Geometry in history

Preface......Page 5
Introduction......Page 6
Contents......Page 14
1 Plato on Geometry and the Geometers......Page 16
 1.1 Introduction......Page 18
  1.2.1 The Concept of Anthyphairesis......Page 27
  1.2.3 Periodic Anthyphairesis......Page 28
  1.2.4 Pythagorean & Theaetetean Incommensurabilities; Theaetetus' Anthyphairetic Theory of Ratios of Lines Commensurable in Square......Page 29
  1.2.5 Theaetetean Incommensurabilities and Anthyphairesis......Page 30
  1.3.2 The Logos Criterion and Anthyphairetic Periodicity......Page 31
  1.3.3 Equalisation of Parts, in Consequence of Periodicity......Page 32
  1.3.7 The True Platonic Being Is Knowable by Name and Logos, Equivalently by Division and Collection......Page 33
 1.4 The Dialectics of the Politeia Is Division and Logos-Collection; True Being in the Politeia Coincides with the One of the Second Hypothesis in the Parmenides......Page 34
   1.4.2.1 Dialectics as Knowledge of True Beings by Division into Kinds and Logos in the Politeia 531d9-e6, 532a5-b3 and d8-e3......Page 35
   1.4.2.2 Division and Logos in the Politeia 534b3-d2......Page 36
   1.4.3.2 Politeia 537b8-c8......Page 37
  1.4.5 Geometric Irrational Lines vs. Dialectic Logos in 534d3-7......Page 38
  1.5.1 Division and Collection in the Sophistes and Politicus......Page 39
   1.5.2.2 Collection......Page 40
  1.5.3 The Very Abbreviated Genus-Species Scheme of Division and Collection......Page 42
   1.5.4.1 Parts of the Division as Hypotheses in the Sophistes......Page 43
   1.5.4.2 Parts of the Division as Hypotheses in the Politicus......Page 44
  1.6.1 Politeia 510b6-9......Page 45
  1.6.3 Politeia 533c7-d3......Page 46
   1.6.3.1 ``tas hupotheseis anairousa'', 533c8: ``anairesis'' Is Used with the Meaning of ``Division'' in Kinds by Aristotle and Commentators and in Two Significant Passages by Proclus......Page 47
   1.6.3.3 ``tas hupotheseis anairousa'', 533c8: In Agreement with ``tas hupotheses dielete'' in the Phaedo 107b7......Page 50
   1.6.3.4 In Conclusion......Page 51
 1.7 Plato's Criticism I of the Geometers......Page 52
  1.7.2 The Indivisible Line Is Accepted by Zeno, Plato  and Xenocrates as a True Being......Page 53
   1.7.2.1 [One and Many] Zeno's Fragment B1 on True Being Agrees with the One Is Many, Namely the Infinite Anthyphairetic Division of the One Being and the Many Are One, Namely the Equalization of Parts in the One of the Second Hypothesis in the Parmenides 142d9-143a3......Page 54
   1.7.2.3 The Identification of Zeno's True Being with Plato's True Being......Page 55
   1.7.2.4 The Description of Plato's True Being as an Indivisible Line......Page 56
  1.7.3 Proclus' Compromise Between Plato and Euclid in In Euclidem 85,1-96,15: Proclus' ``Point'' Is, in Fact, the Indivisible Line; and, the Flow (``rhesis'') of the ``Point'' Is the Motion of the Indivisible Line......Page 57
  1.8.1  Politeia 510c2-d3: Against the Geometers' Axiomatic Method of Mathematics, Because in This Method Logos Cannot Be Provided of Theses Hypotheses......Page 58
   1.8.3.1 Politeia 533b6-c3: Against the Geometers' Hypotheses Because They Are Left Unmoved and `Logos' Is Not Given of Them......Page 60
   1.8.3.2 Politeia 533c3-6: Against the Geometers' Hypotheses Because, Not Having ``Beginning, Between and End'' These Hypotheses Are Not Known......Page 61
  1.8.4 The Nature of Plato's Criticism II Against the Axiomatisation of Mathematics......Page 62
  1.9.2 The Reason That the Geometers Are Compelled to Employ Hypotheses as Principles Is That They Form Their Arguments for the Sake of Actions/Geometric Constructions (527a1-b2)......Page 63
 1.10 The Ascent from Intelligible Hypotheses to the Anhupotheton Generates Intelligible Number, Straight Line, and Circle (Parmenides 137e, 142b-145b, Politeia 522b8-526c8). Plato's Praise of Geometry (Politeia 527b3-c2)......Page 65
  1.10.1 Intelligible Numbers (Parmenides 142b1-145 and Politeia 522b8-526c8). The Mathematical Hypothesis of the Even and the Odd (Definitions VII.1, 2 of Unit and Number in the Elements), and the Ascent by Division and Collection from Hypotheses to Anhupotheton Leading to Intelligible Numbers (Parmenides, Philebus 16c-17a and Politeia 522b8-526c8)......Page 66
  1.10.2 The Intelligible Straight Line Generated by the Division of the One of the Second Hypothesis in the Parmenides (Parmenides 137e, 142b-145b)......Page 67
  1.10.4 The Intelligible Circle Is Generated by the Anthyphairetic Periodicity and Resulting Collection of the One of the Second Hypothesis in the Parmenides......Page 70
  1.11.1 The Three Kinds of Angles in the Elements......Page 72
  1.11.3 Proclus: The Pythagoreans `Provide Logos' for the Three Kinds of Angles......Page 73
  1.11.4 Proclus Connects the Dyad  with the Infinite and the Right Angle with the Finite......Page 74
   1.11.5.1 Anthyphairetic Definition of the Side and Diameter Numbers......Page 75
  1.11.6 Pell Property and the Dyad of Acute and Obtuse Angles......Page 76
  1.11.8 The Fourth Postulate and Its Dialectical Content......Page 77
 1.12 Completion of Criticism II by Proclus: The Geometers' Failure to Derive the Fifth Postulate from a Platonic Principle of Finiteness......Page 78
  1.13.1 The True Opinion of an Intelligible Being Is Identified with an Initial Finite Segment of the Anthyphairesis of the Intelligible Being......Page 79
  1.13.2 The Knowledge of a Sensible Participating in an Intelligible Platonic Being Is Identified with a True Opinion of the Intelligible Being......Page 80
   1.13.3.2 The Fundamental Role of the Two Basic Triangles a and b for the Surfaces of the Four Polyhedral (Timaeus 53c4-54b5)......Page 81
   1.13.4.1 The Incommensurability of the Areas of the Triangles a and b......Page 82
   1.13.4.3 The Anthyphairesis of Ma to Nb, for Any Numbers M,N, Is Periodic......Page 84
  1.13.5 The Problem of Participation of the Sensibles in the Intelligibles......Page 85
  1.13.6 The Purpose of the Receptacle/Hollow Space Is to Transform the Intelligible Anthyphairetic Division into Equivalent Tight Inequalities (Timaeus 56c8-57c6, 57d7-58c4)......Page 86
  1.13.7 The Sensibles Participating in the Intelligible Platonic Being F Are Sometimes F and Sometimes Not-F......Page 90
  1.13.8 The Identification of the ``Receptacle'' (``hupodoche'', 48e-53c) with the ``Hollow Space''......Page 91
  1.14.1 Plato's Criticism III of the Geometers, Employment of Visible Diagrams in Their Study of Geometry......Page 92
  1.14.3 The Perception of a Sensible Is Provoking If It Can Be Considered at the Same Time as Its Opposite, and Non Provoking, If It Cannot Be So Considered (523b8-525a2)......Page 93
  1.14.4 The Perceptions That Are Provoking to the Mind, as Described in the Politeia 522e-525a, Coincide with the True Representations of Sensibles, as Described in the Timaeus 48a-58e......Page 94
  1.14.6 The Representations of the Sensibles That Are Not Provoking Towards Ascent to the Intelligible Are Analogous to the Geometric Hypotheses That Do Not Ascend to the Anhupotheton......Page 96
  1.14.8 The Geometers Form Their Arguments for the Sake of Intelligible Beings (Studied by Means of Dianoia), but About the Visible Diagrams of These Beings (510d5-511a2)......Page 97
 1.15 Criticism IV of the Academy: Criticism of Eudoxus' Theory of Ratios of Magnitudes (Scholion In Euclidem X.2) and Archytas' Duplication of the Cube (527d-528e)......Page 98
 1.16 Epilogue. The Un-Platonic Victory of Axiomatization: From Euclid to Peano and Hilbert......Page 101
 References......Page 102
 2.1 Introduction......Page 104
 2.2 D'Arcy Thompson......Page 109
 2.3 Aristotle, Mathematician and Topologist......Page 112
 2.4 Thom on Aristotle......Page 121
 2.5 On Form......Page 132
 References......Page 139
 Appendix to the article Topology and biology: From Aristotle to Thom by A. Papadopoulos......Page 142
 References......Page 143
 3.1 Introduction......Page 144
 3.2 Brief Summary and Plan of Exposition......Page 146
  3.3.1 Antiquity: Euclidean Geometry, Ptolemy's Epicycles, Antikythera Mechanism......Page 147
  3.3.2 Fourier Sums and Fourier Integrals: Epicycle Calculus......Page 149
  3.3.3 Quantum Amplitudes and Their Interference......Page 150
 References......Page 152
 4.1 Introduction......Page 154
 4.2 Geometry......Page 155
 4.3 Mirrors and Optics......Page 158
 4.4 Billiards in Convex Domains......Page 160
 4.5 Aristotle......Page 162
 4.6 Architecture......Page 164
 References......Page 166
 5.1 Prologue......Page 168
 5.2 From Mathematical Modernity to Mathematical Modernism......Page 169
 5.3 Fundamentals of Mathematical Modernism......Page 185
  5.4.1 Curves as Algebra: Descartes/Fermat/Diophantus......Page 203
  5.4.2 Curves as Surfaces, Surfaces as Curves: Riemann/Riemann/Riemann......Page 205
  5.4.3 Curves as Discrete Manifolds: Grothendieck/Weil/Riemann......Page 208
 5.5 Mathematical Modernism Between Ontologies and Technologies......Page 213
 5.6 Conclusion......Page 223
 References......Page 225
6 From Euclid to Riemann and Beyond: How to Describe the Shape of the Universe......Page 228
 6.1 Ancient Models of the Universe......Page 230
 6.2 What Is Infinity?......Page 233
 6.3 Is the Universe Infinite or Finite?......Page 237
 6.4 From Descartes to Newton and Leibniz......Page 243
 6.5 A New Approach in Classical Geometry......Page 249
 6.6 Euclid's Legacy in Physics and Philosophy......Page 252
 6.7 Gauss: Intrinsic Description of the Universe......Page 255
 6.8 Number Systems......Page 259
 6.9 Euclid's Elements......Page 262
 6.10 The Fifth Postulate......Page 266
 6.11 Discovery of Non-Euclidean Geometry......Page 269
 6.12 Geodesics......Page 271
 6.13 Curvature and Non-Euclidean Geometry......Page 274
 6.14 The Dimension of Space......Page 277
 6.15 Riemann: The Universe as a Manifold......Page 280
 6.16 Hyperbolic and Projective Spaces......Page 285
 6.17 Absolute Differential Calculus......Page 288
 6.18 Cantor's Transfinite Set Theory......Page 294
 6.19 Topological Spaces......Page 297
 6.20 Towards Modern Differential Geometry......Page 299
 6.21 Topology of the Universe......Page 306
 6.22 Right and Left in the Universe......Page 314
 References......Page 318
 7.1 Introduction......Page 320
 7.2 The Precursors......Page 321
 7.3 Christiaan Huygens......Page 327
 7.4 Euler......Page 336
 7.5 Curves in 3-Space......Page 341
 7.6 Curvature of Surfaces: Monge and His School......Page 345
 7.7 Twentieth Century: Return to Euclid......Page 352
 References......Page 363
 8.1 Introduction......Page 369
  8.2.1 Configuration Theorems......Page 371
  8.2.2 The Fundamental Theorem of Projective Geometry......Page 373
  8.2.4 Local Forms of the Desargues and Pappus Theorems......Page 377
   8.2.5.1 Dehn's Question......Page 378
   8.2.5.2 Why pDes and pPapp Stand Out Among All (n3) Configurations......Page 379
   8.2.5.3 Desargues and Pappus as Cayley–Klein Geometries......Page 381
   8.2.5.4 Strambach and Conic Sections......Page 382
  8.2.6 Moufang Planes......Page 383
   8.2.6.1 The Missing Link Between Moufang and Pappus in Fanoian Planes......Page 384
  8.2.7 The Lenz–Barlotti Classification......Page 385
 8.3 The Affine Setting......Page 386
  8.3.1 The Role of the Theorem of Menelaus......Page 390
  8.3.2 Area......Page 391
 8.4 The Effect of the Archimedean Axiom......Page 392
 8.5 Desargues's Axiom as Indicator of a Projective Plane's Embeddability in a Projective Space......Page 393
 8.6 One-Dimensional Characterizations......Page 395
  8.7.2 The Absolute Setting......Page 397
  8.7.3 The Affine Plane with Orthogonality Setting......Page 400
  8.8.2 G-Spaces......Page 401
 References......Page 403
 9.1 Introduction: Classical Configuration Theorems......Page 414
 9.2 Iterated Pappus Theorem and the Modular Group......Page 417
 9.3 Steiner Theorem and the Twisted Cubic......Page 422
 9.4 Pentagram-Like Maps on Inscribed Polygons......Page 425
 9.5 Poncelet Grid, String Construction, and Billiards in Ellipses......Page 430
 9.6 Identities in the Lie Algebras of Motions......Page 437
 9.7 Skewers......Page 439
  9.7.1 Elliptic Approach......Page 442
  9.7.2 Hyperbolic Approach......Page 443
 References......Page 446
 10.1 Introduction......Page 448
 10.2 Hyperbolic Geometry......Page 449
 10.3 Topology-Analysis Situs......Page 451
 10.4 Epistemology of Space and Time......Page 456
 10.5 Spontaneous Philosophy: Conclusion......Page 460
 References......Page 462
 11.1 Introduction......Page 464
 11.2 Elliptic Fixed Points......Page 465
 11.3 Preparation: Poincaré's Theory of Normal Forms......Page 466
  11.4.1 Andronov–Hopf–Neimark–Sacker Bifurcation......Page 468
  11.4.2 Dynamics on the Invariant Curves......Page 471
  11.5.1 Moser's Invariant Curve Theorem ch11:M......Page 472
  11.5.2 Periodic Orbits, Aubry-Mather Sets and Homoclinic Tangles......Page 475
 11.6 When Radial and Angular Behaviours Compete......Page 476
 References......Page 480
  12.1.1 Thom and Smale in 1956–1957......Page 482
  12.1.3 Whitney-Graustein Theorem......Page 484
  12.1.4 The Key Proposition in Smale's Thesis......Page 485
  12.1.6 Hirsch's Definitive Statement......Page 487
  12.1.7 Phillips' Work on Submersions......Page 488
  12.1.8 Differential Relations After M. Gromov......Page 489
  12.1.9 Examples from Singularity Theory......Page 491
  12.2.1 Thom's Point of View in 1959......Page 492
  12.2.2 What Happened Afterwards......Page 493
  12.2.3 The Thom Subdivision......Page 494
  12.2.4 Jiggling Formula......Page 495
  12.2.5 Going Back to Immersions......Page 496
  12.2.6 About the Regularization of Γ-Structures......Page 497
  12.2.7 Regularization of Transversely Geometric Γn-Structures......Page 498
  12.2.9 Final Remark......Page 502
 References......Page 503
 13.2 Symplectic Basics......Page 505
 13.4 Basic Symplectic Problems and Gromov's Alternative......Page 508
 13.6 Advent of Holomorphic Curves......Page 511
  13.6.1 Non-existence of Exact Lagrangians in Cn......Page 512
  13.6.2 Gromov's Non-squeezing Theorem......Page 513
  13.6.3 Packing Inequalities......Page 514
  13.6.4 4-Dimensional Applications......Page 516
  13.7.1 Overtwisted Contact Structures......Page 517
  13.7.3 Guth's Symplectic Embeddings......Page 518
  13.7.4 Bounds on the Number of Double Points of Exact Symplectic Immersions......Page 519
  13.7.6 Construction of Symplectic Cobordisms......Page 520
 13.8 Basic Problems: Where They Stand Now?......Page 521
 References......Page 523
 14.1 Introduction......Page 527
 14.2 Euclidean Manifolds......Page 529
  14.2.1 Riemannian Geometry......Page 530
  14.2.2 The Bieberbach Theorems......Page 531
  14.2.3 Affine Crystallographic Groups......Page 532
 14.3 Geometrization of 3-Manifolds......Page 533
 14.4 Ehresmann Structures......Page 534
  14.4.2 Hierarchy of Structures......Page 536
  14.4.3 The Ehresmann-Weil-Thurston Holonomy Principle......Page 537
  14.4.4 Historical Remarks......Page 538
 14.5 Example: One Real Dimension......Page 539
  14.5.2 Compact Manifolds......Page 540
  14.5.4 Incomplete Affine Structures......Page 541
  14.6.2 Hopf Manifolds......Page 544
   14.7.1.1 Radiant Affine Structures......Page 545
  14.7.2 Complete Affine Structures......Page 547
  14.7.3 Hyperbolic Structures on Surfaces......Page 548
  14.8.1 Projective Structures on Riemann Surfaces......Page 549
  14.8.2 Real-Projective Structures on Surfaces......Page 551
   14.8.3.1 Parallel Volume......Page 552
   14.8.3.2 Hyperbolicity......Page 553
   14.8.3.4 Hyperbolic Torus Bundles......Page 555
  14.8.4 Flat Conformal Structures in Higher Dimensions......Page 556
  14.8.6 Complex Projective Structures in Higher Dimensions......Page 557
 References......Page 558
 15.1 Jets......Page 565
 15.2 Submersions and Fibrations......Page 567
 15.3 Pfaffian Systems and Systems of Partial Differential (in)Equations......Page 571
 15.4 Connections......Page 573
 15.5 Integral of Differential Forms, Pullbacks, Exterior Derivative......Page 575
 15.6 Flows, Lie Derivative and Lie Bracket......Page 579
 15.7 Some Applications of the Cartan Formula......Page 581
 15.8 Curvature......Page 584
 15.9 As a Conclusion......Page 590
 References......Page 591
 16.1 Introduction......Page 593
  16.2.2 Intrinsic Distances and Geodesics......Page 596
  16.2.3 The Exponential Map and Geodesic Completeness......Page 598
  16.3.1 Forward Cut Locus and Forward Conjugate Locus......Page 599
  16.3.2 Geodesic Polar Coordinates......Page 600
  16.3.3 The Whitehead Convexity Theorem......Page 601
  16.4.1 Foots of Closed Sets......Page 605
  16.4.2 The Klingenberg Lemma......Page 607
  16.4.3 The Berger-Omori Lemma......Page 608
  16.4.4 The Rauch Conjecture......Page 610
  16.4.5 Poles......Page 613
  16.4.6 The Continuity of the Injectivity Radius Function......Page 614
  16.4.7 Pointed Blaschke Manifolds......Page 615
  16.5.1 Busemann Functions......Page 618
  16.5.2 Properties of Busemann Functions on (M,F)......Page 619
  16.5.3 Convex Functions......Page 621
  16.5.4 Riemannian and Finslerian Results on Convex Functions......Page 623
  16.5.5 Level Set Configurations......Page 624
  16.5.6 Properness of Exponential Maps......Page 627
  16.5.7 Number of Ends......Page 628
  16.5.8 Isometry Groups......Page 629
 References......Page 631
 17.1 Introduction......Page 634
 17.2 The Poincaré Conjecture......Page 640
 17.3 Some Statements Related with the Poincaré Conjecture......Page 641
 17.4 Statements on Triangulations Related with the Poincaré Conjecture......Page 643
 17.5 Smooth Manifolds......Page 644
 17.6 Closed Simply Connected 4-Manifolds and Smooth Structures......Page 646
 17.7 Homotopy and Fundamental Groups......Page 651
 17.8 The Seifert-van Kampen Theorem......Page 654
 17.9 Closed Surfaces and Their Fundamental Groups......Page 656
 17.10 Heegaard Splittings and Fundamental Groups of Closed 3-Manifolds......Page 657
  17.11.1 Splitting Homomorphisms......Page 660
  17.11.2 The Mapping Class Group......Page 661
 17.12 Connected Sums and Prime Decompositions of 3-Manifolds......Page 662
 17.13 Irreducible Closed and Open Contractible 3-Manifolds......Page 664
 17.14 Compact 3-Manifolds with Finite Fundamental Groups......Page 666
 17.15 Irreducible Closed 3-Manifolds with Infinite Fundamental Group......Page 669
 17.16 Haken Manifolds......Page 671
 17.17 The JSJ-Splitting Theorem for Orientable Haken Manifolds......Page 672
 17.18 The Thurston Geometrization Conjecture......Page 674
 17.19 Triangulations of Topological Manifolds......Page 676
 17.20 The Homeomorphism Problem for Closed n-Manifolds, n≥4......Page 681
 17.21 The Manifold Recognition Problem......Page 682
 17.22 Triangulations and Alexandrov Spaces with Curvature Bounded Above......Page 685
 References......Page 689
 18.1 Introduction......Page 697
 18.2 Geometrical Questions, and Answers......Page 698
 18.3 Casson Handles......Page 703
 18.4 Yang-Mills Theory, From a Topological Viewpoint......Page 706
 18.5 Exotic R4's......Page 708
 References......Page 712
19 Memories from My Former Life: The Making of a Mathematician......Page 714
Index......Page 744