total integral of the curvature on the surface S on the one hand, and 2.s a C f / on the other (because the sum of the angles at a vertex of the triangulation will be 2 in total), whence: Z .VI:7:2/ K.m/ d m D 2.s a C f / S

where here s (resp. a, f ) denotes the number of vertices (resp. sides (edges), triangles (faces)) of the triangulation.

Fig. VI.7.7.

Now the quantity s a C f does not depend on the triangulation but only on the topological type of the surface; it is called the characteristic of the surface, often



denoted by .S/. If the surface is orientable and its genus is (see Sect. VI.1), then D 2.1 /. This formula is clearly of primary importance; here is an example: if the curvature is nonnegative (or else nonpositive) on a torus ( D 0), then it is in fact identically zero and the torus is flat (that is to say, locally Euclidean). Finally we have the formula: Z .VI:7:3/ K.m/ d m D 2.S/: S

It remains to prove that the triangulations exist! We have already spoken about this in Sect. VI.1. Why have we written “so-called Gauss-Bonnet” even though the result is in fact called “Gauss-Bonnet” in the majority of books that treat it? Certainly the proof is not difficult once the existence of a triangulation has been obtained. Historically, to the best of our knowledge, this aspect enters with its proof for the first time in Boy (1903). This is the same mathematician whose name is associated with the surface mentioned in Sect. I.7 representing the real projective RP2 immersed in R3 ; see Karcher and Pinkall (1997). In Berger (2003) we have opted for the name “global theorem of Gauss-Bonnet”. VI.8. What we know how to do with the mean curvature, all about soap bubbles and lead balls We recall that the mean curvature changes sign when we reverse our choice of the unit normal. In everything that follows, we always suppose that we have chosen a normal direction once for all. The curvature K gives very roughly the local shape of the surface (convexity or crossing of the tangent plane), but more essentially the deficiency from being Euclidean: a development limited by the length of the small circles, whereas the second fundamental form is related to the way the surface is situated in space. The geometric interpretation of the mean curvature is the following. Instead of examining the variation of the length of the small circles, we look at the variation of the area of a small domain of the surface when is displaced normally from the surface. We can guess the result: the lines of curvature are, in the small, pieces of circles of radii k11 and k12 , the infinitesimal variation of the area is thus equal to k1 C k2 D 2H. More generally, if we deform the surface over a domain by carrying the function f .m/ along the normals, we obtain the essential formula: Z .VI:8:1/ derivative of the area D 2 H.m/f .m/ d m:

Minimal surfaces. The first question is: What are the surfaces of zero mean curvature? They are called minimal surfaces. In fact, the above formula shows that a local variation of the surface does not change the area to the first order. There is thus a




t f (m)



Fig. VI.8.1. Two one-parameter variations of a surface fragment, the first with constant normal displacement, the second vanishing at the boundary of a domain

Fig. VI.8.2. In spite of appearances, all these minimal surfaces are actually isometric. c M. Spivak Spivak (1999)

certain abuse of language: the area function is critical in the surface, which doesn’t mean that this area is minimum in the class of variations considered, e.g. for those that are restricted to having a given curve as fixed boundary. We nonetheless keep the name “minimal surface”. We find such surfaces (with boundary) by dipping a curve made out of metal wire into a soap solution and withdrawing it slowly; a film is formed that is a minimal surface. We can’t even give the merest idea of the very numerous and detailed results or of the open questions concerning the minimal surfaces in E3 , it being



Fig. VI.8.3. A minimal surface discovered by Costa, Hofmann and Meeks. Wagon c S. Wagon (2009)

an immense subject which has been the constant object of study, almost without interruption subsequent to the memoir of Plateau (1873). Recent references, complete when they appeared, are the detailed books Dierkes, Hildebrandt, Küster, and Wohlrab (1992), Nitsche (1996), the synthesis Osserman (1996) and finally Colding and Minicozzi (1999). The richest in illustrations is by far the first of these. See also the very recent Lazard-Holly and Meeks (2001). We can but quickly treat a few small points. A simple remark: a complete minimal surface of E3 , without boundary, has negative or zero curvature and thus is never compact. In fact, we saw in Sect. VI.5 that every compact surface necessarily has points of positive curvature. Now comes a celebrated example, that of the one-parameter family of minimal surfaces that joins the two most celebrated (complete) minimal surfaces, the helicoid (which is ruled) and the catenoid (which is a surface of revolution): This family of surfaces has the remarkable property that each of its members is minimal and isometric to all other surfaces of the family. Of course, this for the intrinsic metrics of these surfaces and not the restrictions of isometries of E3 . Thus in particular they all have the same principal curvatures k1 and k2 since k1 D k2 and K D k1 k2 is invariant under the deformation (although variable on each surface). Thus we find surfaces having the same principal curvatures, but different in E3 . This doesn’t contradict the fact that the two fundamental forms determine the



Fig. VI.8.4. Longhurst’s model of an Enneper surface, in bubinga wood. Bruning, c R. Longhurst Cantrell, Longhurst, Schwalbe, Wagon (2000)

surface within an isometry of the space (end of Sect. VI.6); what happens here is that the principal directions “turn” during the deformation. For explicit construction of the family of the above example, the simplest thing is to climb the ladder with Weierstrass. Beginning in 1861, Weierstrass gave a way for finding many minimal surfaces with the help of the theory of holomorphic functions of a complex variable. Here are his formulas set on an open subset U of the plane R2 D C, where f .Z/ D f .u C iv/ is an arbitrary holomorphic function (at least initially) that is integrated along a path going from the origin to Z. We then define an immersion .u; v/ 7! .x.u; v/; y.u; v/; z.u; v// of U into R3 by the R formulas: Z x.u; v/ D Re 0 .1 Z2 /f .Z/ d Z ; R Z y.u; v/ D Re i 0 .1 C Z2 /f .Z/ d Z ; R Z z.u; v/ D Re 2 0 Zf .Z/ d Z : It is these formulas and some refinements, steadily in progress, that have recently made way for the construction of complete minimal surfaces of various topological types, such as the one below:



Enneper’s surface is a great classic and here is a sculpture inspired by it: The essential point, once the function f has been chosen in the Weierstrass formulas, is knowing if the surface obtained is embedded and not just immersed, presenting self-intersections. We recall here Nadirashvili’s counterexample from the end of Sect. VI.7: there exist bounded and complete minimal surfaces in space that have no planar points. Soap bubbles and the fabrication of small balls. If we blow a soap bubbles using soapy water and a straw, don’t we obtain magnificent spheres? A physicist can show mathematically the we obtain a compact surface whose area is a local minimum among all those that envelop a domain having a given volume (notion of capillary tension), because a force is exerted that requires the area to be minimal. If additionally this were an absolute minimum, we could have settled the issue with the isoperimetric inequality; see Sect. R VI.11 below. But we only know, thanks to formula (VI.8.1), that here we have D H.m/f .m/ d m D 0, where the functions f are to R be chosen from those that model a variation conserving the volume, i.e. such that M f .m/ d m D 0. A classic argument of the calculus of variations implies that there are enough such f to conclude that “the mean curvature H is a constant function on the surface”. Certainly (round) spheres have constant mean curvature, but are they really the only ones? Is every soap bubble, for which we have just seen that the mean curvature is constant, a sphere? The result that they must indeed be round spheres is true, but awaited (Hopf, 1951) for a proof. And, again, Hopf assumed that the surface had the topology of a sphere; a theorem without this hypothesis on the genus (it must be shown that surfaces of higher genus with constant mean curvature can’t exist in E3 ) awaited A.D. Alexandrov in 1955 . Alexandrov’s proof is geometric; it consists of applying to the surface and its images by symmetry a comparison principle for surfaces that is analogous to the one already see for curves in Sect. V.7. For the sphere Hopf uses functions of a complex variable: the abstract Riemannian sphere must be given a complex structure. What is called a holomorphic differential is then constructed; see Sect. VI.XYZ for complex structures on a surface, which must be zero for reasons of the topology of the sphere, which implies by construction that all the points are umbilics, which can only happen for round spheres (an elementary local result, already expressed in Sect. VI.6). The details can be found in Hopf (1983) and do Carmo (1976). In the industrial realm, the preceding explains how certain balls are made (small balls for more serious balls, see Sects. VII.13.C, II.2, VI.6). There are two methods: in the first, metal is dripped into an appropriate liquid, and these drops take on a perfectly spherical form because of surface tension. In the other method, an arrangement is made for letting drops of liquid loose under conditions of weightlessness; the drops also become spheres, for the same reason. In both cases, the condition given by physics is that the surface area is a local minimum for surfaces enveloping a given constant volume, thus in particular the mean curvature is constant as we have seen above.



For all that, the preceding does not end the study of surfaces with constant mean curvature, for there remains a natural case, that of surfaces that are just immersed and not necessarily embedded. For the non embedded case, where the topological type of the sphere is excluded, the proof of Hopf with holomorphic differentials applies by the same construction. It is not the same for the topological type of the torus, where immersed tori were found for the first time in Wente (1986), contrary to what Hopf had conjectured. These matters are connected with elliptic functions; these tori are completely classified in Pinkall and Sterling (1989).

Fig. VI.8.5. The Wente Torus. At right a portion of the surface has been removed to gain a view of its interior

There is also the difficult problem of the double soap bubble, solved just in 2000; see Chap. 14 of Morgan (1988–2008 (3rd ed., 2000)). But that of the triple bubble problem remains open, and likewise so does the analogous planar case. For the connection between certain soap bubbles with 120 pieces and the regular polytope with 120 faces in E4 , see Sect. VIII.11 and Stillwell (2001).



Fig. VI.8.6. The conjecture was that the double bubble that contained two given (in general different) volumes is composed of three surfaces that are pieces of spheres (that the angles at their intersections are equal to 120ı follows immediately from the first variation formula. Readers may formulate a conjecture for the triple bubble). c Mathematical (a) Sullivan, Technische Universität Berlin. (b) Morgan (2000) Association of America



It’s easy to construct complete surfaces of revolution with constant mean curvature, for which the meridians are provided by an ordinary differential equation; they are called Delaunay surfaces. Those with a geometric bent and who like unexpected relationships will appreciate knowing that they can be obtained by rotating the locus of one focus of a conic that is made to roll on a line (the catenoid with mean curvature zero corresponds to the case of the parabola). Some of these surfaces are merely immersed, but the rest are truly embedded.

Fig. VI.8.7. Examples of meridians of a surface of revolution with constant mean curvature. Readers can study the case of ellipses in more detail

Obtaining surfaces with constant mean curvature of different topological type (noncompact in fact, but with more “pieces” than the surfaces of revolution seen above) had to await (Kapouleas, 1995). Today a classification is still needed to get these examples into a sufficiently general setting. The constructions of Kapouleas are based on a detailed analysis which allows the construction of connected sums. Recent results and a bibliography can be found in Mazzeo, Pacard, and Pollack (2000); see also the numerous computer-generated drawings in Grosse (1997), Grosse-Brauckmann and Polthier (1997) and Kilian, McIntosh, and Schmitt (2000). VI.9. What we don’t entirely know how to do for surfaces Here we give several natural examples of open problems for surfaces in space, sometimes stated around results having positive answers. Various isometries (the compact case). We begin with the case of results that affirm the congruence (identity within an isometry of E3 ) of isometric surfaces. Next we speak about a closely related problem, that of isometric deformation. In 1899 Liebmann showed without much difficulty that the only compact surfaces with constant positive curvature are the round spheres. In 1927 Cohn-Vossen established a much more general result: two isometric compact surfaces with everywhere positive curvature are congruent. The figures below show clearly the absolute necessity of the condition of positive curvature:



Fig. VI.9.1. How to construct two surfaces that are isometric but not congruent, i.e. that are not obtainable from one another by a global isometry of the ambient space

These figures call for two remarks that lead to open problems. The first is that we are dealing with a counterexample of class C1 which is quite flexible. To the present no one has found a counterexample with two real analytic surfaces (class C! ) which are isometric. The second is: can we find a one-parameter family (a deformation) of compact isometric surfaces between them? In at least class C2 it is a completely open question. If such deformations were to be found, the question then arises as to whether the volume they enclose remains constant. See below for those that are merely class C1 and Sect. VIII.6 for the case of polyhedra and the question of what happens to the volume contained by the surface. We can find proofs of Cohn-Vossen’s theorem on p. 46 of Chern (1989) and p. 103, Sect. 6.2.7 of Klingenberg (1973) (Herglotz’s integral formula). In these classic arguments, by isometry, we are able to impose on one surface the second fundamental form of the other; we then compose a shrewd mixture of them that furnishes a nonnegative real number that is zero only in case the two fundamental forms coincide. By integrating this number over the whole of the first surface, a positive number is obtained. But the situation is different when we reverse the roles of the two surfaces; the attached number then changes sign, by the construction. The integral is thus finally zero, hence the coincidence of the two second fundamental forms which, along with the first form (isometry), yields congruence in the space (see the end of Sect. VI.6). Nevertheless, the result seems completely isolated and leaves us hopelessly at the bottom of the ladder. It appears that the first person to have climbed sufficiently high to be able to see clearly was Gromov in 1986, both for the existence problem of Sect. VI.5 and for uniqueness. We find a presentation and some generalizations in Labourie (1989), but we can’t in a reasonable amount of space give an outline (or even a brief insight) of Labourie’s work; we simply say that the theoretical setting used is that of symplectic geometry, which Gromov believes will play a major role in the twenty-first century. Isometries and local isometric deformations (bending, tennis balls and more). If we suppress the compactness conditions while keeping (or not keeping) the completeness condition, then “there is plenty of room for nontrivial isometries and deformations”. For example, there are all the cylinders, that have the topology of the plane or that of a cylinder of revolution. There are no other possibilities, since



there are only two types of curves, the line and the circle. Here is an example of isometric deformation, but only local (no completeness here): take a tennis ball and cut out a small piece. You can then deform it without much effort. It’s a highly isometric deformation because modifying the intrinsic metric requires, as with fabric, the exertion of much force tangential to the surface. These examples lead us to think that, at least locally, every sufficiently small piece of a surface is deformable. By deformation we understand a one-parameter family (continuous, of course) of surfaces all isometric to each other. In fact, this deformability is an extremely complicated and badly understood and is far from being settled at present, even when only infinitesimal deformations are studied, which are naturally a starting point. There is also the question of the differentiability classes that enter into the matter, both for the metric and for the deformation parameter. The author of a (or rather the) bible of differential geometry (Spivak, 1970) apparently gave up the problem in disgust while preparing the book for publication. The problem has been much studied by the Russian school. In Chap. III of Burago and Zalgaller (1992) there is a very complete synthesis that in fact has mostly to do with partial differential equations, but of a type that is generally nonclassical. Geometry enters only little, but it is interesting to remark that mirror symmetry about a parabolic point comes into the picture (see 6.4 of this reference) and is present in an essential way in the proof of the nonexistence of embedded surfaces with constant mean curvature (see Sect. VI.8). We will need to retain just two things: only the parabolic points (those where the total curvature is zero) cause a problem; as soon as the total curvature is nonzero, deformation is possible. On the other hand, in the neighborhood of a parabolic point, a deformation need not exist, even infinitesimally. The other factor that we need take into account is the order to which the curvature is nullified; it’s the large orders that cause difficulties (which was by the way the case in the counterexample of Pogorelov for an abstract Riemannian surface not isometrically embeddable in E3 ; see Sect. VI.5. To allow the complexity of the problem to be felt we give a single example, due to Efimov in 1948: the surface .x; y/ 7! .x; y; x 9 C x 7 y 2 C y 9 /, where the real number is transcendental (see Sect. IX.1 and note the meager geometric character of this condition!), is never deformable locally. We will now look at Bleecker’s crumpling. What occurs in the C1 realm? Completely different things, as we have already partially seen in Sect. VI.5! Let us review the work of Bleecker (1996). There the Nash-Kuiper C1 embedding result is extended to one-parameter families of surfaces. The author applies it to one-parameter embeddings of compact surfaces in E3 and thus obtains examples of surfaces of spherical type which are isometric but not as a consequence of a global isometry of E3 . In view of the beginning of this section, it is good then that we are only in a domain that is C1 and strictly nonconvex, even if we begin with an initial convex surface. For example, we can begin with ellipsoids. The essential interest is that the author shows, if the initial ellipsoid is sufficiently flattened, that the volume of its isometric deformations increases strictly. This result



is surprising, for we will see in Sect. VIII.6 that isometric deformations of polyhedra are necessarily volume preserving. Thus the differential context is an astonishing intermediary between the C2 setting and the purely polyhedral one. We again mention that such surfaces are truly extremely crumpled. The Caratheodory conjecture. We recall that an umbilic is a point of the surface where the two principal curvatures are equal. A surface a torus of revolution for example needn’t have an umbilic (see in Fig. VI.6.2 above how to find the principal curvatures of a surface of revolution). Any points of a surface of revolution that are situated on the axis of revolution are umbilics. More subtle are the umbilics of the ellipsoids encountered in Sect. VI.3. What is the situation in general? We would like to know that umbilics always exist for surfaces of topological type other than that of the torus, and still better: how many must there be? The matter isn’t settled; here is what we know. We give the initial idea and the associated concept. We consider a compact surface (with oriented normals). We suppose the umbilics are finite in number, for otherwise there is nothing to gain in this way, so that what remains after their removal is an open subset O of the surface and our umbilics fxi g are all isolated points. At each point of O there is a well determined principal direction that corresponds to the greater principal curvature, since these two curvatures are never equal. This allows definition on O of a continuous field of tangent directions. For such a field we have the theory of Hopf (1926) at our disposal. It associates an index Ind.x/ with each x of S; it is a half-integer thus defined. We recall the degree theory of mappings S1 ! S1 encountered with respect to curves in Sect. V.3 and also in Sect. V.10. We consider about x a very small circle C on the surface. At each point c of C we associate a direction f .c/ of Tx S (this can be done in numerous ways, for example take the parallel to .c/ in E3 and project it onto the space Tx S/. The set of lines of Tx S is also a circle S1 , a real projective line; see Sect. I.6. Thus f is clearly a mapping S1 ! S1 . We set Ind.x/ D 12 degree.f /. By the invariance of the degree under hom*otopy (end of Sect. V.XYZ) this index does not depend on the small circle chosen in the neighborhood of x and is thus associated with the point x (and with the field of tangent lines considered). Readers should verify that if x 2 O, then its index is zero. The 12 shouldn’t be surprising; it is imposed because the tangent lines considered turn twice about the point when we go once around the small circle. Hopf’s result says that, whatever the continuous field of directions tangent to S and not zero at more than finitely many points fxi g, we always have: X Ind.xi / D .S/ i

where .S/ denotes the characteristic of the surface. For a proof, see for example II.8.4 of [BG]. Thus if .S/ is nonzero, the surface must have at least one umbilic; this is thus the case for all the types of compact surfaces of E3 other than that of the torus (for which we have seen that it can indeed be umbilic free). The points



Index -1


Index 1

Index 2

Fig. VI.9.2. Examples of indexes. When the direction field can be oriented in a coherent manner so as to become a continuous vector field, its index is equal to that of the vector field it’s integral. When the direction field isn’t orientable, its index is a half-integer

of the axis of a surface of revolution have index 1 and each umbilic of an ellipsoid has index 12 . See the figures of the three types of generic umbilics, as classified by Darboux, in Sect. VI.10. The Caratheodory conjecture is that each surface with the topological type of the sphere possesses at least two umbilics. It remains open presently, but has been proved for real analytic surfaces (class C! ). As in all attempts at proving the conjecture, the idea is to show that the index of an umbilic can never exceed the value 1. It is true in the real analytic domain, where the proof is based on the existence of complex coordinates so as to be able to use analytic function theory as well as on ideas taken from the theory of algebraic curves (the Puiseux expansions cited in Sect. V.13). The degrees of rigor of various authors (Hamburger from 1940, Bol) has been improving; see Klotz (1959), Scherbel (1993) and Gutierrez and Sotomayor (1992). But the proof still remains extremely difficult. The Willmore conjecture. This conjecture from 1965 is the following. R For each compact surface S, we introduce the Willmore functional S 7! F .S/ D S H2 .x/ dx, where H is the mean curvature of S. This functional has the remarkable property of being invariant under the conformal group of E3 (for example, under inversion, etc.). We calculate easily F .S/ for tori of revolution and find that it is minimum and equals 2 2 exactly for the tori of revolution for which the conformal type is that of the equidistant Clifford torus in S3 (see Sect. II.6 and as an exercise examine the respective radii of the torus of revolution of this type). In particular this value remains equal to 2 2 for all the conformal images of this special torus. In Willmore (1971) it was conjectured that the inequality F .S/ > 2 2 always holds for surfaces of the topological type of a torus and that equality only holds for the conformal images indicated above. This conjecture has been most intensely studied, but is not yet completely resolved. As a reference, consult Ros (1999). The problem is resolved for many types of toroidal surfaces, whether tubular neighborhoods of space curves or the tori for which the conformal type is not too far from that of the (equidistant) Clifford torus (see Sect. II.6); that is, for which the complex conformal structure is that of an abstract square flat torus. But the most astonishing



thing is that we know that the functional actually attains its minimum value for a torus embedded in E3 that is in addition real analytic. But the value of F isn’t known, for we don’t know the torus in question. It’s a surprising situation, for in the great majority of minimization problems, the greatest difficulty lies in the existence of a minimizing object; its properties are then such that we know enough to finish (see for example the isoperimetric case in Sect. V.12). The situation here is thus completely atypical. The surfaces of constant width. In Chap. V, on curves, we discussed curves of constant width. These are the boundaries of convex subsets of E2 that are likewise said to be of constant width, i.e. each pair of lines that frames the curve has the same breadth. The circle isn’t the only such, for we can include every curve that is the developable of a compact curve having an odd number of cusps. The Reuleaux triangle corresponds to the case where such a curve is degenerate at the three vertices of an equilateral triangle; see 12.10.5 of [B]. Lebesgue showed, using Fourier series, that the Reuleaux triangle is the unique curve of a given width with minimum area. In higher dimensions, starting with d D 3, the convex bodies (or the hypersurfaces) of Rd of constant width are almost a total mystery. If in all dimensions and for a given constant width the balls are those that have a maximum volume (it’s trivial!), for the minimum starting with d D 3 the problem is almost completely open. The spherical harmonics don’t yield any conclusion presently; the state of the problem in 1991 can be found in Problem A.22 of Croft, Falconer, and Guy (1991), along with references. VI.10. Surfaces and genericity Let us recall (see Sects. V.10, VII.13.D, XI.6) the notion of genericity: the generic is what is “real” and what we actually see. An equivalent but more mathematical definition is this: the generic is what is stable under small deformations. In fact nature ignores what isn’t stable, for the attached probability is zero. The subject is not exhausted by classifying the generic objects, because throughout mathematics evolutionary phenomena are encountered, families with one or more parameters. For example, a parameterized curve, e.g. a C1 mapping of S1 into the plane, is generically an immersion (there are no local singularities), and the only global singularities are ordinary crossings. But in a generic one-parameter family, i.e. stable as a family with one parameter, we can encounter cusps, self intersections with contact. Stability here means that we can’t eliminate these singularities with a small deformation of the one-parameter family. In this situation we speak of (unavoidable) singularities of codimension 1.



Fig. VI.10.1. Singularities of codimension 1 for plane curves (immersions of S1 ): cusp (local singularity) and self-contact (global singularity)

This idea of genericity is recent, it began with Whitney in 1955, but it was Thom who got it completely underway in a general setting in a series of articles, then with numerous applications in Thom (1989). Presently it is part of the foundations of geometry and of mathematical physics. The basic reference, for a nontechnical initial exposé, is Thom (1972–1977). For surfaces, genericity can’t be found in the classic works, except for Porteous (1994). For a completely mathematical version, Demazure (2000) or Demazure (1989) can be consulted. For problems concerning the visualization of surfaces, consult the basic reference Koenderink (1993) and the more recent Petitot and Tondut (2000). In Sect. V.16 we already encountered the idea of the relation of the genericity of surfaces to their local order, i.e. in the number of points in which they are intersected by every line in the neighborhood of a given point. This order is 2 for elliptic points (local convexity), 3 for hyperbolic points (called passes or saddle points), 4 for a parabolic point (non planar), 5 for a biinflection point. In this section we will treat three examples (two of these are intimately related) of genericity for the surfaces of E3 . But beforehand we should give a bit more detail about the behavior of the set of normals to a surface. Digression: the focal sheets of a surface. Surfaces enter in an essential way into geometric optics as wave fronts of light rays. We need to remember this: the light rays being (for an instant at least) the normal lines to a surface S, we need to know what happens as time proceeds; i.e. what becomes of the surface when it is allowed to be displaced uniformly with time along these normals (these parallel surfaces are the wave fronts). We have encountered this idea of parallel objects for curves in Sect. V.10. The nature of what we will later call focal sheets of a surface in R3 is a fundamental problem in optics, since it is along these that light rays get concentrated. At a point where the two principal curvatures are nonzero and distinct, the situation is this: the lines of curvature (here line means curve) are exactly the curves of the surface at which the normals admit (locally, of course) an envelope curve, or again generate a surface with total curvature zero, thus a developable surface; see Sect. VI.7. At a generic point these envelope curves do not have singularities and the basic generic figure is thus that shown below (two cases according to the sign of the total curvature):



Fig. VI.10.2. The two types of disposition of the focal sheets of a surface according to the sign of K

The two surfaces of the figure are called the focal sheets of S (we also say caustics). They are common, at least locally, to all the parallel surfaces. The two points where the normal touches them are the points where the light is focused; they are situated on the normal at a distance equal to one of the two radii of principal curvature. The fact that these are different (we are not at an umbilic) interprets the fact that the surface is astigmatic, thus a very bad optical instrument. Such is the case with the eyes of astigmatics; the defect of the surface of their eye is corrected by appropriate lenses. What happens at umbilics or at points where the envelopes of normals have singularities isn’t describable in the single C1 context, but only in the generic context, that will be defined precisely. Now we say only that singularities are expected, fringes, etc., for the focal surfaces. The complete figure in the case of the ellipsoid was already studied in Cayley (1873) and was shown in Fig. VI.3.12, taken from Joets and Ribotta (1999), which studies the case of dimension four for reasons of crystal physics. Figure VI.10.1 is only valid in the generic case where not only are the two principal radii of curvature different, but also the radius of the associated principal curvature varies with a nonzero derivative along the lines of curvature. Beyond this, the focal sheets have a behavior that is still poorly understood; see Porteous (1994): the focal sheets then have cusps, etc. Even before studying the singularities of the focal sheets, we can find where these intersect: it is when the two radii of principal curvature are equal, by definition at an umbilic, so we need first study the behavior of the lines of curvature in the neighborhood of an umbilic. The generic umbilics. This form of the lines of curvature has been studied since Darboux (1887, 1889, 1894, 1896); in a modern book it is found in Porteous (1994). The idea is to look at the Taylor approximation of the surface given as a graph .x; y/ 7! .x; y; z D f .x; y//. At an umbilic, the approximation begins with



k.x 2 C y 2 /; in the generic case the third degree terms will not all be zero, thus of the form ax 3 C 3bx 2 y C 3cxy 2 C dy 3 . The problem is thus in particular to classify the cubic forms under the action of the orthogonal group for dimension 2. This first classification furnishes three types of figures:

Type 1

Type II

Type III

Fig. VI.10.3. The three types of generic umbilics and the lines of curvature in the neighborhood

Fig. VI.10.4.

The index of these umbilics (see Sect. VI.9) is 12 in the first two figures and 12 in the third. But the complete study is more subtle. Moreover, for the focal sheets we best use Thom’s so-called catastrophe theory that we can’t even begin an outline: see details and references in Porteous (1994), which is very concise, and in Koenderink (1993), which is much more detailed. It is noted in Porteous (p. 271) that a detailed study of umbilics had been undertaken by Gullstrand starting in 1904, with the aim of understanding the human eye, (and that this physician by training and



self-taught mathematician and physicist received the Nobel prize in 1911 for this work). Moreover, the book of Porteous tries to remain elementary and doesn’t enter into catastrophe theory, transversality, etc. There is thus a clear link between the singularities of the focal sheets and those of points of the surface, i.e. the umbilics. The focal sheets can also have cusp edges. Here are some figures for the two systems of lines of curvature and the focal sheets that appear in color in the Porteous book, which may be consulted for more information. Platonic readers will also appreciate in Porteous (1994) the brief and informal exposition on the connections between certain types of singularities and the polyhedra in spaces of dimensions 3 and 4, via the groups that are associated with them, called Coxeter groups. Such connections already appear in the theory of curves. We can’t talk about this, even briefly, since we are forced, for lack of space, to treat only singularity-free plane curves. How we view a surface: apparent contours. At first we place ourselves at infinity, in a given direction ı; that is, we study two things: first, the boundary of the projection .S/ of S, onto a plane P perpendicular to ı, by the mapping W E3 ! P defined by the direction ı. Then in fact the “curve” of S that is furnished by this boundary, i.e. the set of points of S where the tangent plane contains the direction ı, can be called its rim, whereas the projected curve is really the apparent contour. The generic case that comes up first is the one we think of naturally, whether it be an ellipsoid, a sphere, a torus (caution! not too inclined!). Here we find the first stable singularity of the mappings R2 ! R2 , called folding, which is quite natural, but see the figure below, which shows realizations in R3 . The points of the visible plane have, outside of the apparent contour, two inverse images. On p. 221 of Porteous (1994) there is a nice little theorem, quite recent, which provides a relation between the Gaussian curvature at a point of the apparent contour of the folded-only type and the curvatures of two curves, one formed by this apparent contour and the other by the section of the surface by vertical normal plane. Now, if we incline a torus more, the interior curve of the apparent contour will develop cusps. This situation is also stable under small deformations: it’s the second stable singularity of the mappings R2 ! R2 and is called a crease. There aren’t any others in the sole setting R2 ! R2 . Here the points of the image, off the boundary curve, have one or three inverse images. But the problem of generic apparent contours doesn’t stop here. The object in which we must study the stable forms is the three-parameter family of apparent contours of S obtained when we vary the viewpoint in R3 . A fold or a crease observed from a certain point will persist if we move over a small neighborhood. Other singularities, in contrast, can only be observed from points of a certain set † R3 , which in general will be a surface, a curve or a point. Taken in isolation, they are thus unstable; but what implies the stability of the family is that a small deformation of the surface allows them to persist, all the while deforming or displacing the set †. The complete study of generic figures of



Fig. VI.10.5. The three types of singularities of the apparent contour of a surface

Fig. VI.10.6. The eleven intrinsic generic situations on a surface. Arnold (1983) c London Mathematical Society and British Library

apparent contours and rims, thus comprised, only dates from 1981. We find it, placed in a general setting, in pp. 145–152 of Arnold (1983), to which readers are referred. We extract several figures from this study. An essential role is played by the curve formed, on a generic surface, by the parabolic points, i.e. those where the Gaussian curvature is zero. This curve will separate the points where the surface is

Fig. VI.10.7.



convex (K > 0) from those where it crosses its tangent plane (K < 0). Moreover, we expect to find supplementary singularities at the points (of negative curvature) for which the projection direction is also an asymptotic direction, i.e. a direction at a point of a surface for which the second fundamental form is zero. There clearly aren’t any unless the Gaussian curvature is negative. Finally, we find eleven generic situations, represented in the figure: We should emphasize that here we are concerned with the theoretical apparent contour, typically as in the case of the torus drawn above. What we really see in this situation is much less, a portion only. Just look at the drawing below. In the sketch of the torus, the apparent contour has been solidly sketched for what is seen, and in dashes for what exists, but is hidden from the eye. We also note that the classification presented doesn’t contain the global generic singularities, such as the appearance of a hidden portion: For a study oriented toward physical reality, and not just mathematics, see the very detailed Koenderink (1993).

VI.11. The isoperimetric inequality for surfaces We will be very brief, and here is why: first, while the problem can be posed in all dimensions, and also while there aren’t any particular properties connected to this problem, unlike certain problems relating the two fundamental forms (see “Higher dimensions” in Sect. VI.XYZ). The isoperimetric inequality is true for surfaces, the minimum of the ratio Area3 =Volume2 being attained by spheres and only by them (see however the hairy spheres in Sect. VII.14); here we have again, as in Sect. V.12, several possible proofs. We refer to Burago and Zalgaller (1988) for a complete exposition; see also the outline in Sect. VII.14. However, we would like to mention two other proofs. First that via Stokes’ formula, given in the planar case in Sect. V.12; here we slice by planes, then the plane sections by lines, and finally the linear sections by points, all the while keeping the ratios of lengths, areas and volumes constant. For more details, if they are wanted, see Berger and Gostiaux (1987) or Berger (1990b), and again the remark on Schmidt’s proof in Sect. V.12. Next the one using GMT (geometric measure theory): this theory shows, by being placed in a sufficiently but not too large category of objects (for example, the submanifolds aren’t sufficiently general) so that there exist such objects minimizing Area3 =Volume2 ; in fact the minimizing object is a true smooth surface (and in a higher dimension their singularities form a set of measure zero). Let S be such a minimizing surface; the differential calculus, in a manner analogous to what we stated for the plane in Sect. V.12, implies that S is of constant mean curvature H, thus it is necessarily a sphere (Hopf’s theorem: see Sect. VI.8). We can also proceed, in any dimension, without this (subtle) result by filling the whole interior of the domain of which S is the boundary with normals, and using a simple inequality in the calculation of this volume. Here are some details, recalling what was said in Sect. V.12.



This method was introduced in Gromov (1980) in the very general setting of Riemannian manifolds; it can also be found in I.5.G of Berger (2003). It is very important to mention it, for it was Gromov who was the first to realize that GMT could be used for isoperimetric problems, the technique being that of filling. The idea is to fill the interior domain D of S by the normals. This is possible because each point d of D possesses at least one foot i.e. a point m of S which is closest to d among all the points of S. But then this condition of a minimum implies (at a smooth point) that the line md is normal to S at m, we thus fill all of D. In the reverse direction, when we start with an m 2 S, it is necessary to stop when the normal ceases to minimize the distance; the length in question is called the cut value m (compare with the theory of the cut-locus in Sect. VI.3), denoted cut.m/. We then calculate the total volume, denoted Volume.D/, as an integral on the surface with the help of Fubini’s theorem, by separating the variables into points of S and distance along the normals.

Fig. VI.11.1. The filling method for proving the isoperimetric inequality

It thus suffices to know the measure along normals, the second fundamental form works here by definition, and we have finally: Z Z cut.S/ .VI:11:1/ Volume.D/ D 1 k1 .m/t 1 k2 .m/t dt d m; S

where the ki are the principal curvatures. There is no difficulty because it is easy to see that the cuts always come before the first focal point k11 (we have assumed k1 6 k2 ): cut.m/ 6

1 . k1

But the elementary inequality xy 6 ..x C y/=2/2 shows

2 that the above integrand is always bounded by .1 ht /2 , where h D k1 Ck is the 2 R dm 1 mean curvature. Whence easily Volume.D/ 6 3 S h.m/ . But we have seen that S must have constant mean curvature h (Formula VI.8.1), where finally Volume.D/ 6 1 Area.S/. This isn’t the inequality we were looking for. The idea for concluding 3h



is that the quotient Area3 =Volume2 is also a minimum. We write that its derivative (first variation) is zero for the variation of the surface obtained by transporting a function f that is constant on the normals to S; we find that this derived formula 1 (Formula VI.8.1) is zero only if we have the equality Volume.D/ D 3h Area.S/. By replacing h by the value that this equality provides, we deduce from this the isoperimetric inequality which was sought. In the optimal isoperimetric inequalities (of all types), the case of equality is always more difficult to treat. See the case of convex sets in Chap. VII. Here the method gives round spheres instantaneously, for if there is equality in the above inequality, then there is equality at each point of the integrand and thus we have everywhere k1 D k2 D is constant, so that all the points are umbilics and we have a round sphere (see above). More simply still, the fact that here the two principal curvatures are constant implies that all the normals pass through a fixed point, Q.E.D. We will give the “established” philosophy of the classical isoperimetric inequality in Sect. VII.13. VI. XYZ The fabrication of surfaces (sequel to Sect. V.7). How are planes, spheres and balls made? In fact, for planes and spheres it suffices to have pieces of them. The practical uses are innumerable, in optics, for example. Note however that if the lenses of ordinary eyeglasses are pieces of spheres, for astigmatics we need to use tori; as for Varilux (lenses with progressive focal length) matters are not yet well understood: it is not yet known what theoretical form will give a very good approximation to the desired result, i.e. good convergence of horizontal lines, with gradual variation of this convergence. In an absolute sense a perfect answer is impossible; readers will convince themselves of this by examining the behavior of the normals indicated in Sect. VI.10. The fabrication of spherical lenses hasn’t changed for a long time, matters are often still just as they were described in Bouasse (1917). What happens when we rub, while turning, two surfaces against one another? If we turn about a single point, we obtain only surfaces of revolution, but if we turn “most everywhere”, we can’t obtain anything but umbilics, and only spheres have their normals passing through a fixed point: study the subgroups of the three-parameter group of rotations of E3 . That is how spherical surfaces are made. That the result is sufficiently good is checked, as in Sect. V.7, by interferometry (study of fringes). As for the straight rulers in Sect. V.7, we obtain good planar surfaces by rubbing three pieces alternatively against one another. See more technical details in Bouasse (1917), whose reading we can’t recommend too strongly, in particular the introduction for its pedagogic reflections. Surfaces are complex curves. We have already seen in several places (twice in Sect. VI.8 and once in Sect. VI.9) that it would be interesting to attempt to introduce functions of a complex variable on a surface so as to be able to use the holomorphic



calculus. To put a complex structure on a Riemannian surface seems simple enough at first glance. We say that a differentiable mapping from one oriented surface M into another N (E2 D C is a particular case) conformal if the differential df W Tm M ! Tf .m/ N is a direct (orientation-preserving) similarity. Since the holomorphic transformations on C coincide with the conformal transformations of E2 , the specification of an atlas made up of conformal charts of M allows us to endow M with a structure of analytic complex manifold of dimension 1 (or, equivalently, of a complex curve). But we can’t be sure a priori that conformal charts of M exist, nor that there are enough of them to compose an atlas. The coordinates associated with a conformal chart m ! .x.m/; y.m// are called isothermal coordinates. Here are equivalent properties: the lines x D const. and y D const. are orthogonal and we have krxk D kryk (we also say that the lines x D const. and y D const. form a lattice of “infinitesimal squares”); we have ds 2 D .x; y/.dx 2 C dy 2 /. The existence of isothermal coordinates is a (not very difficult) classic result (middle of the nineteenth century) in the theory of partial differential equations; see a very quick presentation in Chern et al., 1954). For the geometer, a simple example of the structure of an analytic complex manifold of dimension 1: for the sphere S2 (with its canonical Riemannian structure of a round sphere), an atlas is obtained by lifting by stereographic projection via the north pole and via the south pole, the complex structure of E2 D C. Conformal representation and modules. Can we classify all the Riemannian surfaces? Yes, that has been done in two stages, by the theory of conformal representation beginning at the end of the nineteenth century and by Teichmüller in the 1930s (theory of modules). This is one of the essential points of the theory called Riemann surfaces, objects that are essential in the theory of numbers, functions of a complex variable, differential equations, etc. A Riemann surface is nothing other that an oriented Riemannian surface that is endowed with a complex structure. The fundamental theorem of conformal representation asserts that: On each oriented compact Riemannian surface .S; g/ there exists a unique function f W S ! RC such that the new metric f:g is of constant curvature K. The sign of K is imposed by the topology of S: it is positive for the sphere, zero for the torus and negative in the other cases with more holes. For existence (uniqueness is easy) a result from the theory of partial differential equations had been awaited for a long time. In particular, Klein (the founder of the Erlanger Program in 1872) thought that it didn’t require proof, because its physical rendering is that of electric equilibrium on the surface. The first complete proof dates only from the beginning of the twentieth century. We also encountered Klein in Sect. III.2, where we used a theorem that is due to him.



A first consequence is thus that each oriented compact surface possesses metrics with constant curvature, thus locally isometric to Euclidean geometry, whether spherical or hyperbolic. This result can also be obtained by elementary geometry; see below. The second consequence is that the classification of (oriented) Riemannian surfaces is reduced to those that are of constant curvature. These objects are often called the space forms. It can be shown without difficulty that these are exactly the (compact) geometries for which the metric satisfies a universal relation F.b; c; ˛/ furnishing the length of the third side of a triangle a D F.b; c; ˛/ as a function only of the two other sides b, c and the angle ˛ that they form (recall the formulas of Euclidean space, of the sphere in III.1.1 and of the hyperbolic plane in Fig. II.4.2). Space forms. We can always normalize the constant of the curvature to C1, 0, 1. The case C1 is that of the round sphere; there is but one of them with curvature C1. We will encounter this situation in Sect. XII.5.B. The case 0 is treated thus: we consider the universal covering of the given flat torus T , i.e. the mapping E2 ! T which is everywhere a local isometry between E2 and T . In the reverse direction, T is obtained as the quotient of E2 by a group of translations that we denote by ƒ. This group is a lattice on E2 . The situation is thus exactly what we studied in the digression in Sect. V.14; and there we gave the complete classification of flat tori (within isometry, and with an orientation). The result is that the space of all flat tori (within an isometry and with normalization by an appropriate hom*othety) is a noncompact surface, with two singular points. It is thus in particular an object of dimension 2, our flat tori depending on two parameters if they are taken normalized, on 3 parameters otherwise. Geometrically we may prefer to view the flat tori by slitting them with two cuts, obtaining thus a parallelogram, and inversely by identifying the opposite sides of a parallelogram of E2 , two at a time. But we still don’t have language at our disposal as easy as that of groups. As an exercise, readers may study, according the type of flat torus considered, the shape of the cut-locus of a point. The negative case is quite a different story, and not completely settled. Contrary to the sphere and the flat tori the space forms with negative curvature are not hom*ogeneous spaces, i.e. they are not endowed with a group of isometries that acts transitively, where “all the points are the same”. The first method for studying a space form S with curvature 1 is to copy what was done for the tori. We thus start with a compact surface S of genus > 2 and consider its universal covering; endowed with the locally transported metric; it’s thus the geometry of constant curvature 1, simply connected and complete, i.e. the hyperbolic plane Hyp2 . We obtain S “below” by taking the quotient of Hyp2 by a suitable group of isometries; but the groups are very difficult to “see”, even though they are fundamental and have been much studied (and it hasn’t ended, Maskit, 1988). Let us say only, for the geometer, that just as the tori are obtained by identification of opposite sides of a parallelogram, here we can find in Hyp2 polygonal domains with 4 sides that



yield S by appropriate pairwise identification of their sides. It is, moreover, a way of determining the difficult classification of compact surfaces (see Sect. VI.1) . But there is a much simpler way of constructing all the space forms with negative curvature, it is the pants method. An important result in Hyp2 is to show that there exist hexagons for which the six angles are right angles and for which the three sides x, y, z can be given; then we have formulas that show that the three remaining sides a, b, c are determined.

Fig. VI.XYZ.1. Construction and study of abstract surfaces with constant negative curvature by the pants method

An supplementary exercise, useless here but see Buser (1992) for a prospective proof and important applications, is the concurrence of the three altitudes of all the hexagons with six right angles. We return to the construction; we can then take a pair of such hexagons and glue them along their corresponding sides (see figure) to obtain pants, which is a surface with boundary (but always with constant curvature 1), for which the boundary is formed by three circles (which are geodesics, here hyperbolic lines) of equal lengths. Our boundary curves are finally singularityfree because the angles of these hexagons all equal 2 . But now we can fasten such pants together in arbitrary number to obtain all the types of surfaces of genus > 2, these objects being singularity-free surfaces, truly smooth (i.e. of class C1 ), because the boundary curves are hyperbolic lines and hyperbolic geometry is symmetric with respect to each of its lines. This yields some space forms. But we get them all as follows: to see it, it suffices to incise some a priori space form by periodic geodesics that are pairwise nonintersecting; this is not too difficult. This geometric description allows a classification, for which we sketch the outline. We count the number of parameters on which the hexagons with six right angles and certain equalities involving the sides are dependent; then when we assemble along the boundary circles we can make one



pair of pants turn with respect to the other. A careful study leads finally to this result: the space of all our S of a given genus is a set of dimension equal to 6 6. It is called the Teichmüller space (for genus ) after O. Teichmüller, who was the first to study it in depth, in 1939. A detailed exposition of this description with hexagons, then pants, can be found in Buser (1992); but we can study the structure of Teichmüller spaces much further. This still remains a profound subject; see once more Buser (1992). Higher dimensions. Here is an important remark which, although classic, remains little known. The theory of surfaces we have just considered is an important subject, not only because it is classical and treats surfaces of our ordinary space, but also for the following reason. Take in Ed of dimension d greater than 3 any generic hypersurface (in whatever precise sense desired). Then it is easy to see that its type of abstract isometry (as a Riemannian manifold of dimension d 1) determines its second fundamental form as embedded in Ed . This is why: at each point take a basis fei g that diagonalizes the second fundamental form and let i be the eigenvalues (the principal curvatures); then equations that generalize the theorema egregium assert that the sectional curvatures K.ei ; ej / of the abstract Riemannian metric are given by the formulas K.ei ; ej / D i j for all the i 6D j . But as soon as we know the three products = ab, bc, ca of three nonzero real numbers a, b, c, we can calculate these three numbers. Thus in the case where there is practically but very little sectional curvature zero (which is certainly a generic condition), the Riemannian structure determines the second fundamental form (as many eigendirections as eigenvalues), as well as its derivatives. It is then classical these are the general equations called Gauss-Codazzi that these hypotheses completely determine the hypersurface within a global isometry of the whole space; our hypersurfaces are thus “the same”. The above remark goes back to Killing (1885), it is found in Thomas (1935) and on p. 42 of Volume II of Kobayashi and Nomizu (1963–1969). Matters are completely different when d D 3; for we know only that K D 1 2 and there is thus room for stimulating questions which, as we have seen, are not yet settled. Bibliography [B] Berger, M. (1987, 2009). Geometry I, II. Berlin/Heidelberg/New York: Springer [BG] Berger, M., & Gostiaux, B. (1987). Differential geometry: Manifolds, curves and surfaces. Berlin/Heidelberg/New York: Springer (1991) Biographical Dictionary of Mathematicians. New York: Charles Scribner’s Sons Appell, P., & Lacour, E. (1922). Fonctions elliptiques. Paris: Gauthier-Villars Arnold, V. (1978). Mathematical methods of classical mechanics. Berlin/Heidelberg/New York: Springer Arnold, V. (1983). Singularities of systems of rays. Russian mathematica, 38(2), 83–176 Arnold, V.I. (1994). Topological invariants of plane curves and caustics. Providence, RI: American Mathematical Society



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Chapter VII

Convexity and convex sets

VII.1. History and introduction The history of convexity is rather astonishing, even paradoxical, and we explain why. On the one hand, the notion of convexity is extremely natural, so much so that we find it, for example, in works on art and anatomy without it being defined. Below are two excerpts, one from a book on art (1985) illustrating a modern sculpture; the other, from a classic anatomy reference (Rouvière), describes the extremely subtle overlapping of menisci in the knee. We also find in the same work a description of the aortic arch, also very complex, which uses the words “concave” and “convex” several times. Intra-articular menisci or crescent shaped fibrocartilage. “So arranged, the tibial plateaus are not adapted to the femoral condyles. Congruence is achieved by the interposition between the tibia and the femur of intra-articular menisci or crescent shaped fibrocartilage.

Fig. VII.1.1.

The crescent shaped fibrocartilage is differentiated into medial or lateral. Each of them is a triangular, prismatic lamella curved to form a crescent. They are recognized as follows: a concave upper face in rapport with the femoral condyles; – a lower face, applied to the periphery of the corresponding tibial plateau; an external face or periphery) (base of the prism), convex, very dense, adhering to the articular capsule; – an internal (or central) side, concave, sharp, and whose concavity conforms to the center of the tibial plateau; – finally, two extremities or horns, whence the ligament fasciae” (Rouvière, 1973.)

M. Berger, Geometry Revealed, DOI 10.1007/978-3-540-70997-8_7, c Springer-Verlag Berlin Heidelberg 2010




“Women were a favorite subject of Achipenko. This sculpture reflects the influence of cubism in the optical juxtaposition of concave and convex forms and in the use of the void to give the impression of massiveness.”

Fig. VII.1.2. A. Archipenko, 1914. Woman combing her hair, bronze. Israel Museum, Jerusalem

On the other hand, the notion of convexity didn’t appear “seriously” in mathematics until Minkowski at the turn of the twentieth century. A striking case is that the book Hilbert and Cohn-Vossen (1952), with its astonishing title and which is a



bit our model (see its introduction), doesn’t contain a single word on convexity. For a rather detailed history of the subject, see Chap. 0 of Gruber and Wills (1993). Here we give a brief summary. We find the notion of convexity with Archimedes (250 B.C.), with Newton (1720) with respect to the classification of singularities of algebraic curves, with Poinsot (toward 1800) regarding statics (see Fig.VII.1.3) and the notion of a sustentation polygon, with Fourier where we find the seeds of linear programming. Then comes Minkowski; we will amply encounter this great mathematician in Chaps. IX and X. Apart from specific results, he had an entire program, but he died very young. Between Minkowski and the 1960s, we find little about convexity in the literature, with the notable exception of an important work on polyhedra, Steinitz and Rademacher (1934); see the next chapter. After the 1960s, convexity suddenly took on considerable importance. Here is how Grünbaum, in the introduction to Grünbaum (1993), explains this “hollow”, this abyss. On the one hand, the problems that arose naturally were too difficult for their time there were not enough powerful tools; on the other hand, the demand for such results had been very strong, from at least three directions: harmonic analysis, probability theory and linear programming. We will return to the history in the next chapter. Convexity in physics appears, among other places, in the book Israel (1979); read the introduction there by Whigtman. The ideas and results in this domain are, however, a real jungle, we don’t have the courage to traverse them completely; we will just make a quick safari, the locations chosen by two criteria: that they enter into the “Jacob’s ladder” vision of our book and that they provide aesthetically satisfying examples. We need to emphasize that to a large extent we will work in arbitrary dimension and not just the visible 2 and 3; and even more: we will be interested in how various concepts behave when the dimension becomes very large. One reason for this is that function spaces, in harmonic analysis, are of infinite dimension, and yet it is anticipated that we can grasp them, attain intuition for them, by looking at spaces which are of increasingly large finite dimension. See the representative Giannopoulos and Milman (2004). Another reason is that contemporary applications of linear programming require not just hundreds, but even thousands of variables. We will encounter large dimensions, with applications in the forefront; and, in Chap. X, the very recent proof of Kepler’s conjecture. We add that the idea of convexity has incredible power, its applications are legion; we will see this right away in Sect. VII.2 with applications that are simultaneously spectacular and of disarming simplicity. But we will also see some astonishing illustrations of our theme: results whose statements are elementary but whose proofs (at least to this day) very difficult, and problems whose statements are equally elementary, but which are still unresolved. Regarding references, here first is a partial list of classic works on the subject: [B], Chaps. 11 and 12 of Eggleston (1958), Valentine (1964) and Leichtweiss (1980). But since the discipline has taken off, it is impossible to write a book that treats all aspects. We find an exception in Gruber and Wills (1993), which is of a sort that gives “reports” and is therefore without detailed treatment. It may be consulted regularly



Fig. VII.1.3.

for the nonclassical subjects about which we will speak. At the time of its appearance, this work was a perfectly up-to-date and complete reference; it remains an incomparable source of synthesis. Subsequent to the older works listed, there have appeared several monographs: Gardner (1995) and Schneider (1993). For polytopes more specifically see Chap. VIII we can refer to Ziegler (1995). A good part of what we will treat is very well synthesized in Giannopoulos and Milman (2001). The structure of this chapter, the choices we have made in this immense subject, reflect our taste for this or that topic, result, problem, etc. In Sect. VII.13 (“Miscellaneous”) we very rapidly survey various topics that seem difficult to pass by in silence and say again that Gruber and Wills (1993) is the ideal reference for completing our little excursion when that is needed; also Gruber (1996, 2007). VII.2. Convex functions, examples and first applications A function f (with real values) and defined on an (arbitrary: open, closed, infinite) interval I is called convex if: f .x C .1 /y/ 6 f .x/ C .1 /f .y/ for each x; y 2 I and each 2 Œ0; 1: A function f is called strictly convex if we have strict inequality throughout the above (for x 6D y and 2 0; 1Œ). For the geometry we read this definition on the graph of the function: for any two points of the graph, the portion of the graph between these two points lies below the segment of the line that joins them. The opposing definitions of concave and strictly concave are left to the reader. Three trivialities: a convex function attains its maximum at an extremity of its interval of definition; a strictly convex function attains its minimum at a single point; P P the convexity of f implies P f . i i xi / 6 i i f .xi / for all finite sequences fxi g I and fi g such that i i D 1 and i > 0 for all the i . Two items that are a little less evident, yet simple enough with differential calculus: a convex function is continuous on the interior of its interval of definition (but note that at the endpoints of this interval it can attain any sufficiently large value); a convex function has right and left derivatives at each point on the interior of its interval of definition. Moreover, the points at which it fails to be differentiable are denumerable.



Fig. VII.2.1.

Finally, an essential criterion for convexity: is f is twice differentiable and if f 00 > 0, then f is convex (and mutatis mutandis for strict convexity). Here are two classic and essential examples. The first is f D log x on RC , whose second derivative is x12 and thus clearly positive. For all i as above and the ai > 0 P Q 0 we thus have i i ai > i ai i , for example a C 0 b > a b for C 0 D 1. In 1 particular, for each integer n and taking all the i equal to n , we find a C C a n 1 n > a1 : : : an : .VII:2:1/ n If this inequality (often called Newton’s inequality) is trivial for n D 2, the reader will gain more respect for it starting with n D 3 by attempting it without convexity! We already used it in Sect. VI.11 and will encounter it again in the following section. For the second example we take f D x p (for all real p > 1). A classical calculation, but a bit long and tricky (see as needed the references given) shows that the convexity of x p implies the inequalities: (VII.2.2) Hölder inequality: X i


xi yi 6



1=p X


1 1 C D 1 (positive numbers). p q







When p D 2 (and thus q D 2) we know this inequality well from Euclidean geometry: in En : hx; yi 6 kxkkyk. And here the triangle inequality is generalized by: (VII.2.3) Minkowski’s inequality: 1=p X 1=p X 1=p X jxi C yi jp 6 jxi jp C jyi jp i



(note that the plus sign on the left may be replaced by a minus sign). The above inequalities are nowadays used constantly in practically all of analysis. In fact, first they extend to integrals, which replace the finite sequences, for which we can use For example an arbitrary measure: Z Z 1=p Z 1=q 1 1 p f (VII.2.4) fg 6 gq (always with C D 1). p q For p D q D 2 we find the classical Schwarz inequality. Next, they imply that, for 1=p R is a proper norm. We thus have function spaces at our disposal, functions, f p called Lp spaces, which are indispensable in numerous convergence problems, for example for solutions of partial differential equations. The inequality (VII.2.4) extends trivially to arbitrary finite products; we will use these in an essential way for sections of the cube in Sect. VII.11.C. Here are three geometric examples that illustrate the power of the notion of a convex function. The first is the isoperimetric inequality for spherical triangles; see Sect. III.1. We saw in Sect. III.5 that such an inequality is open starting with 3-dimensional spheres. Here we use the formula (see for example 18.6 of [B]) Area r p pa pb pc tan D tan tan tan tan 4 2 2 2 2 and the fact that the function log ı tan is strictly concave. We obtain tan

pa pb pc p 2 ; tan tan 6 tan 2 2 2 6

with equality if and only if a D b D c. In the second example, we attempt to resolve problems of the “isoperimetric” type for polygons with a fixed number of sides that are inscribed in a given circle (for the considerably more difficult general problem, where there is no such circle, see P Sect. VIII.3). We let ˛i denote the half-angles at the center of these sides, thus i ˛i D . We look for polygons maximal P of maximal area and for those ofP perimeter. The area of the polygon is i cos ˛i sin ˛i and the perimeter 2 i sin ˛i (having normalized the radius of the circle to 1). We obtain the given result by using the strict concavity of the function sin and that of the product sin cos on Œ0; =2: in the two cases the maximum is attained for regular polygons and for them only.



Fig. VII.2.2. Four sides: to maximizePthe perimeter or the area;P six sides: to minimize the perimeter or the area; n sides: ˛i D , perimeter D 2 sin ˛i 6 2n sin n

Thirdly, we pose the same for polygons circ*mscribed about a circle. P problems The area this time equals i tan ˛2i and we proceed in the same fashion. Furthermore, for the perimeter it suffices to remark that, for all polygons circ*mscribed about the unit circle, we have area = 12 perimeter. As to what happens when we pass from inscribed to circ*mscribed, we need to be aware even if we are experts on duality that the areas of the inscribed polygon and its circ*mscribed dual are not reciprocals of each other. For this essential problem of “volume and duality”, see Sect. V.8. VII.3. Convex functions of several variables, an important example Suppose now that we want to define convexity for functions of several variables, i.e. for functions D ! R where D is a subset of Rn . We then want to write a condition in x C .1 /y, where x; y 2 D. This condition cannot therefore have the sense that all x C .1 /y are again in D. Observe that these combinations of points are an affine notion (see Sect. I.1), which leads us entirely naturally to the definition: a subset D of a real affine space A is called convex if x C .1 /y 2 D for arbitrary x; y 2 D and 2 Œ0; 1. The notion of strict convexity for sets is more difficult to define; see Sect. VII.10.A and classical references. In this work, we only consider real affine spaces of finite dimension, thus the spaces Rn . We employ interchangeably the terms convex set and convex body. A function f defined on a convex set D is said to be convex if f .x C .1 /y/ 6 f .x/ C .1 /f .y/ for arbitrary x; y 2 D and 2 Œ0; 1. There is an analogous definition for strict convexity. Convexity is therefore an affine notion, there is no a priori Euclidean structure. As in the single variable case, a strictly convex function attains its minimum at a unique point. A convex function is continuous at each point interior to its domain D



Fig. VII.3.1.

of definition. We will have no need of it, but it is important for a good understanding of the meaning of convexity: a convex function is almost everywhere differentiable in the interior of D, whereas in general there is no second derivative. With respect to convexity criteria, we have here the analogue of the second derivative criterion in the one-variable case: if the function f has on its convex domain of definition D a second derivative f 00 which, in its guise as a quadratic form is positive or zero (resp. strictly positive), then the function f is convex (resp. strictly convex). In 2f coordinate language, the condition is that the matrix of the @x@i @x is symmetric j with nonnegative (resp. strictly positive) eigenvalues; see also Sect. VII.13. A very complete reference for the differentiability of convex functions is Rockafellar and Wets (1998). There is also the classic Roberts and Varberg (1973), then Giles (1982), but the simplest for a first reading is the synthesis Gruber and Wills (1993, Sect. 4.2). We should be wary that things may be worse than in the single variable case. Practically anything at all can happen at the boundary. A typical example consists in taking D to be the unit disk and a function f that is identically zero on the interior and which takes on arbitrary nonnegative values on the boundary. Here is a subtle example essential for the geometry of convex sets of a strictly convex function; see Sect. VII.10.C. The convex set of definition will be the set Q of all positive definite quadratic forms defined on Ed , a part of Rd.d C1/=2 . To each q 2 Q we can attach its determinant det.q/. This is the determinant of the matrix representing q in the canonical basis or any orthonormal basis. It’s also the product of the eigenvalues of q. We will subsequently use the geometric interpretation of this determinant: p Within a constant (dependent only on the dimension), we have: 1= det.q/ D .det.q//1=2 is equal to the volume of the (here always solid) ellipsoid q 1 .Œ0; 1/. P We first verify that: if we diagonalize q in q D i ai xi2 , then q 1 .Œ0; 1/ is an p ellipsoid whose axes have lengthsQequal to the 1= ai and for which we will see in p Sect. VII.6.B have volume ˇ.d / i 1= ai , where ˇ.d / denotes the volume of the unit ball of Ed . This is indeed what we wanted.



Now: the function q 7! det.q/1=2 is strictly convex on Q. To see this we diagonalize simultaneously two forms q and q 0 ; with the obvious prime notation we have, successively: i1=2 1=2 hY det.q C 0 q 0 / D .ai C 0 ai0 / i



ai ai0



0 D .det.q/1=2 .det.q 0 /1=2 6 .det.q/1=2 C 0 .det.q 0 /1=2 ; 0

where we have applied the inequality aC0 b > a b , C0 D 1 from Sect. VII.2 several times. VII.4. Examples of convex sets Here now are some essential examples of convex sets. For all of these we are in an unspecified affine space. First a cautionary example, suggested by the function above. An open disk in the plane to which we adjoin an arbitrary portion of the boundary circle always remains convex. In contrast, we cannot do the same with a square (verify this for yourself!).

Fig. VII.4.1.

This is the moment to remark that a convex set need neither be open nor closed. However, in the majority of cases we will work with closed convex sets; in fact they will most often be compact. Two basic examples: a half-space (open or closed) associated with a hyperplane, and the intersection of two arbitrary convex sets; in fact any intersection of convex sets from any collection, not necessarily countable in number, remains convex. This shows that, whatever the subset D, there exists a smallest convex set that contains D, called the convex envelope of D. For more on the relation to the halfspaces, see Sect. VII.10; we should remember for now that a closed convex set can be realized as a denumerable intersection of closed half-spaces.



Fig. VII.4.2. Convex envelopes

Two examples are important for the sequel, and in a sense they are two extremes. P x2 First, the solid ellipsoids: in each dimension the inequality i ai2 6 1 defines a i

compact convex set, where all the ai are taken positive. To prove it, we give an argument essential in itself, i.e. that from the affine viewpoint (the dimension being fixed) P 2 all ellipsoids are the same. In particular, they are equivalent to the closed ball i xi 6 1, which is convex because of the triangle inequality. The ellipsoids play a pivotal role in the theory of convexity and in its applications. We find them notably in Sect. VII.10.C. They may also be characterized by a number of their properties, typically by having an affine symmetry for each linear direction; see Sect. VII.10.C. We can say that they are the most beautiful of convex sets. All of 11.1.3 of Gruber and Wills (1993) is dedicated to them.

Fig. VII.4.3.

At the other extreme we find the (solid) parallelepipeds: we choose any basis and denote by xi the associated coordinates; then the associated parallelepiped is Q the set f.x1 ; : : : ; xd / W 0 6 xi 6 1 for all i g D i Œ0; 1 D Œ0; 1d . To the affine



eye here too all parallelepipeds are the same and identical (affine-isomorphic) to the cube Œ0; 1d of the Euclidean space Ed . We may consider them as being, in several senses, the “worst” among the convex sets; see Sect. VII.10.D. We have just suggested that the ellipsoids are in some way “the most beautiful” of the convex sets, the “best”, whereas the parallelepipeds would be the least round, the “worst”. We are going to show in the sequel that this is indeed the case, and in several different respects. Nevertheless, for this we will need to restrict ourselves to a fundamental class: the center-symmetric convex sets. These are the convex sets C for which there exists a point p of the affine space such that C is preserved by the affine symmetry with center p. Ellipsoids and cubes are center symmetric, in contrast to the worst (or the simplest and most natural) of convex sets: the simplex. A simplex of Rd is what generalizes a triangle in the plane and the tetrahedra of three-dimensional space: it is by definition the convex envelope of a set of d C 1 points of an affine space of dimension d which are affinely independent, which is to say that if this set is fxi g (i D 0; : : : ; d ), then the vectors xi x0 , i D 1; : : : ; d , form aPbasis of the associated vector space. Otherwise expressed, the two inequalities P x D p and D 1 determine the i uniquely. i i i i i

Fig. VII.4.4.

It is correct to say the simplex of a given dimension, for affinely they are all equivalent. The parallelepipeds and the simplexes are two particular instances of polytopes, which generalize to arbitrary dimensions the polygons in the plane and the polyhedra in space. The definition is trivial: a convex polytope is the envelope of a finite number of points of the space. We devote all of Chap. VIII to them. In fact, despite spectacular progress since 1960, there still remain plenty of open problems concerning polytopes. There is an enormous difference between the center-symmetric convex sets and convex sets in general; we will see this in various places; the polytopes are no exception. Our final example to the Lp . We consider the set of Rd defined, for a P corresponds p real p > 1, by i jxi j 6 1 (denoted in the sequel by the expression Ball.d I p/).



For p D 2 we find the ordinary (closed) Euclidean balls and for p D 1 we find the cocube; we will see in Sect. VII.5 that this is the polytope dual to the cube. In the plane this amounts to a square (but with diagonal verticals), in space to an octahedron (regular in Ed ). For the other p (p > 2), we find sets containing the unit ball and contained therein. It isn’t trivial to see geometrically that they are really convex sets (try it!), but it is an immediate consequence of the Minkowski inequality seen in Sect. VII.2. The cube corresponds to infinite p (in this case we P 1=p replace jxi jp by its limit, which is sup jxi j).

Fig. VII.4.5.

VII.5. Three essential operations on convex sets Geometers too must be knowledgeable and have the courage to introduce invariants and concepts of a more or less algebraic nature in order to make progress. We will identify some rungs of the ladder, three beyond a “zero-th” rung, which is the sectioning of a convex set by an affine subspace (of any dimension): it always yields another convex set. However, we will see in Sect. VII.11 that “slicing” (sectioning by hyperplanes) reveals as many difficulties as surprises, even for the simplest convex sets. Here now are three much less trivial operations. VII.5.A. The (Steiner, Schwarz) symmetrizations The technique of symmetrization has numerous applications; we will see it enter into the isoperimetric inequality in Sect. VII.7. With it we prove the fact (seen in Sect. I.1) that when d C 2 points are thrown at random at a convex set, it is when that set is an ellipsoid that the probability that they form the vertices of a simplex is greatest. The ellipsoids also have an extreme property for the behavior of the product of the volume of a convex set and that of its polar; see Sect. VII.9. In Sect. V.11 we encountered Steiner’s symmetrization for a plane curve and a linear direction. It can be defined just as well in arbitrary dimension and we present it in the affine context. We take hyperplane H and a direction ı transverse to H. To



each portion P of space we associate its (Steiner) symmetrization symH;ı .P/ defined as in the figure:

Fig. VII.5.1.

This says that, for each line D parallel to ı, the part D \ symH;ı .P/ is a segment of D having its midpoint at H \ D and length equal to the length of D \ P. By construction, our symmetrized set is certainly symmetric with respect to the hyperplane H for the direction ı. In the Euclidean case we always tacitly choose ı to be the direction orthogonal to H and use the notation symH . It is easy to see that each Steiner symmetrization preserves convexity: if D is convex, then so is symH;ı .P/. A sketch and the word trapezoid suffice for seeing it.

Fig. VII.5.2. Symmetrization preserves convexity and can only reduce the diameter



The formal result is that in a trapezoid, parallel to the base, the section length is an affine function. It follows that the length of the sections of the convex set P that are parallel to ı, considered as a function defined on H, is clearly a concave function. A brief remark for the purist: there is no need of a Euclidean structure in order to speak of the length of these parallel segments, it suffices to have a unit length in the ı direction. The Schwarz symmetrization is less evident. We place ourselves in dimension 3 and in the Euclidean context. We take a line D of E3 and slice the set P considered by planes H perpendicular to D, which yields convex sets that have an area; we then construct, centered at the point D \ H, the disk which has the same area as P \ H. The union of disks so constructed is a set in E3 , denoted symD .P/ and called the Schwarz symmetrization about D. By construction it is a set of revolution about the axis D, but is it a convex set once P is convex? A bad surprise (that will ultimately become good) awaits us: this seems almost impossible to prove. In fact, we have to show that the function defined on D by the radius of the disks in question is concave. We thus need to see if the function given by the square root of the area of the sections P \ H is concave on the line D.

Fig. VII.5.3.

The little trapezium game seen above now becomes this: let P \ H, P \ H0 be two sections corresponding to two points m, m0 of D. We need to study the section P \ H00 corresponding to the point m00 D m C m0 of D, where C D 1, > 0, > 0. The convexity of P tells us only that P \ H00 contains all the points x Cy, where x ranges over P\H and y over the convex set P\H0 . This operation associated with , and two convex sets (here P\H and P\H0 ) is due to Minkowski and will be the subject of the next subsection. Its motivation thus arises completely naturally. As for our problem of the concavity of the square root of the area, it will be treated in Sect. VII.8: it’s the Brunn-Minkowski Theorem, in our opinion one



of the two fundamental theorems of convexity. It remains difficult to prove, as we will amply see. The second fundamental theorem, simpler to state and deceptively evident, is that of Hahn-Banach; see Sect. VII.10.A. It implies, among innumerable other consequences, the result which states that there is equivalence between the convex envelopes of finite sets of points and the finite intersections (when they are, in addition, compact) of closed half-spaces. There would scarcely be any polyhedra without this result, nor any Chap. VIII. This correspondence is pleasantly precise thanks to polarity; see below. Readers will be able to define without difficulty a general symmetrization symA in Ed which includes those of Steiner and of Schwarz, associated with an affine subspace of any dimension. In fact, a Euclidean structure in any affine direction K transverse to A suffices. Any of these symmetrizations preserves convexity, a consequence of the Brunn-Minkowski inequality (VII.8.1). These symmetrizations, above all when the codimension equals 1 so that the affine subspace is a line and the sections are those by orthogonal hyperplanes are essential in analysis, where they are applied to the graph of a function:

Fig. VII.5.4.

See the applications in Sect. VII.13.C: inequalities of the isoperimetric type, the “fundamental” i.e. lowest frequency of a drum, etc. Observant readers will have noticed that each symmetrization of an arbitrary convex set itself possesses a symmetrization (about the subspaceD). What happens if we continue ad infinitum to symmetrize a convex set, say in the plane, with respect to the directions of different lines? Do we obtain in the limit, supposing that is meaningful, an object that is symmetric with respect to all directions, i.e. a disk? We will see in Sect. VII.10.C that the answer is yes. Only recently has there been interest in the rapidity of convergence of symmetrization. It turns out in fact that very few symmetrizations very quickly produce a convex set that is close to a ball. Things get better as the dimension increases, but readers will be able to program things in the plane and see displayed the rapidity with which almost perfect disks are obtained. For more detail on the evaluation of the rapidity as a function of dimension and of a required precision ", along with proofs, see 7.3 of Giannopoulos and Milman (2001),



Tsolomitis (1996) and Klartag and Milman (2003). This phenomenon is in particular related to that of concentration that will be encountered in Sect. VII.12. VII.5.B. Some algebra of convex sets: Minkowski’s sum Let C and D be two convex sets in the same affine space. Their Minkowski sum C C D is defined as the set fx C y W x 2 C; y 2 Dg; it is trivially convex. This supposes that we have chosen an origin for our affine space so as to make it a vector space. But if we change the origin the result is the same within a translation, thus the “appearance” of the result remains the same. We see that we can define a whole algebra of linear combinations uC C vD, where u, v can be arbitrary real numbers, although the difference C D of two convex sets in particular is much more seldom used than the sum. The figures below give some examples of Minkowski sums.

Fig. VII.5.5. CC 2 is always center-symmetric

Note also that C C D, with C D 1, can be defined intrinsically. With these examples readers will begin to take into cognizance that the Brunn-Minkowski result (Sect. VII.8), which describes the behavior of the volume function under the Minkowski sum, is far from being evident. A basic example, that we present for simplicity in the plane, is the sum of an arbitrary convex set C and a (henceforth



small) disk D.R/ of radius R: then C C D.R/ is nothing other than the tubular neighborhood TubR .C/ of radius R of C, that is to say the set of points in the plane a distance from C less than or equal to R (to see this, vectorize the space at the center of the disk). In fact, in this example C can be an arbitrary portion of the plane, for example a curve. We will encounter these TubR .C/ in all dimensions in dealing with the isoperimetric inequality in Sects. VII.8 and VII.14. As an exercise we propose that readers try to identify the convex set 2C C. The dangers of intuition. If we look at the Minkowski sum of a polygon and a disk, we see that this sum has a regularizing effect. Readers who know about regularization by way of convolution of functions may think that it is the same here. Well, what happens is just opposite, but this wasn’t discovered until Kiselman (1987), where we find the following double assertion in the case of the plane. On the one hand the Minkowski sum of two planar convex sets, for which the boundary is C1 , 4 6 is not C1 in general, for example for the epigraphs of the functions x4 and x6 ; their 6

sum is the epigraph of the function x6 34 jxj20=3 C f , where f is of class C1 ; thus the differentiability here does not exceed 6 and thus is not infinite. It is furthermore the worst possible result, since also the boundary of the sum of two epigraphs of class C1 is always of class C20=3 . (A function is said to be of class CkC˛ , with k positive integral and ˛ 2 0; 1Œ, when its k-th derivative exists and satisfies the Hölder condition of order ˛: jf .k/ .x/ f .k/ .y/j 6 Cjx yj˛ for all x, y and a constant C > 0.)

Fig. VII.5.6.

VII.5.C. A duality: polarity In Sect. IV.2 we encountered polarity with respect to a circle in the Euclidean plane, which we may suppose to be the unit circle of E2 . This extends in any dimension d to Ed without difficulty: we define a bijection between the set of all points other than the origin onto the set of hyperplanes that don’t contain the origin by x 7! fy W x y D 1g, called the polar hyperplane of the point x; and x is called the pole of the corresponding hyperplane. We speak here of polarity with respect to a sphere, here the unit sphere. The incidence properties are trivial in view of the bilinearity of the scalar product. In algebraic geometry see Sect. V.13 we can



thus define the dual curve of a given curve. Here we are going to apply this polarity to compact convex sets whose interior contains the origin. For each set C of Ed we call the polar C0 of C the part of Ed defined by 0 C D fy W x y 6 1 for any x 2 Cg. Thus: If C is compact and contains the origin in its interior, then C0 is a compact convex set containing the origin in its interior. It’s a good duality: .C0 /0 D C for each convex set in the collection of all compact convex sets containing the origin in their interiors. Set inclusions are reversed: if C C0 , then C0 C00 . For the intersection C \ C0 , the polar .C \ C0 /0 is the convex envelope of the union C0 [ C00 .

Fig. VII.5.7. Polarity. Above, we see that the flat parts of K correspond to angular points of K0 , and conversely. Below, detail of the correspondence

Fig. VII.5.8. Polarity. If K doesn’t contain the origin in its interior, then K0 isn’t bounded


Fig. VII.5.9.

P xi2

i ai 6 1

has for polar



2 i ai xi 6 1

Here are some comments on the above figures. Briefly, they borrow certain concepts from Sect. VII.6. If a set is strictly convex, then its polar is also. The essential fact is that the hyperplane polar to a point on the boundary is tangent to the boundary of the polar, and this at a point that is the pole of the hyperplane tangent to the initial convex set at the point of the boundary considered. The situation that is completely opposite the strict case (smooth boundary, see Sect. VII.13.D) is that of (compact) polyhedra: the vertices of a polyhedron correspond to the hyperplane faces of its polar (and vice versa). In the intermediate cases (see Fig. VII.5.7) flat parts of the boundary correspond to the angular points. Finally we give some explicit examples. The unit ball is its own polar, more P x2 P generally the polar of a (solid) ellipsoid i ai2 6 1 is the ellipsoid i ai2 xi2 6 1. i

This is a particular case of the fact that polarity preserves central symmetry. But it remains true that the polar of an ellipsoid not centered at the origin (but still containing it in its interior) is again an ellipsoid. It is less evident, but results from considerations (to be generalized) on duality with respect to conics; look in Sect. IV.4. The polar of the cube Œ1; 1d , centered P at the origin, is the polytope that generalizes the regular octahedron in E3 , i.e. i jxi j 6 1. In the planar case, this remains exceptionally a square, but from dimension 3 on it is another type of polytope essential different, as we will see. We will call it the cocube; (co is a prefix used rather systematically for dual objects and properties); we denote it by Cocubd . The correspondence between faces and vertices is easily described. As for the polar of a simplex, it’s always a simplex, regularity being preserved. This duality between faces and vertices is not difficult to establish. On the other P hand, its generalization to the fact that the polar of i jxi jp 6 1 (for all real p > 1) P is i jxi jq 6 1 with p1 C q1 D 1 requires the use of the Minkowski inequality seen in Sect. VII.2. Don’t fail to observePthat the case of cube and cocube correspond here to p D 1 and q D 1. The i jxi jp 6 1 are called the unit balls for Lp , P because they are the unit balls for the metrics d.x; y/ D . i jxi yi jp /1=p of



Fig. VII.5.10. Dual of the square, dual of the cube

Fig. VII.5.11. Duality of the dodecahedron (20 vertices, 12 faces, 30 edges) and the icosahedron (12 vertices, 20 faces, 30 edges)

Rd ; the classical notation for these spaces is `p or `dp ; we employ subsequently the notation Ball.d I p/ for their unit ball (so as not to overburden readers’ memories; see Fig. VII.4.5). VII.6. Volume and area of (compacts) convex sets, classical volumes: Can the volume be calculated in polynomial time? For compact convex sets, there exists a good notion of volume. We have been diffident up until now about notions of volume, of measure. We will remain so, even though the problem of the notion of measure can be considered part of elementary geometry. But we need to climb a bit up the ladder in order to see things clearly. In the general case, arbitrary parts of Euclidean space (more generally of an affine space, of a sphere, of a Riemannian manifold) are not measurable. But all “reasonable” sets are measurable. The notion of measure languished a very long time before finally being clarified by Lebesgue in 1902 . Let us retain this: for the compact convex sets of a Euclidean space there is a good notion of measure. Moreover, it can be constructed in an elementary manner; see Sect. VII.6.C and, for more details, 12.2.5 and 12.9.3 of [B]. Furthermore, it is an affine notion as soon as we fix a unit, e.g. by choosing a vectorial basis and proclaiming that the cube constructed on it has volume 1.



On the other hand, for wild objects in Ed , e.g. fractals, etc., various notions of measure are possible; the one most used after Lebesgue measure is that of Hausdorff (related to the notion of Hausdorff dimension). Possible references are: Federer (difficult reading), Falconer (1990), Feder (1988), David and Semmes (1997), Morgan (1988–2000). In fact, just as for convex sets, all these possible notions of measure coincide for good objects, typically surfaces in E3 or more generally submanifolds of arbitrary dimension in Ed . Technically, we use the tools and notations ofRintegral calculus, as we have done starting with Sect. I.2. The volume of C will be C dx1 : : : dxd , provided the coordinates fxi g give the unit measure, and will be denoted by Vol.C/. We make free use of Fubini’s theorem and a change of variables, which permits us to make calculations recurrently. The typical example is that of a pyramid C of altitude h and base a convex set B in a hyperplane (but it could be any measurable subset of the hyperplane); thus Z h d 1 h t .VII:6:1/ Vol.C/ D Vol.B/ dt D Vol.B/: h d 0

R d 1 Fig. VII.6.1. Vol.C/ D 0h ht Vol.B/ dt D dh Vol.B/

And, to be sure, the volume is a d -dimensional object if, for each scalar under hom*othety with ratio , we have Vol.C/ D d Vol.C/. What is important, and new in spirit, is that most of the time we will be interested not in exact values of volumes, but in their asymptotic behavior when the dimension tends toward infinity. There are numerous motivations for wanting asymptotics and high dimensions. From the “pure” side, the spaces of analysis (function spaces) are of infinite dimension and we can hope to gain some insight by approximations by spaces of finite, but increasingly greater, dimension. From the “applied” side, physics and in particular statistical mechanics treats spaces whose dimensions are of the order of Avogadro’s number (1023 ). We will also encounter large dimensions in Chap. X. VII.6.A. Volume of cubes, cocubes and simplexes The volume of the cube Cubd D Œ0; 1d is 1 and that of Œ1; 1d is 2d . For a general parallelepiped generated by the vectors fxi g of Ed , it is useful to know that



p the volume is equal to det.xi :xj /, the square root of the Grammian determinant. We might think that the volume of a cube grows quickly with the dimension; this is certainly true, things are exponential, but we will see that this is fairly modest growth in this realm. d In fact the volume of Cocubd (polar of Œ1; 1d ) is equal to 2d Š as can be seen by decomposing it into 2d pieces associated with the various “quadrants”, where each coordinate has a specific sign, and because the formula for the pyramid yields d Š when applied recurrently. This volume tends extremely quickly toward zero when the dimension tends toward infinity; see in fact Stirling’s formula below (VII.6.B.2). We will specify this “roughly” by an appropriate notation in Sect. VII.6.B below. The term in 2d in the formula for the volume of the cocube is thus completely devoured when d becomes very large. For the simplex it is much worse, things are in d1Š purely. Roughly, if the simplex is of “magnitude D”, then its volume is of order

Dd dŠ

. For the regular simplex, all of p

C1 . Readers can make the whose edges have length equal to 1, the volume equals d Š2dd=2 calculation; it will be good to know that very often the best way of calculating with a regular simplex of dimension d is to embed it in Ed C1 , using the end points of the vectors of the canonical basis as vertices (Fig. VII.4.4); for example the equilateral triangle is then seen in R3 as the set of points such that x1 C x2 C x3 D 1 and x1 > 0, x2 > 0, x3 > 0.

VII.6.B. Balls, spheres and ellipsoids At the other extreme from the (regular) polytopes, we have the compact convex sets, the balls. We denote by ˇ.d / the volume of the unit ball of Ed C1 , by ˛.d / the d -dimensional area of the unit sphere (in the literature !.d / is often seen instead of ˇ.d /). The calculation of these constants is always subtle; the result itself will indicate one of the reasons. First note that the formula (VII.6.1) extends infinitesimally, and if the ball is the “pyramid” with base the sphere and constant altitude equal to 1, we will then have ˇ.d / D ˛.dd1/ . It is therefore sufficient that we calculate either the volumes of balls or the “areas” of spheres. Calculation of the volume of spheres. First a classic trick that we cannot resist mentioning even though we will end up not using it; it consists in calculating R I D Rd exp.x12 xd2 / dx1 : : : dxd in two ways, first by using Fubini directly and then changing to spherical coordinates. We obtain Z 1 d Z 2 exp.x 2 / dx D Vol.Sd 1 /: e d 1 d: ID R

All these integrals are known and expressed in terms of the gamma function, whence ˇ.d / D d=2 = . d2 C 1/. Unfortunately the explicit expression for differs according to whether the input is an integer or half an odd integer. We obtain finally:



ˇ.2k/ D

k ; kŠ


ˇ.2k C 1/ D

2kC1 k : .2k C 1/.2k 1/ : : : 3:1

A wonderful calculation, but where the geometer sees nothing. Let us instead calculate the ˛.d / D Vol.Sd / directly, as geometers and by recurrence, staying at first between the north pole and the equator. That is, we regard Sd as a sort of pyramid on Sd 1 , but here along the sections of constant latitude. Things vary along the latitude, not linearly as in an ordinary pyramid, but as sind 1 . As the initial directions to the north pole describe exactly the unit sphere of the tangent space, we have: R Vol.Sd / D Vol.Sd 1 / 0 sind 1 d . This integral is calculated by recurrence, by integrating twice by parts and by knowing that Vol.S0 / D measure of Œ1; 1 D 2 and that Vol.S1 / D 2. We happily get the preceding result and see again that parity plays an inevitable role.

Fig. VII.6.2. The volume at distance from the north pole is give by sind 1 d , and simply by sin d in the case of the ordinary sphere

It seems that we owe to Herman Gluck the best procedure for calculating Vol.Sd /. We generalize the construction given in Sect. II.5 for S3 , obtained starting with two circles and taking all quarter circles whose endpoints trace out these two circles. Here we construct Sd C2 starting with S1 and Sd with such quarter circles (in algebraic topology it is called the join S1 Sd ). We decompose the integral in dimension d C 2 into three coordinates: that of “x” on S1 , that of “y” on Sd and a coordinate on the quarter circles. The integral element, if ˛ denotes the length computed on the quarter circles, is sind ˛ cos ˛, for we have: Z =2 Vol.S1 .cos ˛// Vol.Sd .sin ˛// d˛ Vol.Sd C2 / D 0

D Vol.S1 / Vol.Sd /



sind cos ˛ d˛ D 0

2 Vol.Sd /; d C1

which is the (second order) recurrence we wanted, while the integral in ˛ was trivial to compute.



Fig. VII.6.3. Representation of the sphere Sd C2 as the join of the sphere Sd with the circle S1 : Sd C2 D S1 Sd

Behavior of the volume. Two things are important here, the first being the behavior of ˇ.d /, and thus also that of ˛.d /, when d tends toward infinity. Stirling’s formula gives it immediately as an asymptotic function of d Š, i.e. d d p (VII.6.B.2 Stirling’s formula) d Š d !1 2d . e A first remark is that ˇ.d / tends very fast toward zero when the dimension tends to infinity. The first values are deceptive, up until 6 they are increasing. Here is a table for a selection of values: d 2 3 4 5 6 7 8 9 10

ˇ.d / 3,14159 4,188 4,93 5,263 5,1677 4,124 4,0587 3,298 2,550

d 11 12 13 14 15 16 17 18 19 20

ˇ.d / 1,884 1,335 0,91 0,56

ˇ.100/ 2:1040

ˇ.1000/ D ?

0,235 0,08 0,02 Table VII.6.1

We leave it to readers to fill the gaps with some computations that are convincing for this “surprising?” result. The fact that there is convergence to zero lets us predict the sketch of the graph of sind : we clearly see that the shaded area tends toward



zero when d tends toward infinity, but the sketch tells us nothing of the speed. Some analysis is necessary, here via Stirling. As the Brunn-Minkowski inequality shows us in Sect. VII.8, the most significant quantity for the volumes of convex sets (situated in a space of dimension d ) is their d -th root. Here we find that .ˇ.d //1=d is of order c p1 , where c is a constant d p independent of the dimension, specifically 2e. To clarify this sort of thing, we use the notation f D g to signify equivalence within a constant when the dimension tends toward infinity (i.e. that the ratio of the two functions f and g tends toward a positive constant). We can specify an index for the constant when there is more than one: 1 , 2 , etc. There will be comparable notations for inequalities: f > g, etc. Stirling’s formula shows that ˇ 1=d .d / D d 1=2 , whereas for the cube and the cocube we have: Vol1=d .Cubd / D 1 and Vol1=d .Cocubd / D d 1 . In Sect. X.5 we will need the exact constant: p 2e 1=d .VII:6:B:3/ ˇ .d / d !1 p : d The second matter, to which we will return in much detail in Sect. VII.12, is even more important. It’s the phenomenon of concentration at the equators: the graph of sind 1 shows not only that the volume tends very quickly toward zero with increasing dimension, but above all that it is concentrated more and more about the equator. For a given fixed ", the tubular neighborhood of radius " about the equator will have more and more relative volume. This implies that what remains after the tube is removed is of smaller and smaller measure (compared to the total measure).

Fig. VII.6.4.

Volume of ellipsoids and a natural question. A simple linear change of variables shows at once that:




X x2 i

i ai2

! 6 1 D ˇ.d /


ai :


Here alert readers will begin to feel uneasy. In fact, since we have said P that the cubes, cocubes and balls are nothing other than particular cases of the sets i jxi jp 6 1 in Lp (denoted above by Ball.d I p/), readers have the right to demand a general P d ; formula in p for Vol. i jxi jp 6 1/. It’s a classic: Vol.Ball.d I p// D 2 .1C1=p/

.1Cd=p/ see e.g. the end of the first chapter of Pisier (1989). VII.6.C. Approximation by polytopes, areas of convex sets An abuse of language is involved: we have seen how to define, for a compact convex subset (with nonempty interior) of Ed (in the Euclidean context that we need here), the generalization to all dimensions of the length of a curve (d D 2) and the area of a surface (d D 3). We have pointed out above that such a notion isn’t unique for arbitrary subsets, even if we specify that we want a .d 1/-dimensional measure. When the boundary @C (the difference between the closure and the interior) of the convex set C is a smooth hypersurface (submanifold of codimension 1; see Sect. V.XYZ), the area enclosed by @C exists and is well defined, for all possible notions of measure. It turns out that this works as well for all compact convex sets, and most importantly all this can be done in a completely elementary way. We use two notations, either Area.C/, or else Vold 1 .@C/ when C is in Ed . We give two ways of proceeding; details can be found in [B]. The first way consists in first showing that every compact convex set can be approximated with arbitrary precision by polytopes. The approximation idea is straightforward and is left to the reader (on the other hand, we will see that the question of the number of vertices that are necessary for obtaining a good approximation is quite a different story). For a polytope P of Ed its area Area.P/ is the sum of the canonical Euclidean volumes of its faces (which are polytopes in Ed 1 ). We then define Area.C/ as the limit of the areas of the polytopes that approximate it. It is of course necessary to show that this limit exists and is independent of the approximations chosen, which isn’t very difficult. In the planar case, we obtain the definition of the length of a curve as the limit of the perimeter of the inscribed polygons, but starting with dimension three; however, if we are dealing not with convex sets, but with objects such as polyhedra or polytopes see for example the Venetian lamp (Fig. VII.14.3) difficulties emerge which are in some sense insurmountable. In fact [B] does not use approximation by polytopes in the definition (because it is difficult to show the independence of the limit), but instead that given by Cauchy’s formula, which states that the area of a convex set is the mean of its projections on hyperplanes when we take all possible directions; see 12.10.2 of [B]. The second method is due to Minkowski: the starting point is the observation that, if a domain C of Ed has a smooth hypersurface as its boundary @C, then the calculation of the volume of its tubular neighborhood Tub" .C/ of radius " satisfies



Vol.Tub" .C// Vol.C/ D Area.@C/ "!0 " lim

. Now for convex sets we know how to write Tub" .C/ “a la Minkowski” as Tub" .C/ D C C "B, where B denotes the unit ball of Ed (see Sect. VII.5.B). Minkowski thus chose as definition: Vol.C C "B/ Vol.C/ : "!0 "

Area.@C/ D Vold 1 .@C/ D lim

This definition of the area of the boundary might be called the painter’s definition. Just as we can calculate the number of sheep by dividing the number of their feet by four, we can calculate the area of a surface by evaluating the volume of paint that is need for covering it (the layer of paint is of almost constant small thickness).

Fig. VII.6.5.

It can be shown that the two definitions given coincide, and here we have a second reason for being interested in the behavior of Vol.ACB/ as a function of Vol.A/ and Vol.B/ (Sect. VII.8). In the following three sections we are going to study the behavior of the volume under the three operations on convex sets introduced in the preceding section. Interested readers may reread the preceding while noting that the area is the first term of the Taylor expansion in " of Vol.C C "B/ and then ask the significance of the subsequent terms. The answer will be given in Sect. VII.13.B. VII.6.D. Mission impossible: calculating the volume of a convex set numerically For an “arbitrary” convex set an explicit formula for its volume clearly doesn’t come into question. In return, for practical problems, we might hope to find effective algorithms for calculating the volume of a given convex set numerically to a good degree of accuracy. Practically, convex sets come provided with two oracles: one tells membership, the other tells separation. This is to say that we have one criterion for deciding whether a point belongs to the convex set, and another for knowing



whether it is in a given half-space. We will see a result that is surprising for the non expert: there does not exist a polynomial algorithm that always permits calculation of the volume of a convex set with arbitrary accuracy, even if we have both oracles at our disposal. We sketch the content of Barany and Füredi (1987), where this phenomenon was discovered for the first time.

Fig. VII.6.6.

At first we concern ourselves only with the oracle of belonging. The starting idea, obligatory, is to take (many) points of C and to calculate the volume of the enveloping convex polytope of these points. It is easy to do this polynomially. The problem now is thus to know whether the volume of the inscribed polytopes provide good approximations to the volume of the convex set. If we approximate each convex set by polytopes, we will clearly have approximations, but how good are they? For approximation by polytopes, we have existence itself from 12.9 of [B]. But we can ask more, even much more, of this approximation; a complete summary of the problem is found in 1.10 of Gruber and Wills (1993). The simplest investigation concerns the number of vertices, effectively the number of points that we have “chosen” on the interior of the convex set. For this it suffices to study the MacBeath numbers for C, defined as: o n Vol.P/ McB.C; n/ D sup W P polytope inscribed in C having n vertices : Vol.C/ P We find in MacBeath (1951) the inequality: For every convex set C we have McB.C/ > McB.ball/. The ball is thus the worst among the convex sets with respect to the approximation of volume by polytopes, which isn’t terribly surprising (compare with the Steiner problem in Sect. I.2). The proof is easy using the Steiner symmetrization, we proceed as in the following section: it suffices to know that a Steiner symmetrization, for any hyperplane direction H, reduces the MacBeath number: for each integer n we have McB.symH .C/; n/ 6 McB.C; n/. This established, it remains to find, in place of the exact value which is completely unknown for an arbitrary integer a good approximation for McB.ball; n/.



And the result is a disaster: a bound that is extremely small as soon as the number of points is no longer exponential in the dimension: 2ea log d d=2 2e log n d=2 McB.ball.Ed /; n/ 6 D ; d d where we have thus set n D d a . The proof consists in reasoning that mixes combinatorics with the geometry of triangulation decompositions for polytopes. Thus we are in the exponential realm and without hope of the polynomial; thi