GEOMETRY REVEALED

Selected works of Marcel Berger

RESEARCH

Le spectre d’une variete riemanniene (with Paul Gauduchon and Edmond Mazet),Springer, 1971(Collaboratively under the pseudonym Arthur L. Besse) Manifolds all of whosegeodesics are closed, Springer, 1978(Collaboratively under the pseudonym Arthur L. Besse) Einstein Manifolds, Springer,1978

PEDAGOGY AND POPULARIZATION

Geometry I, II. Springer, 1987, 2009Differential Geometry: Manifolds, Curves and Surfaces (with Bernard Gostiaux),Springer, 1987“Peut-on definir la geometrie aujourd’hui?” , in Results in Mathematics, vol. 40,pp. 37–87, 2001A Panoramic View of Riemannian Geometry, Springer, 2003Cinq siecles de mathematiques en France, Ed. ADPF (Association pour la diffusionde la pensee francaise), 2005Convexite dans le plan et dans l’espace: de la puissance et de la complexite d’unenotion simple (with Pierre Damphousse), Ellipses, Paris, 2006“Geometry in the 20th century” , 2002, in History of mathematics, edited byV.L. Hansen and J.J. Gray, Encyclopedia of Life Support Systems (EOLSS), Devel-oped under the auspices of UNESCO, Eolss Publishers, Oxford, UK

MARCEL BERGER

Geometry Revealed

A Jacob’s Ladder to Modern HigherGeometry

123

AuthorMarcel BergerInsitut des Hautes Etudes Scientifiques (IHES)Bures-sur-YvetteFrance

TranslatorLester J. SenechalProfessor EmeritusDepartment of MathematicsMount Holyoke CollegeSouth Hadley, MA 10475USAlsenecha@mtholyoke.edu

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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 geometrythat is dear to hearts of physicists is related to it. The elegance of his theorem on the1=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 geometryin France. He maintains contact with the association “Arthur Besse” where he andhis students have written several books: the one on the spectra of Riemannian man-ifolds was for a while the bible of the subject. His book on closed geodesics turnedout to be a scientific adventure story because important problems were solved dur-ing the writing process. The one on Einstein manifolds became a best-seller becauseit appeared at an opportune moment and popularized a little-known subject whosemethods were primitive. It showed great flair, for today these manifolds constitutea common area of research for theoretical physicists and mathematicians of all spe-cialties.

Last but not least, Marcel Berger recognized Mikhael Gromov’s talent and con-vinced him to remain in Paris, which has been a determining factor for the devel-opment of geometry in France. Marcel Berger himself had likewise been the benefi-ciary of the support of André Lichnerowicz, a leading figure in differential geometry

V

VI ABOUT THE AUTHOR

and relativistic mechanics, who gave Berger a thesis topic precisely on the holonomygroups 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 ofstudents. His book Geometry remains unequaled as a survey, disclosing and mod-ernizing all the various points of view that comprise elementary geometry, which ofcourse 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 (1974–1985and 1994–1996), Director of IHES (1985–1994). He was President of the FrenchMathematical Society 1979–1981.

Introduction

Jacob had a dream: and behold, a ladder was placed upon theearth, and it reached to heaven; and behold, the angels of God as-cended 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 thisplace and I did not know it”. . . So Jacob arose early in the morningand took the stone that had served as his pillow and set it up as apillar and poured oil upon it (Genesis 28: 16,18)

Numerous problems of geometry that are quite visual and can be presented in avery 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 � theyrequire 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 pur-poses.

In this work we present a whole series of such problems while showing thenecessity of abstract concepts and how they enter progressively into the solution.These are conceptual notions, each built “above” the preceding and permitting anincrease 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” thatmay seem a priori astonishing, but less so once we understand the totality of the ef-forts and conceptual progress needed for solving similar problems. These classicalproblems are ever the object of vigorous research, all the while mathematical re-search is constantly suggesting new ones. And thus so-called elementary geometryis indeed very much alive and at the very heart of the work of numerous contempo-rary mathematicians.

This book pursues another goal: to show of course the unceasingly renewedvigor of the spirit of geometry, but also to offer readers the elements of a moderngeometric culture. For mathematical instruction in our time presents a disquietingparadox. On the one hand, geometry is increasingly present in daily life; we livein a civilization of images. Virtual reality, robotic vision, aerial navigation and theconquest of space require more and more specialists: engineers, aerial controllers,navigators in space, etc. But at the same time geometry � in any case spatial geom-etry � is almost completely missing from the instructional programs of schools and

VII

VIII INTRODUCTION

universities. The small amount of three-dimensional geometry that nominally stillbelongs to those programs is in fact practically never treated.

It should seem obvious that geometry is of relatively large importance in thewhole of mathematics, but the reality is quite different: the language of geometry,geometrical metaphors, have taken hold throughout as an expedient in modern math-ematics. The most banal of these metaphors consists of calling a set of objects of thesame 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 towarm 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 suchprogress. The impact that it produces seems to come from what might be calleda transfer of intuition,” writes Jean Dieudonné in an article entitled Domination uni-verselle de la géometrie. He adds: “in breaking out of its traditional boundaries, [ge-ometry] has revealed its hidden powers, its flexibility and its extraordinary adapta-tion abilities, thus becoming one of the most used and universal tools in all branchesof mathematics.”

It also happens that a theory breathes new life by renewing contact with ge-ometry in a rather unexpected way, so as Antaeus revived his strengths by contactwith mother Earth: it’s been the case, in recent history, with probability theory, withtopology under the geometrization of Thurston and Gromov, and with that part offunctional analysis which, under Alain Connes’ influence has become noncommu-tative geometry.

For Alain Connes, “a geometer is a person with sufficient vision to be able tocreate sufficient mental images that permit treating varied mathematical problems.”For “what is difficult and essential in mathematics is the creation of enough mentalimages to allow the brain to function.”

An attempt at explanation was given by Michael Atiyah in his 2000 Fields Lec-ture Mathematics in the twentieth century: “Vision ... uses up something like 80 or90 percent of the cortex of the brain. There are about 17 different centers in thebrain, ... some parts are concerned with vertical, some parts with horizontal, someparts with color, perspective, finally some parts are concerned with meaning andinterpretation. Understanding, and making sense of, the world that we see is a veryimportant part of our evolution. Therefore spatial intuition or spatial perception isan enormously powerful tool, and that is why geometry is actually such a powerfulpart of mathematics � not only for things that are obviously geometrical, but evenfor things that are not.”

There remains in the cortex a place for algebra, which Atiyah associates withtime, with the succession of events, of operations. For the geometer, this one-dimensionality evokes the necessity of complying with the logical rules for proof,of metaphorically projecting our intuition onto a single axis. Geometers are oftentempted to reject these rules when it is perceived that they bully the intuition, butknow from experience that it’s precisely these that lead to pushing our imagination

INTRODUCTION IX

beyond its limits so as to create completely new mathematical objects, as we attemptto show in this work.

As for the book’s structure, we have on the one hand grouped these problems bytheir nature, affinity and similarity. For the most part the chapter can be read inde-pendently. Nevertheless, together with our book Geometry [B] for details beyond theconceptual, this work can serve for a course in geometry as seen from the culturalaspect. We can in fact perceive the various chapters as extensions of [B], illuminat-ing it with some very recent results. We have therefore used [B] as a systematicreference for “elementary” geometry. Although biased, this choice is justified sinceonly [B] treats all the notions used here; conversely, for each particular subject thereare 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 havebeen placed at the ends of chapters under the rubric XYZ. Only the crucial ideas andabove all the abstract concepts introduced for attaining these results are elucidated.

In this respect we follow in the steps inaugurated by the absolutely remarkablebook Geometry and the Imagination by Hilbert & Cohn-Vossen (original German,1932, English translation, 1952), which filled a need for modern and easily acces-sible cultural geometry. It is this book that we hope to emulate, for it seems to usthat a modern version is now much needed. This can only be at the price of a hugeincrease in size, given the exponential growth of results since the appearance in Ger-man 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 fromthe first to the last page, but that we rather hoped that the reader would open it atrandom and page through it and plunge into this or that chapter with some pleasuredepending on intuition and inclination. Will we likewise here be able to transmitour conviction that geometry is especially alive and that there are still innumerableways to be explored and concepts to be created?

It is important to state that we have by no means covered all the directions ofcontemporary geometry. Thus we have made but little room for geometric prob-ability, very little for combinatorics and none at all for some recent extensionsof the notions of space and point. For this Cartier (1998) and Chap. 3 of Gro-mov (1999) may be consulted. A good reference for combinatorial geometry isPach and Argawal (1995); we also find much that is well presented in Aignerand Ziegler (1998). A good idea of several new directions in mathematics andcontemporary geometry can be obtained from the recent Carbone, Gromov andPrusinkiewisz (2000).

We should add that mathematics today is advancing extremely fast. We musttherefore alert the reader � and especially the researcher � that we are certainly notcompletely up-to-date in all subjects treated.

This book began as a course given at the University of Pennsylvania in the fall of1994, 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 possiblefor us to offer courses which diverged very noticeably from traditional instruction.

X INTRODUCTION

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 andcarried out important work in its realization. From the scientific point of view,practically all chapters have been painstakingly reviewed and edited by outside ex-perts. These generous colleagues are: Patrick Popescu-Pampu for Chap. VII, DanielMeyer for Chaps. VI and VIII, Pierre Arnoux and Sylvain Gallot for Chaps. XI andXII. Their criticisms and additions, sometimes very detailed, have been essential.André Bellaïche gave the entire text a final critical reading with special attention toChaps. I and IX, where he rewrote several passages. Donal O’Shea provided manyvaluable comments on the early chapters during the translation process. To thesefriends I add with pleasure the name of Lester Senechal, who made the present En-glish translation for Springer-Verlag with great dispatch, considering the compassof the book, and in the course of translation offered numerous remarks, correctionsand criticisms important for the completion of this work.

Bibliography

[B] Berger, M. (1987, 2009) Geometry I,II. Berlin/Heidelberg/New York: SpringerAigner, M., & Ziegler, G. (1998, 4th ed. 2010). Proofs from the book. Berlin/Heidelberg/New York:

SpringerAtiyah, M. (2002). Mathematics in the 20th century. The Bulletin of the London Mathematical

Society, 34, 1–15Carbone, A., Gromov, M., & Prusinkiewisz, P. (2000). Pattern formation in biology and dynamics.

Alghero: World ScientificCartier, P. (1998). La folle journée, de Grothendieck à Kontsevich. Bulletin of the American Math-

ematical Society, 38, 389–408Dieudonné, J. (1980). The universal domination of geometry. Berkeley: International Congress of

Mathematical Education IVDieudonné, J. (1981). Domination universelle de la géométrie (traduction du précédent). IREM de

Paris-NordGromov, M. (1999), Metric structures for Riemannian and non-Riemannian spaces. Basel:

BirkhäuserHilbert, D., & Cohn-Vossen, S. (1932, 1996). Anschauliche geometrie. Berlin/Heidelberg/New

York: SpringerHilbert, D., & Cohn-Vossen, S. (1952). Geometry and the imagination (English translation). New

York: ChelseaPach, J., & Argawal, P. (1995). Combinatorial geometry. New York: Wiley

Table of Contents

About the Author V

Introduction VII

Chapter I. Points and lines in the plane 1

I.1. In which setting and in which plane are we working? And right awayan utterly simple problem of Sylvester about the collinearity of points 1

I.2. Another naive problem of Sylvester, this time on the geometricprobabilities of four points : : : : : : : : : : : : : : : : : : : : : : 6

I.3. The essence of affine geometry and the fundamental theorem : : : : : 12I.4. Three configurations of the affine plane and what has happened to them:

Pappus, Desargues and Perles : : : : : : : : : : : : : : : : : : : : : 17I.5. The irresistible necessity of projective geometry and the construction

of the projective plane : : : : : : : : : : : : : : : : : : : : : : : : : 23I.6. Intermezzo: the projective line and the cross ratio : : : : : : : : : : : 28I.7. Return to the projective plane: continuation and conclusion : : : : : : 31I.8. The complex case and, better still, Sylvester in the complex case:

Serre’s conjecture : : : : : : : : : : : : : : : : : : : : : : : : : : : 40I.9. Three configurations of space (of three dimensions): Reye, Möbius

and Schläfli : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 43I.10. Arrangements of hyperplanes : : : : : : : : : : : : : : : : : : : : : 47I. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 48Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 57

Chapter II. Circles and spheres 61

II.1. Introduction and Borsuk’s conjecture : : : : : : : : : : : : : : : : : 61II.2. A choice of circle configurations and a critical view of them : : : : : 66II.3. A solitary inversion and what can be done with it : : : : : : : : : : 78II.4. How do we compose inversions? First solution: the conformal group

on the disk and the geometry of the hyperbolic plane : : : : : : : : : 82II.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 appearanceof the first fractals : : : : : : : : : : : : : : : : : : : : : : : : : : 87

II.6. Inversion in space: the sextuple and its generalization thanksto the sphere of dimension 3 : : : : : : : : : : : : : : : : : : : : : 91

II.7. Higher up the ladder: the global geometry of circles and spheres : : : 96II.8. Hexagonal packings of circles and conformal representation : : : : : 103

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XII TABLE OF CONTENTS

II.9. Circles of Apollonius : : : : : : : : : : : : : : : : : : : : : : : : : 113II. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 116Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 137

Chapter III. The sphere by itself: can we distribute points on it evenly? 141

III.1. The metric of the sphere and spherical trigonometry : : : : : : : : : 141III.2. The Möbius group: applications : : : : : : : : : : : : : : : : : : : : 147III.3. Mission impossible: to uniformly distribute points on the sphere S2:

ozone, electrons, enemy dictators, golf balls, virology, physics ofcondensed matter : : : : : : : : : : : : : : : : : : : : : : : : : : : 149

III.4. The kissing number of S2, alias the hard problem of the thirteenth sphere 170III.5. Four open problems for the sphere S3 : : : : : : : : : : : : : : : : : 172III.6. A problem of Banach–Ruziewicz: the uniqueness of canonical measure 174III.7. A conceptual approach for the kissing number in arbitrary dimension 175III. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 177Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 178

Chapter IV. Conics and quadrics 181

IV.1. Motivations, a definition parachuted from the ladder, and why : : : 181IV.2. Before Descartes: the real Euclidean conics. Definition and some

classical properties : : : : : : : : : : : : : : : : : : : : : : : : : : 183IV.3. The coming of Descartes and the birth of algebraic geometry : : : : 198IV.4. Real projective theory of conics; duality : : : : : : : : : : : : : : : 200IV.5. Klein’s philosophy comes quite naturally : : : : : : : : : : : : : : 205IV.6. Playing with two conics, necessitating once again complexification : 208IV.7. Complex projective conics and the space of all conics : : : : : : : : 212IV.8. The most beautiful theorem on conics: the Poncelet polygons : : : : 216IV.9. The most difficult theorem on the conics: the 3264 conics of Chasles 226IV.10. The quadrics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 232IV. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 242Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 245

Chapter V. Plane curves 249

V.1. Plain curves and the person in the street: the Jordan curve theorem, theturning tangent theorem and the isoperimetric inequality : : : : : : : 249

V.2. What is a curve? Geometric curves and kinematic curves : : : : : : : 254V.3. The classification of geometric curves and the degree of mappings

of the circle onto itself : : : : : : : : : : : : : : : : : : : : : : : : 257V.4. The Jordan theorem : : : : : : : : : : : : : : : : : : : : : : : : : : 259V.5. The turning tangent theorem and global convexity : : : : : : : : : : 260V.6. Euclidean invariants: length (theorem of the peripheral boulevard)

and curvature (scalar and algebraic): Winding number : : : : : : : : 263

TABLE OF CONTENTS XIII

V.7. The algebraic curvature is a characteristic invariant: manufactureof rulers, control by the curvature : : : : : : : : : : : : : : : : : : : 269

V.8. The four vertex theorem and its converse; an application to physics : 271V.9. Generalizations of the four vertex theorem: Arnold I : : : : : : : : : 278V.10. Toward a classification of closed curves: Whitney and Arnold II : : : 281V.11. Isoperimetric inequality: Steiner’s attempts : : : : : : : : : : : : : : 295V.12. The isoperimetric inequality: proofs on all rungs : : : : : : : : : : : 298V.13. Plane algebraic curves: generalities : : : : : : : : : : : : : : : : : : 305V.14. The cubics, their addition law and abstract elliptic curves : : : : : : 308V.15. Real and Euclidean algebraic curves : : : : : : : : : : : : : : : : : 320V.16. Finite order geometry : : : : : : : : : : : : : : : : : : : : : : : : : 328V. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 331Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 336

Chapter VI. Smooth surfaces 341

VI.1. Which objects are involved and why? Classification of compactsurfaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 341

VI.2. The intrinsic metric and the problem of the shortest path : : : : : : 345VI.3. The geodesics, the cut locus and the recalcitrant ellipsoids : : : : : 347VI.4. An indispensable abstract concept: Riemannian surfaces : : : : : : 357VI.5. Problems of isometries: abstract surfaces versus surfaces of E3 : : : 361VI.6. Local shape of surfaces: the second fundamental form, total curvature

and mean curvature, their geometric interpretation, the theoremaegregium, the manufacture of precise balls : : : : : : : : : : : : : 364

VI.7. What is known about the total curvature (of Gauss) : : : : : : : : : 373VI.8. What we know how to do with the mean curvature, all about soap

bubbles and lead balls : : : : : : : : : : : : : : : : : : : : : : : : 380VI.9. What we don’t entirely know how to do for surfaces : : : : : : : : 386VI.10. Surfaces and genericity : : : : : : : : : : : : : : : : : : : : : : : 391VI.11. The isoperimetric inequality for surfaces : : : : : : : : : : : : : : 397VI. XYZ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 399Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 403

Chapter VII. Convexity and convex sets 409

VII.1. History and introduction : : : : : : : : : : : : : : : : : : : : : : 409VII.2. Convex functions, examples and first applications : : : : : : : : : 412VII.3. Convex functions of several variables, an important example : : : : 415VII.4. Examples of convex sets : : : : : : : : : : : : : : : : : : : : : : 417VII.5. Three essential operations on convex sets : : : : : : : : : : : : : : 420VII.6. Volume and area of (compacts) convex sets, classical volumes:

Can the volume be calculated in polynomial time? : : : : : : : : : 428VII.7. Volume, area, diameter and symmetrizations: first proof

of the isoperimetric inequality and other applications : : : : : : : : 437

XIV TABLE OF CONTENTS

VII.8. Volume and Minkowski addition: the Brunn-Minkowski theoremand a second proof of the isoperimetric inequality : : : : : : : : : 439

VII.9. Volume and polarity : : : : : : : : : : : : : : : : : : : : : : : : 444VII.10. The appearance of convex sets, their degree of badness : : : : : : : 446VII.11. Volumes of slices of convex sets : : : : : : : : : : : : : : : : : : 459VII.12. Sections of low dimension: the concentration phenomenon

and the Dvoretsky theorem on the existence of almostspherical sections : : : : : : : : : : : : : : : : : : : : : : : : : : 470

VII.13. Miscellany : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 477VII.14. Intermezzo: can we dispose of the isoperimetric inequality? : : : : 493Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 499

Chapter VIII. Polygons, polyhedra, polytopes 505

VIII.1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 505VIII.2. Basic notions : : : : : : : : : : : : : : : : : : : : : : : : : : : : 506VIII.3. Polygons : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 508VIII.4. Polyhedra: combinatorics : : : : : : : : : : : : : : : : : : : : : 513VIII.5. Regular Euclidean polyhedra : : : : : : : : : : : : : : : : : : : 518VIII.6. Euclidean polyhedra: Cauchy rigidity and Alexandrov existence : 524VIII.7. Isoperimetry for Euclidean polyhedra : : : : : : : : : : : : : : : 530VIII.8. Inscribability properties of Euclidean polyhedra; how to encage

a sphere (an egg) and the connection with packings of circles : : : 532VIII.9. Polyhedra: rationality : : : : : : : : : : : : : : : : : : : : : : : 537VIII.10. Polytopes (d > 4): combinatorics I : : : : : : : : : : : : : : : : 539VIII.11. Regular polytopes (d > 4) : : : : : : : : : : : : : : : : : : : : : 544VIII.12. Polytopes (d > 4): rationality, combinatorics II : : : : : : : : : : 550VIII.13. Brief allusions to subjects not really touched on : : : : : : : : : : 555Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 558

Chapter IX. Lattices, packings and tilings in the plane 563

IX.1. Lattices, a line in the standard lattice Z2 and the theory of continuedfractions, an immensity of applications : : : : : : : : : : : : : : : 563

IX.2. Three ways of counting the points Z2 in various domains: pickand Ehrhart formulas, circle problem : : : : : : : : : : : : : : : : 567

IX.3. Points of Z2 and of other lattices in certain convex sets: Minkowski’stheorem and geometric number theory : : : : : : : : : : : : : : : : 573

IX.4. Lattices in the Euclidean plane: classification, density, Fourier analysison lattices, spectra and duality : : : : : : : : : : : : : : : : : : : : 576

IX.5. Packing circles (disks) of the same radius, finite or infinite in number,in the plane (notion of density). Other criteria : : : : : : : : : : : : 586

IX.6. Packing of squares, (flat) storage boxes, the grid (or beehive) problem 593IX.7. Tiling the plane with a group (crystallography). Valences, earthquakes 596IX.8. Tilings in higher dimensions : : : : : : : : : : : : : : : : : : : : : 603

TABLE OF CONTENTS XV

IX.9. Algorithmics and plane tilings: aperiodic tilings and decidability,classification of Penrose tilings : : : : : : : : : : : : : : : : : : : 607

IX.10. Hyperbolic tilings and Riemann surfaces : : : : : : : : : : : : : : 617Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 620

Chapter X. Lattices and packings in higher dimensions 623

X.1. Lattices and packings associated with dimension 3 : : : : : : : : : : 623X.2. Optimal packing of balls in dimension 3, Kepler’s conjecture at last

resolved : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 629X.3. A bit of risky epistemology: the four color problem and the Kepler

conjecture : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 639X.4. Lattices in arbitrary dimension: examples : : : : : : : : : : : : : : : 641X.5. Lattices in arbitrary dimension: density, laminations : : : : : : : : : 648X.6. Packings in arbitrary dimension: various options for optimality : : : : 654X.7. Error correcting codes : : : : : : : : : : : : : : : : : : : : : : : : : 659X.8. Duality, theta functions, spectra and isospectrality in lattices : : : : : 667Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 673

Chapter XI. Geometry and dynamics I: billiards 675

XI.1. Introduction and motivation: description of the motion of two particlesof equal mass on the interior of an interval : : : : : : : : : : : : : 675

XI.2. Playing billiards in a square : : : : : : : : : : : : : : : : : : : : : 679XI.3. Particles with different masses: rational and irrational polygons : : : 689XI.4. Results in the case of rational polygons: first rung : : : : : : : : : : 692XI.5. Results in the rational case: several rungs higher on the ladder : : : 696XI.6. Results in the case of irrational polygons : : : : : : : : : : : : : : 705XI.7. Return to the case of two masses: summary : : : : : : : : : : : : : 710XI.8. Concave billiards, hyperbolic billiards : : : : : : : : : : : : : : : : 710XI.9. Circles and ellipses : : : : : : : : : : : : : : : : : : : : : : : : : 713XI.10. General convex billiards : : : : : : : : : : : : : : : : : : : : : : : 717XI.11. Billiards in higher dimensions : : : : : : : : : : : : : : : : : : : : 728XI.XYZ Concepts and language of dynamical systems : : : : : : : : : : : 730Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 735

Chapter XII. Geometry and dynamics II: geodesic flow on a surface 739

XII.1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 739XII.2. Geodesic flow on a surface: problems : : : : : : : : : : : : : : : : 741XII.3. Some examples for sensing the difficulty of the problem : : : : : : 743XII.4. Existence of a periodic trajectory : : : : : : : : : : : : : : : : : : 751XII.5. Existence of more than one, of many periodic trajectories;

and can we count them? : : : : : : : : : : : : : : : : : : : : : : : 757XII.6. What behavior can be expected for other trajectories?

Ergodicity, entropies : : : : : : : : : : : : : : : : : : : : : : : : : 772

XVI TABLE OF CONTENTS

XII.7. Do the mechanics determine the metric? : : : : : : : : : : : : : : : 779XII.8. Recapitulation and open questions : : : : : : : : : : : : : : : : : : 781XII.9. Higher dimensions : : : : : : : : : : : : : : : : : : : : : : : : : : 781Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 782

Selected Abbreviations for Journal Titles 785

Name Index 789

Subject Index 795

Symbol Index 827