Geometry Revealed (eBook)

Marcel Berger is Ancien Professeur of the University of Paris and emeritus director of research at the Centre National de la Recherche Scientifique (CNRS), from 1979 to 1981 he was president of the French Mathematical Society and from 1985 to 1994 director of the Institut des Hautes Études Scientifiques (IHÉS) in Bures-sur-Yvette.

About the Author 6
Introduction 8
Bibliography 11
Table of Contents 12
I Points and lines in the plane 18
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 18
I.2 Another naive problem of Sylvester, this time on the geometric probabilities of four points 23
I.3 The essence of affine geometry and the fundamental theorem 29
I.4 Three configurations of the affine plane and what has happened to them: Pappus, Desargues and Perles 34
I.5 The irresistible necessity of projective geometry and the construction of the projective plane 40
I.6 Intermezzo: the projective line and the cross ratio 45
I.7 Return to the projective plane: continuation and conclusion 48
I.8 The complex case and, better still, Sylvester in the complex case: Serre's conjecture 57
I.9 Three configurations of space (of three dimensions): Reye, Möbius and Schläfli 60
I.10 Arrangements of hyperplanes 64
I. XYZ 65
Bibliography 74
II Circles and spheres 77
II.1 Introduction and Borsuk's conjecture 77
II.2 A choice of circle configurations and a critical view of them 82
II.3 A solitary inversion and what can be done with it 94
II.4 How do we compose inversions? First solution: the conformal group on the disk and the geometry of the hyperbolic plane 98
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 appearanceof the first fractals 103
II.6 Inversion in space: the sextuple and its generalization thanksto the sphere of dimension 3 107
II.7 Higher up the ladder: the global geometry of circles and spheres 112
II.8 Hexagonal packings of circles and conformal representation 119
II.9 Circles of Apollonius 129
II. XYZ 132
Bibliography 153
III The sphere by itself: can we distribute points on it evenly? 156
III.1 The metric of the sphere and spherical trigonometry 156
III.2 The Möbius group: applications 162
III.3 Mission impossible: to uniformly distribute points on the sphere S2: ozone, electrons, enemy dictators, golf balls, virology, physics of condensed matter 164
III.4 The kissing number of S2, alias the hard problem of the thirteenth sphere 185
III.5 Four open problems for the sphere S3 187
III.6 A problem of Banach--Ruziewicz: the uniqueness of canonical measure 189
III.7 A conceptual approach for the kissing number in arbitrary dimension 190
III. XYZ 192
Bibliography 193
IV Conics and quadrics 196
IV.1 Motivations, a definition parachuted from the ladder, and why 196
IV.2 Before Descartes: the real Euclidean conics. Definition and some classical properties 198
IV.3 The coming of Descartes and the birth of algebraic geometry 213
IV.4 Real projective theory of conics duality
IV.5 Klein's philosophy comes quite naturally 220
IV.6 Playing with two conics, necessitating once again complexification 223
IV.7 Complex projective conics and the space of all conics 227
IV.8 The most beautiful theorem on conics: the Poncelet polygons 231
IV.9 The most difficult theorem on the conics: the 3264 conics of Chasles 241
IV.10 The quadrics 247
IV. XYZ 257
Bibliography 260
V Plane curves 263
V.1 Plain curves and the person in the street: the Jordan curve theorem, the turning tangent theorem and the isoperimetric inequality 263
V.2 What is a curve? Geometric curves and kinematic curves 268
V.3 The classification of geometric curves and the degree of mappings of the circle onto itself 271
V.4 The Jordan theorem 273
V.5 The turning tangent theorem and global convexity 274
V.6 Euclidean invariants: length (theorem of the peripheral boulevard) and curvature (scalar and algebraic): Winding number 277
V.7 The algebraic curvature is a characteristic invariant: manufacture of rulers, control by the curvature 283
V.8 The four vertex theorem and its converse an application to physics
V.9 Generalizations of the four vertex theorem: Arnold I 292
V.10 Toward a classification of closed curves: Whitney and Arnold II 295
V.11 Isoperimetric inequality: Steiner's attempts 309
V.12 The isoperimetric inequality: proofs on all rungs 312
V.13 Plane algebraic curves: generalities 319
V.14 The cubics, their addition law and abstract elliptic curves 322
V.15 Real and Euclidean algebraic curves 334
V.16 Finite order geometry 342
V. XYZ 345
Bibliography 350
VI Smooth surfaces 355
VI.1 Which objects are involved and why? Classification of compact surfaces 355
VI.2 The intrinsic metric and the problem of the shortest path 359
VI.3 The geodesics, the cut locus and the recalcitrant ellipsoids 361
VI.4 An indispensable abstract concept: Riemannian surfaces 371
VI.5 Problems of isometries: abstract surfaces versus surfaces of E3 375
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 378
VI.7 What is known about the total curvature (of Gauss) 387
VI.8 What we know how to do with the mean curvature, all about soap bubbles and lead balls 394
VI.9 What we don't entirely know how to do for surfaces 400
VI.10 Surfaces and genericity 405
VI.11 The isoperimetric inequality for surfaces 411
VI. XYZ 413
Bibliography 417
VII Convexity and convex sets 422
VII.1 History and introduction 422
VII.2 Convex functions, examples and first applications 425
VII.3 Convex functions of several variables, an important example 428
VII.4 Examples of convex sets 430
VII.5 Three essential operations on convex sets 433
VII.5.A The (Steiner, Schwarz) symmetrizations 433
VII.5.B Some algebra of convex sets: Minkowski's sum 437
VII.5.C A duality: polarity 438
VII.6 Volume and area of (compacts) convex sets, classical volumes: Can the volume be calculated in polynomial time? 441
VII.6.A Volume of cubes, cocubes and simplexes 442
VII.6.B Balls, spheres and ellipsoids 443
VII.6.C Approximation by polytopes, areas of convex sets 447
VII.6.D Mission impossible: calculating the volume of a convex set numerically 448
VII.7 Volume, area, diameter and symmetrizations: first proof of the isoperimetric inequality and other applications 450
VII.8 Volume and Minkowski addition: the Brunn-Minkowski theorem and a second proof of the isoperimetric inequality 452
VII.9 Volume and polarity 457
VII.10 The appearance of convex sets, their degree of badness 459
VII.10.A How to generate a convex set 459
VII.10.B Topology of complex sets and their boundaries 461
VII.10.C The John-Loewner ellipsoid and its applications 461
VII.10.D A first metric space formed of all center-symmetric convex sets: the compact set of Banach-Mazur 464
VII.10.E The Rogalski conjecture and a mapping of isoperimetric type 467
VII.10.F Badness test for a convex set using its moments of inertia: the ellipsoids of Legendre and Binet (ellipsoid of inertia), the inertial invariant and the grand conjecture on convex sets (first formulation) 469
VII.11 Volumes of slices of convex sets 472
VII.11.A Slicing by lines, Hammer's X-ray problem 473
VII.11.B Slicing with hyperplanes: general questions, the grand conjecture 474
VII.11.C The hyperplane sections of the cube 478
VII.12 Sections of low dimension: the concentration phenomenonand the Dvoretsky theorem on the existence of almostspherical sections 483
VII.12.A Statement of the result 483
VII.12.B The concentration phenomenon of Paul Lévy 486
VII.12.C The proof 488
VII.13 Miscellany 490
VII.13.A Projections 490
VII.13.B Steiner-Minkowski formula and mixed volume 491
VII.13.C Convex sets and mathematical physics: the floating body that loses its head, the fundamental frequency, the Poinsot motion, Newtonian gravitation, the destiny of the rolling stones 492
VII.13.D The appearance of the boundary of a convex set the space of all convex sets
VII.13.E Immobilization of a convex set 505
VII.14 Intermezzo: can we dispose of the isoperimetric inequality? 506
Bibliography 512
VIII Polygons, polyhedra, polytopes 518
VIII.1 Introduction 518
VIII.2 Basic notions 519
VIII.3 Polygons 521
VIII.4 Polyhedra: combinatorics 526
VIII.5 Regular Euclidean polyhedra 531
VIII.6 Euclidean polyhedra: Cauchy rigidity and Alexandrov existence 537
VIII.7 Isoperimetry for Euclidean polyhedra 543
VIII.8 Inscribability properties of Euclidean polyhedra how to encage a sphere (an egg) and the connection with packings of circles
VIII.9 Polyhedra: rationality 550
VIII.10 Polytopes (d4): combinatorics I 552
VIII.11 Regular polytopes (d4) 557
VIII.12 Polytopes (d4): rationality, combinatorics II 563
VIII.13 Brief allusions to subjects not really touched on 568
Bibliography 571
IX Lattices, packings and tilings in the plane 575
IX.1 Lattices, a line in the standard lattice Z2 and the theory of continued fractions, an immensity of applications 575
IX.2 Three ways of counting the points Z2 in various domains: pick and Ehrhart formulas, circle problem 579
IX.3 Points of Z2 and of other lattices in certain convex sets: Minkowski's theorem and geometric number theory 585
IX.4 Lattices in the Euclidean plane: classification, density, Fourier analysis on lattices, spectra and duality 588
IX.5 Packing circles (disks) of the same radius, finite or infinite in number, in the plane (notion of density). Other criteria 598
IX.6 Packing of squares, (flat) storage boxes, the grid (or beehive) problem 605
IX.7 Tiling the plane with a group (crystallography). Valences, earthquakes 608
IX.8 Tilings in higher dimensions 615
IX.9 Algorithmics and plane tilings: aperiodic tilings and decidability, classification of Penrose tilings 619
IX.10 Hyperbolic tilings and Riemann surfaces 629
Bibliography 632
X Lattices and packings in higher dimensions 635
X.1 Lattices and packings associated with dimension 3 635
X.2 Optimal packing of balls in dimension 3, Kepler's conjecture at last resolved 641
X.3 A bit of risky epistemology: the four color problem and the Kepler conjecture 651
X.4 Lattices in arbitrary dimension: examples 653
X.5 Lattices in arbitrary dimension: density, laminations 660
X.6 Packings in arbitrary dimension: various options for optimality 666
X.7 Error correcting codes 671
X.8 Duality, theta functions, spectra and isospectrality in lattices 679
Bibliography 685
XI Geometry and dynamics I: billiards 687
XI.1 Introduction and motivation: description of the motion of two particles of equal mass on the interior of an interval 687
XI.2 Playing billiards in a square 691
XI.2.A The dichotomy and continued fractions 692
XI.2.B Counting periodic trajectories 697
XI.2.C Introduction of the language of dynamical systems 700
XI.3 Particles with different masses: rational and irrational polygons 701
XI.4 Results in the case of rational polygons: first rung 704
XI.5 Results in the rational case: several rungs higher on the ladder 708
XI.5.A The nature of nonperiodic trajectories 709
XI.5.B Counting periodic trajectories 715
XI.6 Results in the case of irrational polygons 717
XI.7 Return to the case of two masses: summary 722
XI.8 Concave billiards, hyperbolic billiards 722
XI.9 Circles and ellipses 725
XI.10 General convex billiards 729
XI.10.A Very smooth and strictly convex billiards: caustics 729
XI.10.B Three strange phenomena 731
XI.10.C Generic billiards 735
XI.10.D Periodic trajectories 737
XI.10.E Billiards and duality 739
XI.11 Billiards in higher dimensions 740
XI.XYZ Concepts and language of dynamical systems 742
XI.XYZ.A Ergodicity and mixing 742
XI.XYZ.B The various notions of entropy 745
Bibliography 747
XII Geometry and dynamics II: geodesic flow on a surface 750
XII.1 Introduction 750
XII.2 Geodesic flow on a surface: problems 752
XII.3 Some examples for sensing the difficulty of the problem 754
XII.3.A The spheres 754
XII.3.B The surfaces of revolution: the Zoll surfaces 754
XII.3.C Weinstein's counterexample 757
XII.3.D Ellipsoids with three axes 758
XII.3.E The flat tori 761
XII.4 Existence of a periodic trajectory 762
XII.4.A The torus and surfaces of higher genus 762
XII.4.B The sphere, Birkhoff's result 763
XII.5 Existence of more than one, of many periodic trajectories and can we count them?
XII.5.A The case of the torus 769
XII.5.B Surfaces of higher genus 771
XII.5.C The sphere: the three Lusternik-Schnirelman geodesics. 775
XII.5.D The sphere: an infinity of periodic geodesics 778
XII.6 What behavior can be expected for other trajectories?Ergodicity, entropies 783
XII.6.A Surfaces of higher genus 783
XII.6.B The entropies 785
XII.6.C The case of the sphere. The example of Osserman-Donnay 786
XII.6.D Entropy and the length of geodesics joining two given points 789
XII.7 Do the mechanics determine the metric? 790
XII.8 Recapitulation and open questions 792
XII.9 Higher dimensions 792
Bibliography 793
Selected Abbreviations for Journal Titles 795
Name Index 798
Subject Index 804
Symbol Index 836