Hargittai Symmetry 2

Unifying Human Understanding
1. Auflage 2016
ISBN: 978-1-4832-9949-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 18, 1084 Seiten, Web PDF

Reihe: International Series in Modern Applied Mathematics and Computer Science

Symmetry 2 aims to present an overview of the contemporary status of symmetry studies, particularly in the arts and sciences, emphasizing both its role and importance. Symmetry is not only one of the fundamental concepts in science, but is also possibly the best unifying concept between various branches of science, the arts and other human activities. Whereas symmetry has been considered important for centuries primarily for its aesthetic appeal, this century has witnessed a dramatic enhancement of its status as a cornerstone in the sciences. In addition to traditionally symmetry-oriented fields such as crystallography and spectroscopy, the concept has made headway in fields as varied as reaction chemistry, nuclear physics, and the study of the origin of the universe. The book was initiated in response to the success of the first volume, which not only received good reviews, but received the award for 'The Best Single Issue of a Journal' by the Association of American Publishers for 1986. The second volume extends the application of symmetry to new fields, such as medical sciences and economics, as well as investigating further certain topics introduced in Symmetry. The book is extensively illustrated and with over 64 contributions from 16 countries presents an international overview of the nature and diversity of symmetry studies today.

Autoren/Hrsg.

Hargittai, I.

Inhaltsverzeichnis

1;Front Cover;1
2;Symmetry 2: Unifying Human Understanding;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Part 1: SYMMETRY 2 Unifying Human Understanding ;12
6.1;CHAPTER 1. SYMMETRY REVISITED;16
6.1.1;1. INTRODUCTION;16
6.1.2;2. CLASSICAL SYMMETRY;18
6.1.3;3. REREADING WEYL;21
6.1.4;4. "ENLARGING OUR DEFINITION OF HOMOGENEITY";24
6.1.5;REFERENCES;27
6.2;CHAPTER 2. SYMMETRY AT THE FOUNDATIONS OF SCIENCE;28
6.2.1;1. INTRODUCTION;28
6.2.2;2. SYMMETRY;28
6.2.3;3. REPRODUCIBILITY AS SYMMETRY;29
6.2.4;4. PREDICTABILITY AS SYMMETRY;30
6.2.5;5. CONCLUSION;30
6.2.6;BIBLIOGRAPHY;30
6.3;CHAPTER 3. SYMMETRY AND CHAOS;32
6.3.1;1. INTRODUCTION;32
6.3.2;SECTION 2;33
6.3.3;SECTION 3;33
6.3.4;SECTION 4;35
6.3.5;SECTION 5;36
6.3.6;SECTION 6;36
6.3.7;SECTION 7;37
6.3.8;SECTION 8;39
6.3.9;SECTION 9;39
6.3.10;SECTION 10;42
6.3.11;SECTION 11;44
6.3.12;REFERENCES;46
6.4;CHAPTER 4. THE MAGIC OF THE PENTANGLE: DYNAMIC SYMMETRY FROM MERLIN TO PENROSE;48
6.4.1;1. THE BEETLES;48
6.4.2;2. DYNAMIC SYMMETRY;52
6.4.3;3. THE SQUARE GNOMON;53
6.4.4;4. FIBONACCI'S NUMBERS;54
6.4.5;5. THE GOLDEN TRIANGLE;56
6.4.6;6. SIR GAWAIN;58
6.4.7;7. MERLIN UNLEASHED?;61
6.4.8;8. EPILOGUE;62
6.4.9;REFERENCES;62
6.5;CHAPTER 5. SYMMETRY, GROUPOIDS AND HIGHER-DIMENSIONAL ANALOGUES!;64
6.5.1;REFERENCES;72
6.6;CHAPTER 6. TRISECTING AN ORTHOSCHEME;74
6.6.1;1. FRIEZE PATTERNS;74
6.6.2;2. THE GENERAL ORTHOSCHEME IN ABSOLUTE 3-SPACE;74
6.6.3;3. THE DISSECTION INTO THREE ORTHOSCHEMES;75
6.6.4;4. TWO LINES CROSSING AT RIGHT ANGLES;76
6.6.5;5. AN OCTAHEDRON DISSECTED INTO 16 OR 48 ORTHOSCHEMES;77
6.6.6;6. POLYGONOMETRY;77
6.6.7;7. LOBACHEVSKY'S "PYRAMID";80
6.6.8;8. TRISECTING A EUCLIDEAN ORTHOSCHEME;82
6.6.9;9. A CHALLENGING DEFINITE INTEGRAL;84
6.6.10;REFERENCES;86
6.7;CHAPTER 7. DUALITY AND THE DESCARTES DEFICIENCY;88
6.7.1;1. INTRODUCTION;88
6.7.2;2. POLYHEDRA;90
6.7.3;3. DUAL POLYHEDRAt;94
6.7.4;4. DESCARTES ANGULAR DEFICIENCY . AND ITS DUAL .';96
6.7.5;5. THE RELATIONSHIP BETWEEN ., . AND .;98
6.7.6;6. EXAMPLES;101
6.7.7;7. CONCLUDING REMARKS;103
6.7.8;REFERENCES;103
6.8;CHAPTER 8. EXTENDING THE BINOMIAL COEFFICIENTS TO PRESERVE SYMMETRY AND PATTERN;104
6.8.1;1. INTRODUCTION;104
6.8.2;2. EXTENDING THE DOMAIN OF THE BINOMIAL COEFFICIENTS OVER THE FULL RANGE OF INTEGERS;106
6.8.3;3. SOME PATTERNS IN THE PASCAL HEXAGON;110
6.8.4;4. WEIGHTS IN THE PASCAL FLOWER;114
6.8.5;5. EXTENDING THE DOMAIN OF THE BINOMIAL COEFFICIENTS OVER THE COMPLEX NUMBERS;115
6.8.6;REFERENCES;117
6.9;CHAPTER 9. SYMMETRY IN THE SIMPLEST CASE: THE REAL LINE;118
6.9.1;POINTS OF (LOCAL) SYMMETRY;118
6.9.2;SYMMETRIC AND LOCALLY SYMMETRIC SETS;118
6.9.3;IS ABSENCE OF SYMMETRY RELATED TO DISORDER?;118
6.9.4;POINTS OF NON-SYMMETRY;118
6.9.5;THREE WAYS IN WHICH SYMMETRY OF A SET MAY BE TRANSGRESSED;119
6.9.6;ANTISYMMETRIC SETS;119
6.9.7;SYMMETRY OF A SET A WITH RESPECT TO POINTS WHICH DO NOT BELONG TO A;120
6.9.8;MORE SOPHISTICATED EXAMPLES OF SYMMETRIC SETS;120
6.9.9;STRUCTURAL IDENTITY OF SYMMETRY AND ANTISYMMETRY;121
6.9.10;HOW RESTRICTIVE ARE ASYMMETRIC SETS?;121
6.9.11;DISORDER MEANS COMBINATION OF SYMMETRY AND NON-SYMMETRY;121
6.9.12;SYMMETRY MODULO A NEGLIGIBLE SET;122
6.9.13;STRONG SYMMETRY;122
6.9.14;STRONG ASYMMETRY AND STRONG ANTISYMMETRY;123
6.9.15;SYMMETRIC CONTINUITY;124
6.9.16;OTHER ASPECTS OF SYMMETRIC CONTINUITY;124
6.9.17;LOCALLY SYMMETRIC FUNCTIONS;125
6.9.18;NON-TRIVIAL SYMMETRIC CONTINUITY IS NEVER UNIFORM;125
6.9.19;SYMMETRY STRUCTURE OF A SET AND OF A FUNCTION: AN OPEN PROBLEM;126
6.9.20;SYMMETRIC DERIVATIVES: THEIR HISTORICAL ORIGIN;126
6.9.21;SYMMETRIC DERIVATIVE;127
6.9.22;SYMMETRIC DERIVATIVE VERSUS ORDINARY DERIVATIVE;127
6.9.23;SYMMETRIC DERIVATIVE AND MONOTONICITY;127
6.9.24;SCHWARTZ SYMMETRIC FUNCTIONS;128
6.9.25;OTHER RESULTS, PROBLEMS AND SUGGESTIONS;128
6.9.26;SOME CONCEPTUAL AND PHILOSOPHICAL DIFFICULTIES;129
6.9.27;REFERENCES;130
6.10;CHAPTER 10. DELICATE SYMMETRY;132
6.10.1;1. INTRODUCTION;132
6.10.2;2. TOTAL CHAOS IS IMPOSSIBLE;132
6.10.3;3. FRAGILE SYMMETRY;134
6.10.4;4. FLEXIBLE SYMMETRY;136
6.10.5;REFERENCES;139
6.11;CHAPTER 11. HALLEY MAPS FOR A TRIGONOMETRIC AND RATIONAL FUNCTION;140
6.11.1;REFERENCES;147
6.12;CHAPTER 12. A CLASS OF SYMMETRIC POLYTOPES;148
6.12.1;INTRODUCTION;148
6.12.2;REFERENCES;151
6.13;CHAPTER 13. ARRANGEMENTS OF MINIMAL VARIANCEMULTIDIMENSIONAL SCALING IN THE SYMMETRICAL CASE;152
6.13.1;1. ARRANGEMENT OF POINTS IN THE PLANE WITH DISTANCE OF MINIMAL VARIANCE;152
6.13.2;2. MULTIDIMENSIONAL SCALING;155
6.13.3;3. MULTIPLE MDS;156
6.13.4;4. MMDS IN THE SYMMETRICAL CASE;159
6.13.5;5. MMDS OF CONGENITAL ABNORMALITIES;159
6.13.6;REFERENCES;160
6.14;CHAPTER 14. TILINGS BY REGULAR POLYGONS—II A CATALOG OF TILINGS;162
6.14.1;INTRODUCTION;162
6.14.2;REGULARITY PROPERTIES;162
6.14.3;THE TILINGS;164
6.14.4;REFERENCES;180
6.15;CHAPTER 15. SYMMETRY AND POLYHEDRAL STELLATION—Iaf;182
6.15.1;INTRODUCTION;182
6.15.2;THE STRUCTURE OF THE ASYMMETRICAL STELLATION PATTERN;182
6.15.3;ANALYSIS OF THE STELLATION PATTERN;186
6.15.4;REFERENCES;190
6.16;CHAPTER 16. SYMMETRY AND POLYHEDRAL STELLATION—Ib;192
6.16.1;MODEL MAKING BY MEANS OF THE STELLATION PATTERN;192
6.16.2;SYMMETRICAL COMPOUNDS;203
6.16.3;APPLICATIONS;206
6.16.4;REFERENCES;208
6.17;CHAPTER 17. SYMMETRY AND POLYHEDRAL STELLATION—II;210
6.17.1;INTRODUCTION;210
6.17.2;THE STRUCTURE OF A SYMMETRICAL STELLATION PATTERN;210
6.17.3;REFERENCES;216
6.18;CHAPTER 18. THE COMPLETE SET OF JITTERBUG TRANSFORMERS AND THE ANALYSIS OF THEIR MOTION;218
6.18.1;1. INTRODUCTION;218
6.18.2;2. MATHEMATICAL APPROACH;218
6.18.3;3. POLYGON-DIPOLYGON-DIPOLYGONID;219
6.18.4;4. JITTERBUG AND DIPOLYGONIDS;227
6.18.5;5. FINITE DIPOLYGONIDS;227
6.18.6;6. CLASSIFICATION OF THE FINITE DIPOLYGONIDS;230
6.18.7;7. ASPECTS OF UNIFORM MOTION;237
6.18.8;8. EXTENDED GROUPS OF ISOMETRIES;238
6.18.9;9. REGULAR DIPOLYGONIDS;241
6.18.10;10. EQUIRADIAL DIPOLYGONIDS;248
6.18.11;11. PAIRS OF CHIRAL DIPOLYGONIDS;248
6.18.12;12. THE USE OF DIPOLYGONIDS;259
6.18.13;REFERENCES;265
6.19;CHAPTER 19. A GEOMETRICAL ANALOGUE OF THE PHASE TRANSFORMATION OF CRYSTALS;266
6.19.1;REFERENCES;269
6.20;CHAPTER 20. COMPUTERS AND GROUP-THEORETICAL METHODS FOR STUDYING STRUCTURAL PHASE TRANSITIONS;270
6.20.1;1. INTRODUCTION;270
6.20.2;2. GROUP-THEORETICAL METHODS FOR ANALYSIS OF PHASE TRANSITIONS;270
6.20.3;3. OBTAINING FULL IR'S OF THE SPACE GROUPS;275
6.20.4;4. OBTAINING A SET OF NONEQUIVALENT STATIONARY VECTORS;276
6.20.5;5. IDENTIFICATION OF SUBGROUPS OF A SPACE GROUP;282
6.20.6;6. OBTAINING THE COMPLETE CONDENSATE OF STATIONARY VECTORS;287
6.20.7;7. CONSTRUCTION OF BASIS FUNCTIONS OF THE IR's OF SPACE GROUPS;288
6.20.8;8. SPLITTING OF THE ORBITS OF A SPACE GROUP INDUCED BY A PHASE TRANSITION;289
6.20.9;9. A COMPARISON BETWEEN VARIOUS APPROACHES TO THE GROUP-THEORETICAL ANALYSIS OF PHASE TRANSITIONS IN CRYSTALS;289
6.20.10;REFERENCES;291
6.21;CHAPTER 21. STRUCTURAL THEORY OF SPACE-TIME AND INTRAPOINT SYMMETRY;294
6.21.1;INTRODUCTION;294
6.21.2;THE UNIVERSE AS A MULTILEVEL SYSTEM OF CONTACTING NATURAL REGIONS;295
6.21.3;THE UNIVERSE AS MULTILEVEL MATERIAL SPACE;301
6.21.4;THE UNIVERSE AS TIME AND LIGHT;308
6.21.5;REFERENCES;313
6.22;CHAPTER 22. VISUAL AND HIDDEN SYMMETRY IN GEOMETRY‡;316
6.22.1;1. CRITICAL POINTS OF SMOOTH FUNCTIONS ON THREE-DIMENSIONAL MANIFOLDS;316
6.22.2;2. TWO-ADIC SOLENOID;316
6.22.3;3. ALGEBRAIC SURFACE IN TWO-DIMENSIONAL COMPLEX SPACE;316
6.22.4;4. TWO-DIMENSIONAL SPHERE, POISSON SPHERE, NUTATION AND PRECESSION;317
6.22.5;5. HEEGARD DIAGRAMS OF THREE-DIMENSIONAL MANIFOLDS;317
6.22.6;6. ALGEBRAIC FUNCTIONS AND FUNDAMENTAL DOMAINS OF ACTION OF DISCRETE GROUPS;317
6.22.7;7. CRYSTAL STRUCTURES. THE PARTITIONING OF THE SPACE INTO SIMPLEXES;326
6.22.8;8. CRYSTAL STRUCTURES. CUBIC PARTITIONINGS OF POLYHEDRA;326
6.22.9;9. SYMMETRIC SPACES;326
6.22.10;10. SYMMETRY OF PLANE WAVES;326
6.22.11;11. CUBIC LATTICES;326
6.22.12;12. CHAOS;327
6.22.13;13. HIDDEN SYMMETRICS;327
6.22.14;15. MATHEMATICAL FANTASY;335
6.22.15;REFERENCES;335
6.23;CHAPTER 23. MATHEMATICS AND BEAUTY—VIII TESSELATION AUTOMATA DERIVED FROM A SINGLE DEFECT;336
6.23.1;INTRODUCTION;336
6.23.2;MOTIVATION;337
6.23.3;METHOD AND OBSERVATIONS;337
6.23.4;SUMMARY AND CONCLUSIONS;344
6.23.5;REFERENCES;350
6.23.6;APPENDIX;351
6.24;CHAPTER 24. INTERPRETATION OF SO-CALLED ICOSAHEDRAL AND DECAGONAL QUASICRYSTALS OF ALLOYS SHOWING APPARENT ICOSAHEDRAL SYMMETRY ELEMENTS AS TWINS OF AN 820-ATOM CUBIC CRYSTAL;352
6.24.1;REFERENCES;354
6.25;CHAPTER 25. LOCAL PSEUDOSYMMETRY IN SIMPLE LIQUIDS;356
6.25.1;1. INTRODUCTION;356
6.25.2;2. BUILDING UP A LIQUID FROM ITS ELEMENTARY UNIT;358
6.25.3;3. SHORT-RANGE ORDER IN SIMPLE LIQUIDS;358
6.25.4;4. LOCAL SUB-UNITS IN CLOSE-PACKED STRUCTURES;359
6.25.5;5. A TWO-DIMENSIONAL ANALOGUE—KAWAMURA'S MODEL;360
6.25.6;6. PACKING TETRAHEDRA AND OCTAHEDRA IN THREE DIMENSIONS;361
6.25.7;7. PACKINGS OF SOFT SPHERES;362
6.25.8;8. EVIDENCE FROM MODEL STUDIES;363
6.25.9;9. INTERMEDIATE RANGE ORDER;365
6.25.10;10. SUMMARY;367
6.25.11;REFERENCES;369
6.26;CHAPTER 26. TOPOLOGICAL ASPECTS OF BENZENOIDS AND CORONOIDS, INCLUDING "SNOWFLAKES" AND "LACEFLOWERS";370
6.26.1;1. INTRODUCTION;370
6.26.2;2. EXAMPLES OF THE RESEARCH PROGRESS;371
6.26.3;3. SOME DEFINITIONS AND THE INVARIANT .;373
6.26.4;4. FURTHER DEFINITIONS AND THE "neo" CLASSIFICATION;374
6.26.5;5. ENUMERATION AND CLASSIFICATION OF BENZENOIDS WITH HEXAGONAL SYMMETRY;376
6.26.6;6. ANALYSING SINGLE CORONOIDS: THE "rheo" AND "r/o" CLASSIFICATIONS;377
6.26.7;7. COMBINATORIAL K FORMULAS FOR CLASSES OF SOME CORONOIDS WITH HEXAGONAL SYMMETRY;384
6.26.8;8. MULTIPLE CORONOIDS WITH HEXAGONAL SYMMETRY;388
6.26.9;REFERENCES;388
6.27;CHAPTER 27. A SET THEORETIC APPROACH TO THE SYMMETRY ANALYSIS OF HEXADECAHEDRANE;390
6.27.1;INTRODUCTION;390
6.27.2;SYMMETRY ELEMENTS AND SYMMETRY OPERATIONS;391
6.27.3;SOME SET THEORETIC CONCEPTS;391
6.27.4;SYMMETRY POINT SETS AND GENERATOR SETS;392
6.27.5;MOLECULAR AND LOCAL SYMMETRY POINT SETS;396
6.27.6;INTERSECTION OF LPS (INTERSECTION SETS);397
6.27.7;INTERCHANGE SETS;397
6.27.8;SYMMETRY POINT SETS AND SYMMETRY POINT GROUPS;397
6.27.9;APPLICATIONS OF SYMMETRY POINT SETS;398
6.27.10;CONSTRUCTION OF THE MPS OF HEXADECAHEDRANE;398
6.27.11;CONTRIBUTIONS OF THE LPG TO THE MPG OF HEXADECAHEDRANE;407
6.27.12;CONCLUSIONS;408
6.27.13;REFERENCES;409
6.28;CHAPTER 28. CARBON AND ITS NETS;412
6.28.1;1. INTRODUCTION;412
6.28.2;2. THE MOST PRECIOUS ATOMS IN THE UNIVERSE, OR WHY C?;412
6.28.3;3. THE TWO KNOWN ALLOTROPIC FORMS OF C: GRAPHITE AND DIAMOND, OR THE SOFTEST AND HARDEST CRYSTALS;413
6.28.4;4. CARBON MOLECULES;417
6.28.5;5. HYPOTHETICAL ALTERNATIVE C ALLOTROPES (INFINITE NETS);418
6.28.6;6. RELATED NETS WITH OTHER ATOMS INSTEAD OF C;424
6.28.7;7. SYMMETRY OF FRAGMENTS FROM THE GRAPHITE LATTICE;425
6.28.8;8. SYMMETRY OF FRAGMENTS FROM THE DIAMOND LATTICE;427
6.28.9;9. SYMMETRY AND STABILITY OF MOLECULES;429
6.28.10;10. CONCLUSION;430
6.28.11;REFERENCES;430
6.29;CHAPTER 29. CB 6o BUCKMINSTERFULLERENE, OTHER FULLERENES AND THE ICOSPIRAL SHELL;432
6.29.1;INTRODUCTION;432
6.29.2;THE ICOSPIRAL PARTICLE AND THE GIANT FULLERENES;435
6.29.3;SUMMARY OF SYMMETRY ASPECTS;437
6.29.4;REFERENCES;438
6.30;CHAPTER 30. ASYMMETRY THROUGH THE EYES OF A(NOTHER) CHEMIST!;440
6.30.1;SYMMETRY AND CHIRALITY—PRINCIPLE OF APPROPRIATENESS;440
6.30.2;DOES NATURE TEND TOWARD SYMMETRY OR ASYMMETRY?;442
6.30.3;INTERACTIONS AMONG ASYMMETRIC (CHIRAL) MOLECULES AND CHIRAL SYSTEMS;443
6.30.4;MANIFESTATION OF ASYMMETRY IN MOLECULAR DIMENSIONS;444
6.30.5;SEPARATION OF OPTICAL ISOMERS;447
6.30.6;THE THREE-POINT INTERACTION MODEL OF CHIRAL DISCRIMINATION;449
6.30.7;MACROSCOPIC DIFFERENCES AND WEAK SECOND ORDER INTERACTIONS;450
6.30.8;WHY IS TARTARIC ACID A GOOD RESOLVING AGENT?;451
6.30.9;SYMMETRY VERSUS CHIRALITY AND THE PRINCIPLE OF COMPLEMENTARITY;452
6.30.10;REFERENCES;455
6.31;CHAPTER 31. CHEMICAL KINETICS AND THERMODYNAMICS A HISTORY OF THEIR RELATIONSHIP;458
6.31.1;1. WHAT IS CHEMICAL AFFINITY?;458
6.31.2;2. HOW CHEMICAL REACTIONS PROCEED AND WHY?;459
6.31.3;3. WHAT IS PROCESS THERMODYNAMICS?;460
6.31.4;4. THE UNIFIED FIELD THEORY OF THERMODYNAMICS;462
6.31.5;5. LINEARLY INDEPENDENT CHEMICAL REACTIONS AND THEIR ENTROPY PRODUCTION;463
6.31.6;6. EXTENSION BEYOND LOCAL EQUILIBRIUM;464
6.31.7;7. EXTENSION TO NON-LINEAR PHENOMENOLOGICAL EQUATIONS;464
6.31.8;8. WHAT ARE THE FORCES DRIVING CHEMICAL REACTIONS TO ADVANCE? HOW IS THE EVOLUTION OF CHEMICAL KINETIC SYSTEMS GOVERNED?;465
6.31.9;9. HOW DOES A CHEMICAL KINETIC SYSTEM EVOLVE IN TIME?;466
6.31.10;10. THE ROLE OF SYMMETRY IN THE STUDY OF IRREVERSIBLE PROCESSES;468
6.31.11;REFERENCES;469
6.32;CHAPTER 32. SYMMETRIES AND THE NOTION OF ELEMENTARITY IN HIGH ENERGY PHYSICS;472
6.32.1;1. INTRODUCTION;472
6.32.2;2. SUCCESSFUL METHODS, BROKEN SYMMETRIES AND THE QUESTION OF SIMPLICITY;473
6.32.3;3. THE NOTION OF OBSERVABILITY;475
6.32.4;4. CONCLUSION;478
6.32.5;REFERENCES;479
6.33;CHAPTER 33. SYMMETRY-BREAKING PATTERN FORMATION IN SEMICONDUCTOR PHYSICS: SPATIO-TEMPORAL CURRENT STRUCTURES DURING AVALANCHE BREAKDOWN;482
6.33.1;1. INTRODUCTION;482
6.33.2;2. SYSTEM;482
6.33.3;3. RESULTS;483
6.33.4;4. MODEL;486
6.33.5;5. CONCLUDING REMARKS;487
6.33.6;REFERENCES;487
7;Part 2: SYMMETRY 2 Unifying Human Understanding;490
7.1;CHAPTER 34. THE PERCEPTUAL VALUE OF SYMMETRY;494
7.1.1;1. INTRODUCTION;494
7.1.2;2. GLOBAL DETECTION OF SYMMETRY;496
7.1.3;3. VISUAL EXPLORATION OF SYMMETRICAL STIMULI;497
7.1.4;4. AROUSAL POTENTIAL OF SYMMETRICAL COMPOSITIONS;501
7.1.5;5. ENCODING SYMMETRY;502
7.1.6;6. CONCLUSION;502
7.1.7;REFERENCES;503
7.2;CHAPTER 35. MIND'S EYE;504
7.2.1;INTRODUCTION;504
7.2.2;VISION AS PROCESS: GENERAL CONSIDERATIONS;505
7.2.3;SYMMETRY: SOME GENERAL OBSERVATIONS;508
7.2.4;ROBOTIC VISION: ANALOGIES AND ANALYSIS;511
7.2.5;GENERAL IMPLICATIONS: RELATION TO SCIENTIFIC PROCESS;519
7.2.6;CONCLUSION: NEW WINE FROM NEW WINESKINS;521
7.2.7;REFERENCES;522
7.3;CHAPTER 36. NON-EUCLIDEAN GEOMETRIES AND ALGORITHMS OF LIVING BODIES!;524
7.3.1;1. THE ERLANGER PROGRAM AND GEOMETRIZATION IN BIOLOGY;524
7.3.2;2. THE HISTORY OF RESEARCH IN THE SYMMETRIES AND ALGORITHMS OF THE MORPHOLOGICAL SELF-ORGANIZATION IN BIOLOGICAL OBJECTS;526
7.3.3;3. NON-EUCLIDEAN GEOMETRIES AND ITERATIVE ALGORITHMS IN ORGANIC FORMS;528
7.3.4;4. EUCLIDEAN AND NON-EUCLIDEAN ITERATIVE ALGORITHMS IN THE KINEMATICS OF BIOLOGICAL MOVEMENTS AND PHYSIOLOGICALLY NORMAL POSTURES;533
7.3.5;5. THE KINEMATICS OF THE THREE-DIMENSIONAL GROWTH OF BIOLOGICAL BODIES WHOSE PARTS ARE NOT CYCLOMERICALLY POSITIONED;538
7.3.6;6. AUTONOMOUS AUTOMATA AND GROUP INVARIANT PROPERTIES OF LIVING ORGANISMS;542
7.3.7;7. APERIODIC ITERATIVE BIORHYTHMS;543
7.3.8;8. THE PROTOSYSTEM OF BIOLOGICAL REGULATION AND PSYCHOPHYSICAL MATTERS;544
7.3.9;9. CONCLUSIONS;549
7.3.10;REFERENCES;550
7.3.11;APPENDIX;551
7.4;CHAPTER 37. SPIRAL PHYLLOTAXIS;554
7.4.1;REFERENCES;557
7.5;CHAPTER 38. A SYMMETRY-ORIENTED MATHEMATICAL MODEL OF CLASSICAL COUNTERPOINT AND RELATED NEUROPHYSIOLOGICAL INVESTIGATIONS BY DEPTH EEG;558
7.5.1;1. INTRODUCTIONTOTHEPROBLEM;558
7.5.2;2. THE GEOMETRY OF INTERVAL DICHOTOMIES;566
7.5.3;3. COUNTERPOINT AS AN INFINITESIMAL VARIATION OF GREGORIAN CHORAL;574
7.5.4;4. LOCAL SYMMETRIES IN COUNTERPOINT;579
7.5.5;5. SOME MUSICOLOGICAL COMMENTS;583
7.5.6;6. METHODS, TEST MATERIAL AND ANALYSIS OF THE NEUROPHYSIOLOGICAL INVESTIGATIONS;585
7.5.7;7. GENERAL OBSERVATIONS OF EVOKED EEG RESPONSES;597
7.5.8;8. DEFECTIVE COUNTERPOINT;603
7.5.9;9. SPECTRAL DATA FROM ISOLATED CONSONANCES AND DISSONANCES;604
7.5.10;10. TESTING THE SYMMETRY OF AUTOCOMPLEMENTARITY ON THE INTERVAL SET;609
7.5.11;11. DISCUSSION;611
7.5.12;REFERENCES;612
7.6;CHAPTER 39. SQUARE SPIRALS, DIMENSIONALITY AND BIOPOLYMERS;614
7.6.1;REFERENCES;630
7.7;CHAPTER 40. THE GOLDEN SECTION IN THE MEASUREMENT THEORY;632
7.7.1;1. INTRODUCTION;632
7.7.2;2. THE GOLDEN SECTION;632
7.7.3;3. HOW DO RABBITS MULTIPLY?;635
7.7.4;4. SOME FIBONACCI TRIGONOMETRY;637
7.7.5;5. PASCAL'S TRIANGLE OR ANOTHER WAY OF RABBITS MULTIPLYING;638
7.7.6;6. HOW MANY GOLDEN SECTIONS CAN BE FOUND?;641
7.7.7;7. HOW THE FIRST CRISIS IN THE FOUNDATIONS OF MATHEMATICS WAS OVERCOME;643
7.7.8;8. CAN THE MEASUREMENT BE COMPLETED FOR THE INFINITE PERIOD OF TIME OR THE CONSTRUCTIVE APPROACH TO MEASUREMENT?;645
7.7.9;9. GEOMETRIC PROBLEM PUT FORWARD BY N. I. LOBACHEVSKI;645
7.7.10;10. THE PROBLEM OF CHOOSING THE BEST SYSTEM OF SCALE WEIGHTS OR HOW THE BINARY NOTATION WAS INVENTED;647
7.7.11;11. NEUGEBAUER'S HYPOTHESIS;648
7.7.12;12. ASYMMETRY PRINCIPLE OF MEASUREMENT;648
7.7.13;13. THE FUNDAMENTAL RESULTS OF THE ALGORITHMIC MEASUREMENT THEORY;649
7.7.14;14. UNEXPECTED RELATIONS;652
7.7.15;15. FIBONACCI CODES;653
7.7.16;16. THE GOLDEN p-RATIO CODES;655
7.7.17;17. CONCLUSIONS;656
7.7.18;REFERENCES;657
7.8;CHAPTER 41. BUCKLING PATTERNS OF SHELLS AND SPHERICAL HONEYCOMB STRUCTURES;658
7.8.1;1. INTRODUCTION;658
7.8.2;2. CIRCLES ON A SPHERE;659
7.8.3;3. BUCKLING OF COMPLETE SPHERICAL SHELLS;663
7.8.4;4. ANALOGIES IN NATURE;664
7.8.5;5. CONCLUSIONS;669
7.8.6;REFERENCES;670
7.9;CHAPTER 42. SYMMETRY IN FREE MARKETS;672
7.9.1;INTRODUCTION;672
7.9.2;A GAME OF ROULETTE;673
7.9.3;THE ROULETTE MARKET;673
7.9.4;SYMMETRY IN THE ROULETTE MARKET;674
7.9.5;A SYMMETRY VIOLATION IN THE BLACKJACK MARKET;675
7.9.6;THE PARI MUTUEL MARKET;676
7.9.7;SYMMETRY IN THE PARI MUTUEL MARKET;677
7.9.8;THE STOCK MARKET;679
7.9.9;SYMMETRY IN THE STOCK MARKET;680
7.9.10;SYMMETRY VIOLATION IN THE STOCK MARKET;682
7.9.11;SYMMETRY IN CONSUMER MARKETS;683
7.9.12;THE ROLE OF GOVERNMENT IN SYMMETRIC MARKETS;685
7.9.13;REFERENCES;687
7.9.14;APPENDIX A;687
7.9.15;APPENDIX B;688
7.9.16;APPENDIX C;688
7.10;CHAPTER 43. SYMMETRIES IN MUSIC TEACHING;690
7.10.1;SYMMETRIES IN MUSIC TEACHING;690
7.10.2;SYMMETRY OF INTENSITY;690
7.10.3;SYMMETRY OF DURATION —SYMMETRY OF RHYTHM;690
7.10.4;PITCH SYMMETRY —MELODY SYMMETRY;692
7.10.5;SYMMETRY OF HARMONY;697
7.10.6;SYMMETRY OF TONALITY;701
7.10.7;SYMMETRY OF FORM;706
7.10.8;GOLDEN SECTION;709
7.10.9;FCGDAEH;711
7.11;CHAPTER 44. UNCERTAINTY PRINCIPLE AND SYMMETRY IN METAPHORS;716
7.11.1;REFERENCES;725
7.12;CHAPTER 45. MODERN SYMMETRY;728
7.12.1;REFERENCES;732
7.13;CHAPTER 46. SYMMETRY AND TECHNOLOGY IN ORNAMENTAL ART OF OLD HUNGARIANS AND AVAR-ONOGURIANS FROM THE ARCHAEOLOGICAL FINDS OF THE CARPATHIAN BASIN, SEVENTH TO TENTH CENTURY A.D.;734
7.13.1;INTRODUCTION;734
7.13.2;OLD HUNGARIAN ORNAMENTAL ART;735
7.13.3;AVAR-ONOGURIAN ORNAMENTAL ART;740
7.13.4;THE SYMMETRY METHOD IN ARCHAEOLOGY;747
7.13.5;REFERENCES;748
7.14;CHAPTER 47. THE GEOMETRY OF DECORATION ON PREHISTORIC PUEBLO POTTERY FROM STARKWEATHER RUIN;750
7.14.1;MATHEMATICAL BACKGROUND;750
7.14.2;LITERATURE SURVEY;754
7.14.3;THE STARKWEATHER SITE;758
7.14.4;COMMENTARY AND ANALYSIS;760
7.14.5;THE VALUE AND MEANING OF SYMMETRY ANALYSIS;765
7.14.6;CONCLUSION;766
7.14.7;REFERENCES;766
7.15;CHAPTER 48. IN THE TOWER OF BABEL: BEYOND SYMMETRY IN ISLAMIC DESIGN;770
7.15.1;THEORETIC RELATIONSHIPS AND DESIGN STRUCTURE;783
7.15.2;THE NEED FOR A SCIENTIFIC LANGUAGE AND METHODOLOGY TO UNDERSTAND AND SYSTEMATICALLY CATEGORIZE AND DESCRIBE ISLAMIC GEOMETRIC PATTERNS;790
7.15.3;AN EXAMPLE OF THE PROCESS OF ISLAMIC GEOMETRIC DESIGN;795
7.15.4;CONCLUSION;807
7.15.5;REFERENCES;808
7.16;CHAPTER 49. RECONSTRUCTION AND EXTENSION OF LOST SYMMETRIES: EXAMPLES FROM THE TAMIL OF SOUTH INDIA;810
7.16.1;INTRODUCTION: TAMIL THRESHOLD DESIGNS;810
7.16.2;ANALYSIS AND RECONSTRUCTION OF A PA VITRAM-DESIGN;810
7.16.3;A SECOND EXAMPLE;811
7.16.4;TRANSFORMATION RULES;814
7.16.5;RECONSTRUCTION OF THE "ROSEWATER SPRINKLER" "SWINGING BOARD" DESIGNS AND;814
7.16.6;THE PAVITRAM-DESIGN REVISITED;815
7.16.7;EXTENSIONS IN THRESHOLD DESIGNS;817
7.16.8;EXTENSIONS OF THE RECONSTRUCTED P A VITR AM-DESIGN;821
7.16.9;EXAMINATION OF A BRAHMA'S KNOT;823
7.16.10;SOURCE OF INSPIRATION;829
7.16.11;CONCLUDING REMARKS;831
7.16.12;REFERENCES;832
7.17;CHAPTER 50. THE ROLE OF SYMMETRY IN JAVANESE BATIK PATTERNS;834
7.17.1;INTRODUCTION;834
7.17.2;SYMBOLIC CONTENT AND SYMMETRY OF TRADITIONAL PATTERNS;835
7.17.3;SYMMETRY ELEMENTS AND TRADITION;842
7.17.4;CONCLUSIONS;844
7.17.5;REFERENCES;845
7.18;CHAPTER 51. SYMMETRY IN THE MOVEMENTS OF T'AI CHI CHUAN;846
7.18.1;1. INTRODUCTION;846
7.18.2;2. TYPES OF SYMMETRY;847
7.18.3;3. CONCLUDING REMARKS;852
7.18.4;REFERENCES;852
7.18.5;APPENDIX;853
7.19;CHAPTER 52. SYMMETRY ASPECTS OF BOOKBINDINGS;856
7.19.1;INTRODUCTION;856
7.19.2;BOOKBINDING IN THE MIDDLE AGES;858
7.19.3;RENAISSANCE BOOKBINDINGS;873
7.19.4;FROM BAROQUE TO MODERN BINDINGS;893
7.19.5;CONCLUSIONS;902
7.19.6;REFERENCES;903
7.20;CHAPTER 53. 515—A SYMMETRIC NUMBER IN DANTE;906
7.20.1;EN GIRO TORTE SOL CICLOS ET ROTOR IGNE,;907
7.20.2;REFERENCES;916
7.21;CHAPTER 54. SYMMETRY IN CHRISTIAN TIME AND SPACE;918
7.21.1;REFERENCES;925
7.22;CHAPTER 55. URBAN SYMMETRY;926
7.23;CHAPTER 56. SYMMETRY AND WAYWARD NATURE;932
7.23.1;INTRODUCTION;932
7.23.2;SYMMETRY AND THE ARTIST;932
7.23.3;ATTEMPTING TO INTEGRATE VARIOUS APPROACHES;937
7.23.4;REFERENCES;938
7.24;CHAPTER 57. SYMMETRY IN PICTURES BY YOUNG CHINESE CHILDREN;940
7.24.1;1. INTRODUCTION;940
7.24.2;2. TYPES OF SYMMETRY IN CHINESE CHILDREN'S DRAWINGS;940
7.24.3;3. ORDER OF DEVELOPMENT OF THE VARIOUS KINDS OF SYMMETRY IN CHINESE CHILDREN'S PICTURES;947
7.24.4;4. WHAT CAUSES CHINESE CHILDREN TO DEVELOP THESE FORMS OF SYMMETRY IN THEIR DRAWINGS?;947
7.25;CHAPTER 58. FROM GEOMETRICAL RIGOR TO VISUAL EXPERIENCE;950
7.26;CHAPTER 59. ORNAMENTAL BRICKWORK THEORETICAL AND APPLIED SYMMETROLOGY AND CLASSIFICATION OF PATTERNS;974
7.26.1;1. INTRODUCTION;974
7.26.2;2. ROSETTES, FRIEZES AND PLANE PATTERNS;974
7.26.3;3. CLASSIFICATION OF PATTERNS;976
7.26.4;4. PLAIN BRICKWORK;985
7.26.5;5. ORNAMENTAL BRICKWORK TECHNIQUES;987
7.26.6;6. SELECTED PATTERN FAMILIES;990
7.26.7;7. COLOURED PATTERNS;999
7.26.8;8. ORDER-DISORDER (OD) PHENOMENA AND TWINNING;1003
7.26.9;9. AFFINE AND OTHER TRANSFORMATIONS;1006
7.26.10;10. CASE STUDY 1;1008
7.26.11;11. CASE STUDY 2;1014
7.26.12;12. EPILOGUE;1016
7.26.13;REFERENCES;1017
7.27;CHAPTER 60. INFLUENCES OF THE IDEAS OF JAY HAMBIDGE ON ART AND DESIGN;1020
7.27.1;INTRODUCTION;1020
7.27.2;PRINCIPLES OF DYNAMIC SYMMETRY;1021
7.27.3;THE GOLDEN SECTION AND HUMAN MOVEMENT;1022
7.27.4;THE MODULAR;1023
7.27.5;CONCLUSIONS;1026
7.27.6;REFERENCES;1026
7.28;CHAPTER 61. SYMMETRY IN CHINESE ARTS AND CRAFTS;1028
7.28.1;1. INTRODUCTION;1028
7.28.2;2. SYMMETRICAL CHARACTERISTICS OF CHINESE ARTS AND CRAFTS AS SEEN IN THE TAI JI PATTERN;1028
7.28.3;3. SYMMETRY FORMS IN CHINESE ARTS AND CRAFTS;1032
7.28.4;4. HOW ARE THE CHINESE SYMMETRICAL PATTERNS DEVELOPED IN ARTS AND CRAFTS;1036
7.28.5;SPACER COLORS SYMMETRY;1046
7.28.6;STUDY OF A BEAM OF LIGHT;1046
7.28.7;CUTTING OF PLANES;1053
7.28.8;LINEAR PERSPECTIVE—IMPRESSION OF VACANCY;1053
7.28.9;STRIATIONS IN A LOGARITHMIC SCALE;1058
7.28.10;AD INFINITUM;1060
7.28.11;GLOSSARY;1060
7.28.12;SCOPE OF THE SEARCH;1060
7.28.13;PREFACE;1062
7.28.14;FORM IN ARCHITECTURE;1062
7.28.15;METHOD;1062
7.28.16;TECHNIQUES;1063
7.28.17;AD INFINITUM;1068
7.28.18;CONCLUSIONS;1084
7.28.19;REFERENCES;1085
7.29;CHAPTER 62. THE JOY OF SYMMETRY;1086