Front Cover 1
Symmetry: Unifying Human Understanding 4
Copyright Page 5
Table of Contents 6
FOREWORD 10
PREFACE 12
Part 1 16
CHAPTER 1. LIMITS OF PERFECTION 20
REFERENCES 36
CHAPTER 2. BUT WHAT IS SYMMETRY? 38
CHAPTER 3. GENERALISED CRYSTALLOGRAPHY 40
LITERARY STRUCTURE 40
INTRODUCTION 41
ATOMS AND REDUCTIONISM 42
GROUP THEORY 42
CRYSTALLOGRAPHY 43
FROM SYMMETRY GROUPS TO PROGRAMS 44
INFORMATION 44
CELLULAR AUTOMATA 45
CONCLUSIONS 46
REFERENCES 55
CHAPTER 4. SEEN AND UNSEEN SYMMETRIES: A PICTURE ESSAY 58
ENDNOTES 80
CHAPTER 5. SYMMETRY AND MODULARITY 82
INTRODUCTION 82
REGULARLY SPACED POINTS 83
TILINGS 84
SPACE FILLERS 87
THE RHOMBIC DODECAHEDRON 90
THE TRUNCATED OCTAHEDRON 90
CONCLUSIONS 94
REFERENCES 94
CHAPTER 6. SYMMETRY AS AN AESTHETIC FACTOR 96
CHAPTER 7. SYMMETRY INDUCED BY ECONOMY 102
1. INTRODUCTION 102
2. POLYHEDRA AND SPHERICAL CONFIGURATIONS 102
3. ARRANGEMENTS IN THE PLANE 106
4. ARRANGEMENTS IN THE SPACE 109
REFERENCES 109
CHAPTER 8. MAGIC RECTANGLES IN THE TABLE OF THE 32 CLASSES OF SYMMETRY 112
REFERENCES 115
CHAPTER 9. AN EXTENSION OF THE NEUMANN-MINNIGERODE-CURIE PRINCIPLE 116
1. THE HISTORY OF THE NEUMANN-MINNIGERODE-CURIE PRINCIPLE 116
2. THE NMC-PRINCIPLE FOR THE TOTALLY SYMMETRIC REPRESENTATION 116
3. THE EXTENDED NMC-PRINCIPLE FOR THE NON-TOTALLY SYMMETRIC REPRESENTATIONS 117
4. APPLICATION TO THE CALCULATION OF THE TENSORS FOR THE DYNAMICAL PROPERTIES WITH RESPECT TO THE 32 CLASSICAL CRYSTALLOGRAPHIC POINT GROUPS 118
5. APPLICATION TO THE CLASSICAL CURIE LIMITING GROUPS 118
6. DYNAMICAL PROPERTIES AND MAGNETIC GROUPS 118
REFERENCES 119
CHAPTER 10. SYMMETRY AND ITS "LOVE–HATE" ROLE IN MUSIC 120
SYMMETRICAL OPERATIONS IN MUSIC 121
SYMMETRY CREATED, SYMMETRY DENIED: LOVE-HATE 124
REFERENCES 131
CHAPTER 11. THE COMPUTERIZED SOMA CUBE 132
1. INTRODUCTION 132
2. MATHEMATICAL REPRESENTATION 132
3. TOTAL NUMBER OF POSITIONS 132
4. CONGRUENT SOLUTIONS 136
5. SOLUTION OF THE PROBLEM 136
REFERENCES 140
CHAPTER 12. THE SYMMETRY OF M. C. ESCHER'S"IMPOSSIBLE" IMAGES 142
1. WHAT IS IMPOSSIBLE? 142
2. DIFFERENT VIEWPOINTS COMBINED IN ONE PRINT 142
3. PLANE AND SPACE 146
4. CHANGING THE SCALE 147
5. IMPOSSIBLE BUILDINGS 152
6. FINAL REMARKS 156
REFERENCES 157
CHAPTER 13. HYPERBOLIC SYMMETRY 158
1. INTRODUCTION 158
2. HYPERBOLIC GEOMETRY 158
3. REPEATING HYPERBOLIC PATTERNS 158
4. SYMMETRY GROUPS 160
5. CREATING A MOTIF 160
6. REPLICATING THE PATTERN 163
7. COLOR SYMMETRY 168
8. CONCLUSION 171
REFERENCES 171
CHAPTER 14. ASYMMETRY AND SYMMETRY IN CELLULAR ORGANIZATION 174
1. ASYMMETRY OF THE CELL SURFACE ORGANIZATION 175
2. ASYMMETRIES IN THE GENETIC INFORMATION SYSTEM 178
3. THE ASYMMETRY OF CELLULAR MECHANICAL WORK 181
REFERENCES 186
CHAPTER 15. SYMMETRY AND SPONTANEOUSLY BROKEN SYMMETRY IN THE PHYSICS OF ELEMENTARY PARTICLES 188
1. INTRODUCTION 188
2. HISTORICAL BACKGROUND 189
3. SYMMETRY AS A GROUP-THEORETICAL CLASSIFICATION FOR THE ELEMENTARY PARTICLES 190
4. LOCAL SYMMETRY 192
5. SPONTANEOUSLY BROKEN SYMMETRY 193
6. THE WEINBERG-SALAM-GLASHOW THEORY OF THE WEAK INTERACTIONS 196
7. GRANDUNIFICATION 198
8. SUPERUNIFICATION 199
REFERENCES 200
APPENDIX: LISTING OF THE MAIN ELEMENTARY PARTICLES 201
CHAPTER 16. SYMMETRY OF POINT IMPERFECTIONS IN SOLIDS 204
1. INTRODUCTION 204
2. CRYSTALLOGRAPHIC POINT GROUPS 204
3. GROUP THEORY AND QUANTUM MECHANICS 205
4. POINT IMPERFECTIONS 207
5. APPLICATION OF GROUP THEORY TO POINT IMPERFECTIONS 208
6. THE JAHN–TELLER EFFECT 211
7. EPILOGUE 214
REFERENCES 215
CHAPTER 17. ON SYMMETRY AND ASYMMETRY IN LITERATURE 216
INTRODUCTION 216
BASIC CONCEPTS 219
SYMMETRY AND ASYMMETRY OF LETTERS AND WORDS 226
SYMMETRY AND ASYMMETRY IN LITERARY WORKS 229
CONCLUSIONS 241
REFERENCES 242
CHAPTER 18. SYMMETRY RULES FOR CHEMICAL REACTIONS 248
INTRODUCTION 248
APPLICATION OF PERTURBATION THEORY 249
THE ORBITAL CORRELATION METHOD 252
ORBITAL TOPOLOGY 254
REFERENCES 255
CHAPTER 19. SYMMETRY OF BIOLOGICAL MACROMOLECULES AND THEIR ASSOCIATIONS 256
1. GENERAL PRINCIPLES. CHIRALITY OF BIOMOLECULES 256
2. SYMMETRY OF BIOSYSTEMS AT DIFFERENT LEVELS OF MOLECULAR ORGANIZATION 261
REFERENCES 287
CHAPTER 20. MATCHING AND SYMMETRY OF GRAPHS 290
1. INTRODUCTION 290
2. PERFECT MATCHING 293
3. IMPERFECT MATCHING AND TOPOLOGICAL INDEX 298
4. ORTHOGONAL POLYNOMIALS 301
REFERENCES 308
CHAPTER 21. SYMMETRY-MAKING AND -BREAKING IN VISUAL ART 310
THE PROBLEM OF THE PERCEPTION OF SYMMETRY 310
THE PROBLEM OF THE AESTHETIC ASPECT OF SYMMETRY 311
REFERENCES 320
CHAPTER 22. SYMMETRY AND NOTATION: REGULARITY AND SYMMETRY IN NOTATED COMPUTER GRAPHICS 322
FRAMES OF REFERENCE 322
ESHKOL-WACHMAN MOVEMENT NOTATION AND SHAPE DESCRIPTION 323
COMPUTER LANGUAGE AND A LANGUAGE FOR SHAPE 324
REGULARITY, SYMMETRY 325
SUMMARY 333
REFERENCES 333
CHAPTER 23. SYMMETRY IN MATHEMATICS 334
INTRODUCTION 334
1. SYMMETRY IN GEOMETRY 335
2. SYMMETRY IN ALGEBRA 341
3. SYMMETRY IN MATHEMATICAL DEDUCTION 343
4. SYMMETRY IN THE FORMULATION OF MATHEMATICAL CONCEPTS 345
REFERENCES 347
CHAPTER 24. MOIRÉS 348
1. INTRODUCTION 348
2. THE MATHEMATICAL MOIRÉ 348
3. PSYCHOLOGICAL MOIRÉS 349
4. SCIENTIFIC APPLICATIONS 350
5. SPATIAL MOIRÉS 363
REFERENCES 378
APPENDIX 1 378
APPENDIX 2 379
CHAPTER 25. MOIRÉS 382
APPENDIX 384
CHAPTER 26. SYMMETRY OF SYSTEM AND SYSTEM OF SYMMETRY 398
INTRODUCTION 398
GENERAL THEORY OF ISOMERISM. ISOMERISM AND SYMMETRY 409
THE SYSTEM OF SYMMETRY 422
REFERENCES 424
CHAPTER 27. THE MOSAIC PATTERNS OF H. J. WOODS 426
REFERENCES 430
CHAPTER 28. BILATERAL SYMMETRY IN INSECTS: COULD IT DERIVE FROM CIRCULAR ASYMMETRIES DURING EARLY EMBRYOGENESIS? 432
REFERENCES 437
CHAPTER 29. FORMATION OF SYMMETRIC AND ASYMMETRIC STRUCTURES DURING DEVELOPMENT OF HIGHER ORGANISMS 438
INTRODUCTION 438
INSECTS AS A MODEL SYSTEM 439
ONE PEAK—TWO SLOPES: THE BASIS OF MANY NATURALLY OCCURRING SYMMETRIC PATTERNS 439
THE GENERATION OF POSITIONAL INFORMATION BY AUTOCATALYSIS AND LONG-RANGING INHIBITION 439
THE DORSOVENTRAL AXIS AND THE FORMATION OF STRIPES 441
SYMMETRICAL MALFORMATIONS: TWO SEPARATE PEAKS PROVIDE TWO SETS OF POSITIONAL INFORMATION 443
SYMMETRY AND ASYMMETRY IN LIMB FORMATION 445
SYMMETRICAL MALFORMATIONS OF LEGS AND WINGS 447
PROBLEMS IN GENERATING CARDINAL AXES ON A SPHERE 448
TWO OR THREE CARDINAL AXES? 450
SIAMESE TWINS ARE A PARTICULAR TYPE OF SYMMETRIC MALFORMATIONS 451
CONCLUSION 451
REFERENCES 451
CHAPTER 30. THE VISUALIZATION OF MUSIC: SYMMETRY AND ASYMMETRY 454
REFERENCES AND NOTES 482
CHAPTER 31. ON THE SYMMETRY OF PERIODIC STRUCTURES IN TWO DIMENSIONS 484
1. TWO-DIMENSIONAL SYMMETRY: HYDROCARBON CHEMISORPTION AND WALL DECORATION 484
2. MOLECULAR CLOSE-PACKING AND SYMMETRY 486
3. TWO IN TWO: THE PLANE TWICE-PERIODIC GROUPS G22 487
4. THREE IN TWO: THE LAYER GROUPS G32 489
5. MATRIX REPRESENTATION 490
6. CONCLUSIONS 495
REFERENCES 495
CHAPTER 32. A SPACE VIEW OF A SYMMETRIC OBJECT 496
1. INTRODUCTION 496
2. IMAGES FROM THE LANDSAT THEMATIC MAPPER EXPERIMENT 496
3. SCENE DESCRIPTION 497
4. COLOR ENLARGEMENT OF THE STADIUM 500
5. INDIVIDUAL BANDS 500
6. CONCLUSION 501
REFERENCES 504
CHAPTER 33. CLASSIFICATION, SYMMETRY AND THE PERIODIC TABLE 506
EXPLANATION AND CLASSIFICATION: STAMP COLLECTING RECONSIDERED 506
SORTING MAPS AND CLASSIFICATION 509
SYMMETRY AND CLASSIFICATION: THE PERIODIC TABLE 511
CHOOSING A MODERN PERIODIC TABLE 516
ADVANTAGES OF THE STEP-PYRAMID TABLE 518
ABUSES OF SYMMETRY 527
REFERENCES 528
Part 2 530
CHAPTER 34. CRYSTALLOGRAPHIC PATTERNS 534
REFERENCES 552
CHAPTER 35. SYMMETRY IN A NATURAL FRACTURE PATTERN: THEORIGIN OF COLUMNAR JOINT NETWORKS 554
INTRODUCTION 554
THE SYMMETRY OF COLUMNAR NETWORKS 554
JUNCTIONS IN COLUMNAR NETWORKS 557
ORIGIN OF JUNCTIONS 557
SUMMARY 567
REFERENCES 567
CHAPTER 36. CHIRALITY IN THE WORLD OF STEREOCHEMISTRY 570
INTRODUCTION 570
SYMMETRY AND CHIRALITY IN STEREOCHEMISTRY 571
ALGEBRAIC PROPERTIES OF CHEMICAL SETS [4, 11] 571
CHIRAL SETS [4, 12] 573
THE RUCH THEOREM AND HOMOCHIRALITY 575
CHIRALITY, CONFORMATION, CONFIGURATION, AND THEIR INTERRELATIONSHIP 579
THE GEOMETRICAL THEORY OF CHIRALITY[22] 580
THE NOETHER THEOREM AND CONFIGURATION 584
CONCLUSIONS 584
REFERENCES 585
CHAPTER 37. GEOMETRY AND CRYSTAL SYMMETRY 588
1. INTRODUCTION 588
2. LOCAL SYMMETRY AND GLOBAL STRUCTURE 590
3. THE STRUCTURE OF A REGULAR SYSTEM OF POINTS 593
4. LATTICES AND POLYHEDRA 594
5. CRYSTAL TWINS 598
REFERENCES 601
CHAPTER 38. THE GENERALIZED PETERSEN GRAPH G(24, 5) 602
1. INTRODUCTION 602
2. THE PARTIAL PLANE 83 602
3. ROTATION GROUPS IN 4 DIMENSIONS 604
4. A NEW ROLE FOR THE GRAPH G(24/5) 604
REFERENCES 606
CHAPTER 39. AN INTERPLAY BETWEEN THE PHENOMENON OF CHEMICAL ISOMERISM AND SYMMETRY REQUIREMENTS: A PERENNIAL SOURCE OF STIMULI FOR MOLECULAR-STRUCTURE CONCEPTS, AS WELL AS FOR ALGEBRAIC AND COMPUTATIONAL CHEMISTRY 608
1. INTRODUCTION 608
2. THE CONCEPT OF POTENTIAL-ENERGY HYPERSURFACE 609
3. SYMMETRY PROPERTIES OF STATIONARY POINTS 610
4. ENUMERATION OF ISOMERIC STRUCTURES 618
5. TOPOLOGICAL REDUCTION AND CHARACTERIZATION OF POTENTIAL HYPERSURFACES 621
6. THE GRAPH-LIKE STATE OF MOLECULES 623
7. INTERPLAY OF ISOMERS OF DIFFERENT SYMMETRIES 625
8. CONCLUSIONS 631
REFERENCES 631
CHAPTER 40. MILTON'S MATHEMATICAL SYMBOL OF THEODICY 640
ENDNOTES 649
CHAPTER 41. SYMMETRY IN COURT AND COUNTRY DANCE 652
INTRODUCTION 652
RHYTHM 652
FLOOR PLANS 654
SYMMETRY RELATIONS BETWEEN DANCERS 657
CONTRA- AND COUNTRY DANCES 658
SCOTTISH COUNTRY DANCES 660
CONCLUSIONS 662
REFERENCES 662
CHAPTER 42. SYMMETRY IN MOORISH AND OTHER ORNAMENTS 664
LEGENDS FOR COLOR FIGURES ON PAGES 645–646 667
REFERENCES 675
CHAPTER 43. THE GEOMETRY OF COASTLINES: A STUDY IN FRACTALS 678
1. INTRODUCTION 678
2. HISTORICAL PERSPECTIVE 678
3. COASTLINES 679
4. A GEOMETRIC MODEL OF A COASTLINE 682
5. A STATISTICAL MODEL OF A COASTLINE 689
6. CONCLUSION 693
REFERENCES 693
APPENDIX 694
CHAPTER 44. IN BLACK AND WHITE: HOW TO CREATE PERFECTLY COLORED SYMMETRIC PATTERNS 696
1. ISOMETRIES AND THE GENERATION OF DESIGNS AND TILINGS 696
2. SYMMETRY GROUPS AND THE CLASSIFICATION OF DESIGNS 701
3. PERFECT COLORING IN BLACK AND WHITE 708
4. PERFECTLY COLORING DESIGNS AND TILINGS WITH THREE OR MORE COLORS 716
REFERENCES 717
CHAPTER 45. SYMMETRY, HYBRIDIZATION AND BONDING IN MOLECULES 720
INTRODUCTION 720
HYBRIDIZATION OF ATOMIC ORBITALS 722
HYBRIDIZATION AND MOLECULAR PROPERTIES 731
SYMMETRY, HYBRID ORBITALS AND MORE INVOLVED MOLECULAR WAVEFUNCTIONS 740
FINAL REMARKS AND CONCLUSIONS 741
REFERENCES 742
CHAPTER 46. SYMMETRY IN MUSLIM ARTS 748
REFERENCES 773
CHAPTER 47. NON-SPACE-GROUP SYMMETRY IN CRYSTALLOGRAPHY 774
1. INTRODUCTION: THE THREE-DIMENSION ALLY PERIODIC CRYSTAL 774
2. GENERALIZATIONS OF SPACE GROUPS AND EXAMPLES OF NON-SPACE-GROUP SYMMETRIES 775
3. PARTIAL SYMMETRIES IN MOLECULAR CRYSTALS 780
4. POLYTYPIC STRUCTURES, THE OD THEORY OF DORNBERGER-SCHIFF 780
5. DESCRIPTION OF THE SYMMETRY OF AN OD STRUCTURE 782
6. GENERALIZATIONS OF THE DEFINITION OF A CRYSTAL 784
REFERENCES 784
CHAPTER 48. VORONOI POLYHEDRA AND THE DOCKING PROBLEM 786
REFERENCES 789
CHAPTER 49. PATTERN SYMMETRY AND COLORED REPETITION IN CULTURAL CONTEXTS 790
1. INTRODUCTION 790
2. SYMMETRY IN CULTURAL ANALYSES 790
3. SYMMETRY AND COLORED REPETITION 791
SUMMARY 802
REFERENCES 802
CHAPTER 50. SYMMETRY IN ARCHAEOLOGY 806
1. THE PROBLEMS CONCERNING MATHEMATICS AND ART 806
2. STEPS IN THE DEFINITION OF "SYMMETRY" 806
3. ORNAMENTS OF THE TELL HALAF CERAMICS 808
4. THE MOST GENERAL DEFINITION OF SYMMETRY 808
5. SCIENCE AND ART 809
REFERENCES 809
CHAPTER 51. STRANGE PATTERNS GENERATED FROM SIMPLE RULES WITH POSSIBLE RELEVANCE TO CRYSTALLIZATION AND OTHER PHENOMENA 812
BASIC PRINCIPLES 812
PRELIMINARY OBSERVATIONS OF SOME PROPERTIES 812
MATHEMATICAL REPRESENTATION 813
COMPUTERIZATION 813
ABUNDANCIES OF THE NUMBERS 813
THE PATTERN OF FIVES 813
REGULAR PATTERNS 814
SIMPLIFIED PROCEDURES 818
PHYSICAL ANALOGIES 818
EFFECT OF DIFFERENT INTEGERS AT ORIGIN 818
EFFECTS OF MISPLACED INTEGERS NEAR THE ORIGIN WHILE R(0, 0) = 1 819
EFFECTS OF MISPLACED INTEGERS NEAR ORIGIN WHILE R(0, 0) = 5 822
CONCLUSION 824
CHAPTER 52. TOWARDS A GRAMMAR OF INORGANIC STRUCTURE 826
INTRODUCTION 826
A GRAMMAR OF STRUCTURE 828
THE CONCEPT OF CURVATURE: THE GENERALISED SPHEROMETER 829
MINIMAL SURFACES 832
MATHEMATICAL DESCRIPTION 834
THE OCCURRENCE OF MINIMAL SURFACES 835
MINIMAL SURFACES AND ELECTROSTATIC POTENTIAL 837
NUMERICAL CALCULATION OF MINIMAL SURFACES 839
THE RELATIVE STABILITY OF SURFACES 840
PENTAGONAL SYMMETRY: SHORT-RANGE vs LONG-RANGE ORDER 842
CONCLUSIONS 844
REFERENCES 845
CHAPTER 53. SYMMETRIES OF SOAP FILMS 848
REFERENCES 857
CHAPTER 54. NOTES ON A VISUAL PHILOSOPHY 858
CHAPTER 55. SYMMETRY IN POLYMER SCIENCE 872
1. ISOMERS AND DIMENSION 872
2. STEREOREGULARITY IN POLYPROPYLENE 874
3. CONFORMATION IN POLY(VINYLIDENE FLUORIDE) 878
REFERENCES 880
CHAPTER 56. KEKULÉ STRUCTURES AND THEIR SYMMETRY PROPERTIES 882
INTRODUCTION 882
BENZENOID SYSTEMS: BASIC CONCEPTS 884
KEKULÉAN AND NON-KEKULÉAN SYSTEMS 886
KEKULÉ STRUCTURES AND GRAPH THEORY 886
SYMMETRY OF KEKULÉ STRUCTURES 888
CATA-CONDENSED BENZENOIDS 891
SOME CLASSES OF RETICULAR BENZENOIDS 893
MULTIPLE ZIG-ZAG CHAINS 893
REFERENCES 899
CHAPTER 57. THE GAPON FORMALISM 900
INTRODUCTION 900
1. CLASSIFICATION OF ELECTRONIC STATES 900
2. PAIRS OF HIGHER ORDER 901
3. SYMMETRY BREAKING AND MULTIPLETS 903
REFERENCES 904
CHAPTER 58. ON SYMMETRIES IN THE GRAPHIC ART OF HORST BARTNIG 906
REFERENCES 916
CHAPTER 59. COLORED SYMMETRIES IN SPACE-TIME 918
INTRODUCTION 918
FORM FOLLOWS PROCESS 919
GENESIS OF THE ELEMENT 921
TRIHEDRAL FORMATION 921
THE NEED FOR COLOR 924
EMERGENCE OF A DECAGON 926
EMERGENCE OF TRIHEDRON 929
COLORED SYMMETRIES IN SPACE-TIME 930
CONCLUSION 934
REFERENCES 934
CHAPTER 60. A COURSE IN THE MATHEMATICS OF DESIGN 936
1. INTRODUCTION 936
2. GRAPH THEORY 937
3. POLYHEDRA 940
4. TILINGS OF THE PLANE 944
5. TILING OF THREE-DIMENSIONAL SPACE 949
6. SIMILARITY, PROPORTION AND THE GOLDEN MEAN 956
7. TRANSFORMATIONS 960
8. SYMMETRY 964
9. VECTORS 965
10. CONCLUSION 968
REFERENCES 969
APPENDIX: STUDENT RESPONSES TO WRITING PROJECTS 970
CHAPTER 61. SYMMETROLOGY OF ART: COLOURED AND GENERALIZED SYMMETRIES 972
INTRODUCTION 972
SOME FUNDAMENTAL REMARKS 972
THE CATEGORY OF COLOR 973
DICHROIC (BLACK-AND-WHITE) GROUPS 976
COLORED SYMMETRY 978
GENERALIZATIONS OF SYMMETRY 987
ADDITIONAL CATEGORIES AND COMPLICATIONS 996
EPILOGUE 1001
REFERENCES 1002
CHAPTER 62. A RECURRING GEOMETRICAL PATTERN IN THE EARLY RENAISSANCE IMAGINATION 1004
REFERENCES 1019
CHAPTER 63. SYMMETRY IN CHEMICAL STRUCTURES AND REACTIONS 1022
INTRODUCTION 1022
SYMMETRY AT THE ATOMIC LEVEL: ATOMS, ELECTRONS AND THE PERIODIC SYSTEM 1023
SYMMETRY AT THE MOLECULAR LEVEL: POLYHEDRA, SOLID ANGLES AND CHEMISTRY 1025
ISOMERISM AND PÓLYA'S THEOREM 1028
JAHN-TELLER EFFECTS 1031
SYMMETRY OF LINEAR MACROMOLECULES 1033
SYMMETRY AT THE SUPRAMOLECULAR LEVEL: CONSERVATION OF ORBITAL SYMMETRY, AND REACTIONS WITH AROMATIC TRANSITION STATES 1034
REACTION GRAPHS HEURISTIC SYMMETRY OF (3, .)-CAGES
SPHERES AND CHEMISTRY 1038
REFERENCES 1040
CHAPTER 64. THE VSEPR MODEL OF MOLECULAR GEOMETRY 1044
1. INTRODUCTION 1044
2. THE BASIC POSTULATE 1046
3. MOLECULAR SHAPES AND BOND ANGLES 1048
4. TRIGONAL BIPYRAMIDAL CONFIGURATIONS 1050
5. MORE THAN FIVE ELECTRON PAIRS 1052
6. TESTING THE APPLICABILITY OF THE MODEL 1053
7. CONCLUDING REMARKS 1058
REFERENCES 1060
CHAPTER 65. THE MATHEMATICAL THEORY OF CHAOS 1062
1. ONE-DIMENSIONAL CHAOS: INTERVAL MAPS 1062
2. TWO-DIMENSIONAL CHAOS: HORSESHOES AND SOLENOIDS 1065
3. THREE-DIMENSIONAL CHAOS: THE LORENZ EQUATION 1067
REFERENCES 1068