Symmetry 2 (eBook)

Unifying Human Understanding

I. Hargittai (Herausgeber)

eBook Download: PDF

2016 | 1. Auflage
1084 Seiten
Elsevier Science (Verlag)
978-1-4832-9949-5 (ISBN)

Infos

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.

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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 "e;The Best Single Issue of a Journal"e; 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.

Inhalt

Front Cover 1
Symmetry 2: Unifying Human Understanding 4
Copyright Page 5
Table of Contents 6
Preface 10
Part 1: SYMMETRY 2 Unifying Human Understanding 12
CHAPTER 1. SYMMETRY REVISITED 16
1. INTRODUCTION 16
2. CLASSICAL SYMMETRY 18
3. REREADING WEYL 21
4. "ENLARGING OUR DEFINITION OF HOMOGENEITY" 24
REFERENCES 27
CHAPTER 2. SYMMETRY AT THE FOUNDATIONS OF SCIENCE 28
1. INTRODUCTION 28
2. SYMMETRY 28
3. REPRODUCIBILITY AS SYMMETRY 29
4. PREDICTABILITY AS SYMMETRY 30
5. CONCLUSION 30
BIBLIOGRAPHY 30
CHAPTER 3. SYMMETRY AND CHAOS 32
1. INTRODUCTION 32
SECTION 2 33
SECTION 3 33
SECTION 4 35
SECTION 5 36
SECTION 6 36
SECTION 7 37
SECTION 8 39
SECTION 9 39
SECTION 10 42
SECTION 11 44
REFERENCES 46
CHAPTER 4. THE MAGIC OF THE PENTANGLE: DYNAMIC SYMMETRY FROM MERLIN TO PENROSE 48
1. THE BEETLES 48
2. DYNAMIC SYMMETRY 52
3. THE SQUARE GNOMON 53
4. FIBONACCI'S NUMBERS 54
5. THE GOLDEN TRIANGLE 56
6. SIR GAWAIN 58
7. MERLIN UNLEASHED? 61
8. EPILOGUE 62
REFERENCES 62
CHAPTER 5. SYMMETRY, GROUPOIDS AND HIGHER-DIMENSIONAL ANALOGUES! 64
REFERENCES 72
CHAPTER 6. TRISECTING AN ORTHOSCHEME 74
1. FRIEZE PATTERNS 74
2. THE GENERAL ORTHOSCHEME IN ABSOLUTE 3-SPACE 74
3. THE DISSECTION INTO THREE ORTHOSCHEMES 75
4. TWO LINES CROSSING AT RIGHT ANGLES 76
5. AN OCTAHEDRON DISSECTED INTO 16 OR 48 ORTHOSCHEMES 77
6. POLYGONOMETRY 77
7. LOBACHEVSKY'S "PYRAMID" 80
8. TRISECTING A EUCLIDEAN ORTHOSCHEME 82
9. A CHALLENGING DEFINITE INTEGRAL 84
REFERENCES 86
CHAPTER 7. DUALITY AND THE DESCARTES DEFICIENCY 88
1. INTRODUCTION 88
2. POLYHEDRA 90
3. DUAL POLYHEDRAt 94
4. DESCARTES ANGULAR DEFICIENCY . AND ITS DUAL .' 96
5. THE RELATIONSHIP BETWEEN ., . AND . 98
6. EXAMPLES 101
7. CONCLUDING REMARKS 103
REFERENCES 103
CHAPTER 8. EXTENDING THE BINOMIAL COEFFICIENTS TO PRESERVE SYMMETRY AND PATTERN 104
1. INTRODUCTION 104
2. EXTENDING THE DOMAIN OF THE BINOMIAL COEFFICIENTS OVER THE FULL RANGE OF INTEGERS 106
3. SOME PATTERNS IN THE PASCAL HEXAGON 110
4. WEIGHTS IN THE PASCAL FLOWER 114
5. EXTENDING THE DOMAIN OF THE BINOMIAL COEFFICIENTS OVER THE COMPLEX NUMBERS 115
REFERENCES 117
CHAPTER 9. SYMMETRY IN THE SIMPLEST CASE: THE REAL LINE 118
POINTS OF (LOCAL) SYMMETRY 118
SYMMETRIC AND LOCALLY SYMMETRIC SETS 118
IS ABSENCE OF SYMMETRY RELATED TO DISORDER? 118
POINTS OF NON-SYMMETRY 118
THREE WAYS IN WHICH SYMMETRY OF A SET MAY BE TRANSGRESSED 119
ANTISYMMETRIC SETS 119
SYMMETRY OF A SET A WITH RESPECT TO POINTS WHICH DO NOT BELONG TO A 120
MORE SOPHISTICATED EXAMPLES OF SYMMETRIC SETS 120
STRUCTURAL IDENTITY OF SYMMETRY AND ANTISYMMETRY 121
HOW RESTRICTIVE ARE ASYMMETRIC SETS? 121
DISORDER MEANS COMBINATION OF SYMMETRY AND NON-SYMMETRY 121
SYMMETRY MODULO A NEGLIGIBLE SET 122
STRONG SYMMETRY 122
STRONG ASYMMETRY AND STRONG ANTISYMMETRY 123
SYMMETRIC CONTINUITY 124
OTHER ASPECTS OF SYMMETRIC CONTINUITY 124
LOCALLY SYMMETRIC FUNCTIONS 125
NON-TRIVIAL SYMMETRIC CONTINUITY IS NEVER UNIFORM 125
SYMMETRY STRUCTURE OF A SET AND OF A FUNCTION: AN OPEN PROBLEM 126
SYMMETRIC DERIVATIVES: THEIR HISTORICAL ORIGIN 126
SYMMETRIC DERIVATIVE 127
SYMMETRIC DERIVATIVE VERSUS ORDINARY DERIVATIVE 127
SYMMETRIC DERIVATIVE AND MONOTONICITY 127
SCHWARTZ SYMMETRIC FUNCTIONS 128
OTHER RESULTS, PROBLEMS AND SUGGESTIONS 128
SOME CONCEPTUAL AND PHILOSOPHICAL DIFFICULTIES 129
REFERENCES 130
CHAPTER 10. DELICATE SYMMETRY 132
1. INTRODUCTION 132
2. TOTAL CHAOS IS IMPOSSIBLE 132
3. FRAGILE SYMMETRY 134
4. FLEXIBLE SYMMETRY 136
REFERENCES 139
CHAPTER 11. HALLEY MAPS FOR A TRIGONOMETRIC AND RATIONAL FUNCTION 140
REFERENCES 147
CHAPTER 12. A CLASS OF SYMMETRIC POLYTOPES 148
INTRODUCTION 148
REFERENCES 151
CHAPTER 13. ARRANGEMENTS OF MINIMAL VARIANCEMULTIDIMENSIONAL SCALING IN THE SYMMETRICAL CASE 152
1. ARRANGEMENT OF POINTS IN THE PLANE WITH DISTANCE OF MINIMAL VARIANCE 152
2. MULTIDIMENSIONAL SCALING 155
3. MULTIPLE MDS 156
4. MMDS IN THE SYMMETRICAL CASE 159
5. MMDS OF CONGENITAL ABNORMALITIES 159
REFERENCES 160
CHAPTER 14. TILINGS BY REGULAR POLYGONS—II A CATALOG OF TILINGS 162
INTRODUCTION 162
REGULARITY PROPERTIES 162
THE TILINGS 164
REFERENCES 180
CHAPTER 15. SYMMETRY AND POLYHEDRAL STELLATION—Iaf 182
INTRODUCTION 182
THE STRUCTURE OF THE ASYMMETRICAL STELLATION PATTERN 182
ANALYSIS OF THE STELLATION PATTERN 186
REFERENCES 190
CHAPTER 16. SYMMETRY AND POLYHEDRAL STELLATION—Ib 192
MODEL MAKING BY MEANS OF THE STELLATION PATTERN 192
SYMMETRICAL COMPOUNDS 203
APPLICATIONS 206
REFERENCES 208
CHAPTER 17. SYMMETRY AND POLYHEDRAL STELLATION—II 210
INTRODUCTION 210
THE STRUCTURE OF A SYMMETRICAL STELLATION PATTERN 210
REFERENCES 216
CHAPTER 18. THE COMPLETE SET OF JITTERBUG TRANSFORMERS AND THE ANALYSIS OF THEIR MOTION 218
1. INTRODUCTION 218
2. MATHEMATICAL APPROACH 218
3. POLYGON-DIPOLYGON-DIPOLYGONID 219
4. JITTERBUG AND DIPOLYGONIDS 227
5. FINITE DIPOLYGONIDS 227
6. CLASSIFICATION OF THE FINITE DIPOLYGONIDS 230
7. ASPECTS OF UNIFORM MOTION 237
8. EXTENDED GROUPS OF ISOMETRIES 238
9. REGULAR DIPOLYGONIDS 241
10. EQUIRADIAL DIPOLYGONIDS 248
11. PAIRS OF CHIRAL DIPOLYGONIDS 248
12. THE USE OF DIPOLYGONIDS 259
REFERENCES 265
CHAPTER 19. A GEOMETRICAL ANALOGUE OF THE PHASE TRANSFORMATION OF CRYSTALS 266
REFERENCES 269
CHAPTER 20. COMPUTERS AND GROUP-THEORETICAL METHODS FOR STUDYING STRUCTURAL PHASE TRANSITIONS 270
1. INTRODUCTION 270
2. GROUP-THEORETICAL METHODS FOR ANALYSIS OF PHASE TRANSITIONS 270
3. OBTAINING FULL IR'S OF THE SPACE GROUPS 275
4. OBTAINING A SET OF NONEQUIVALENT STATIONARY VECTORS 276
5. IDENTIFICATION OF SUBGROUPS OF A SPACE GROUP 282
6. OBTAINING THE COMPLETE CONDENSATE OF STATIONARY VECTORS 287
7. CONSTRUCTION OF BASIS FUNCTIONS OF THE IR's OF SPACE GROUPS 288
8. SPLITTING OF THE ORBITS OF A SPACE GROUP INDUCED BY A PHASE TRANSITION 289
9. A COMPARISON BETWEEN VARIOUS APPROACHES TO THE GROUP-THEORETICAL ANALYSIS OF PHASE TRANSITIONS IN CRYSTALS 289
REFERENCES 291
CHAPTER 21. STRUCTURAL THEORY OF SPACE-TIME AND INTRAPOINT SYMMETRY 294
INTRODUCTION 294
THE UNIVERSE AS A MULTILEVEL SYSTEM OF CONTACTING NATURAL REGIONS 295
THE UNIVERSE AS MULTILEVEL MATERIAL SPACE 301
THE UNIVERSE AS TIME AND LIGHT 308
REFERENCES 313
CHAPTER 22. VISUAL AND HIDDEN SYMMETRY IN GEOMETRY‡ 316
1. CRITICAL POINTS OF SMOOTH FUNCTIONS ON THREE-DIMENSIONAL MANIFOLDS 316
2. TWO-ADIC SOLENOID 316
3. ALGEBRAIC SURFACE IN TWO-DIMENSIONAL COMPLEX SPACE 316
4. TWO-DIMENSIONAL SPHERE, POISSON SPHERE, NUTATION AND PRECESSION 317
5. HEEGARD DIAGRAMS OF THREE-DIMENSIONAL MANIFOLDS 317
6. ALGEBRAIC FUNCTIONS AND FUNDAMENTAL DOMAINS OF ACTION OF DISCRETE GROUPS 317
7. CRYSTAL STRUCTURES. THE PARTITIONING OF THE SPACE INTO SIMPLEXES 326
8. CRYSTAL STRUCTURES. CUBIC PARTITIONINGS OF POLYHEDRA 326
9. SYMMETRIC SPACES 326
10. SYMMETRY OF PLANE WAVES 326
11. CUBIC LATTICES 326
12. CHAOS 327
13. HIDDEN SYMMETRICS 327
15. MATHEMATICAL FANTASY 335
REFERENCES 335
CHAPTER 23. MATHEMATICS AND BEAUTY—VIII TESSELATION AUTOMATA DERIVED FROM A SINGLE DEFECT 336
INTRODUCTION 336
MOTIVATION 337
METHOD AND OBSERVATIONS 337
SUMMARY AND CONCLUSIONS 344
REFERENCES 350
APPENDIX 351
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
REFERENCES 354
CHAPTER 25. LOCAL PSEUDOSYMMETRY IN SIMPLE LIQUIDS 356
1. INTRODUCTION 356
2. BUILDING UP A LIQUID FROM ITS ELEMENTARY UNIT 358
3. SHORT-RANGE ORDER IN SIMPLE LIQUIDS 358
4. LOCAL SUB-UNITS IN CLOSE-PACKED STRUCTURES 359
5. A TWO-DIMENSIONAL ANALOGUE—KAWAMURA'S MODEL 360
6. PACKING TETRAHEDRA AND OCTAHEDRA IN THREE DIMENSIONS 361
7. PACKINGS OF SOFT SPHERES 362
8. EVIDENCE FROM MODEL STUDIES 363
9. INTERMEDIATE RANGE ORDER 365
10. SUMMARY 367
REFERENCES 369
CHAPTER 26. TOPOLOGICAL ASPECTS OF BENZENOIDS AND CORONOIDS, INCLUDING "SNOWFLAKES" AND "LACEFLOWERS" 370
1. INTRODUCTION 370
2. EXAMPLES OF THE RESEARCH PROGRESS 371
3. SOME DEFINITIONS AND THE INVARIANT . 373
4. FURTHER DEFINITIONS AND THE "neo" CLASSIFICATION 374
5. ENUMERATION AND CLASSIFICATION OF BENZENOIDS WITH HEXAGONAL SYMMETRY 376
6. ANALYSING SINGLE CORONOIDS: THE "rheo" AND "r/o" CLASSIFICATIONS 377
7. COMBINATORIAL K FORMULAS FOR CLASSES OF SOME CORONOIDS WITH HEXAGONAL SYMMETRY 384
8. MULTIPLE CORONOIDS WITH HEXAGONAL SYMMETRY 388
REFERENCES 388
CHAPTER 27. A SET THEORETIC APPROACH TO THE SYMMETRY ANALYSIS OF HEXADECAHEDRANE 390
INTRODUCTION 390
SYMMETRY ELEMENTS AND SYMMETRY OPERATIONS 391
SOME SET THEORETIC CONCEPTS 391
SYMMETRY POINT SETS AND GENERATOR SETS 392
MOLECULAR AND LOCAL SYMMETRY POINT SETS 396
INTERSECTION OF LPS (INTERSECTION SETS) 397
INTERCHANGE SETS 397
SYMMETRY POINT SETS AND SYMMETRY POINT GROUPS 397
APPLICATIONS OF SYMMETRY POINT SETS 398
CONSTRUCTION OF THE MPS OF HEXADECAHEDRANE 398
CONTRIBUTIONS OF THE LPG TO THE MPG OF HEXADECAHEDRANE 407
CONCLUSIONS 408
REFERENCES 409
CHAPTER 28. CARBON AND ITS NETS 412
1. INTRODUCTION 412
2. THE MOST PRECIOUS ATOMS IN THE UNIVERSE, OR WHY C? 412
3. THE TWO KNOWN ALLOTROPIC FORMS OF C: GRAPHITE AND DIAMOND, OR THE SOFTEST AND HARDEST CRYSTALS 413
4. CARBON MOLECULES 417
5. HYPOTHETICAL ALTERNATIVE C ALLOTROPES (INFINITE NETS) 418
6. RELATED NETS WITH OTHER ATOMS INSTEAD OF C 424
7. SYMMETRY OF FRAGMENTS FROM THE GRAPHITE LATTICE 425
8. SYMMETRY OF FRAGMENTS FROM THE DIAMOND LATTICE 427
9. SYMMETRY AND STABILITY OF MOLECULES 429
10. CONCLUSION 430
REFERENCES 430
CHAPTER 29. CB 6o BUCKMINSTERFULLERENE, OTHER FULLERENES AND THE ICOSPIRAL SHELL 432
INTRODUCTION 432
THE ICOSPIRAL PARTICLE AND THE GIANT FULLERENES 435
SUMMARY OF SYMMETRY ASPECTS 437
REFERENCES 438
CHAPTER 30. ASYMMETRY THROUGH THE EYES OF A(NOTHER) CHEMIST! 440
SYMMETRY AND CHIRALITY—PRINCIPLE OF APPROPRIATENESS 440
DOES NATURE TEND TOWARD SYMMETRY OR ASYMMETRY? 442
INTERACTIONS AMONG ASYMMETRIC (CHIRAL) MOLECULES AND CHIRAL SYSTEMS 443
MANIFESTATION OF ASYMMETRY IN MOLECULAR DIMENSIONS 444
SEPARATION OF OPTICAL ISOMERS 447
THE THREE-POINT INTERACTION MODEL OF CHIRAL DISCRIMINATION 449
MACROSCOPIC DIFFERENCES AND WEAK SECOND ORDER INTERACTIONS 450
WHY IS TARTARIC ACID A GOOD RESOLVING AGENT? 451
SYMMETRY VERSUS CHIRALITY AND THE PRINCIPLE OF COMPLEMENTARITY 452
REFERENCES 455
CHAPTER 31. CHEMICAL KINETICS AND THERMODYNAMICS A HISTORY OF THEIR RELATIONSHIP 458
1. WHAT IS CHEMICAL AFFINITY? 458
2. HOW CHEMICAL REACTIONS PROCEED AND WHY? 459
3. WHAT IS PROCESS THERMODYNAMICS? 460
4. THE UNIFIED FIELD THEORY OF THERMODYNAMICS 462
5. LINEARLY INDEPENDENT CHEMICAL REACTIONS AND THEIR ENTROPY PRODUCTION 463
6. EXTENSION BEYOND LOCAL EQUILIBRIUM 464
7. EXTENSION TO NON-LINEAR PHENOMENOLOGICAL EQUATIONS 464
8. WHAT ARE THE FORCES DRIVING CHEMICAL REACTIONS TO ADVANCE? HOW IS THE EVOLUTION OF CHEMICAL KINETIC SYSTEMS GOVERNED? 465
9. HOW DOES A CHEMICAL KINETIC SYSTEM EVOLVE IN TIME? 466
10. THE ROLE OF SYMMETRY IN THE STUDY OF IRREVERSIBLE PROCESSES 468
REFERENCES 469
CHAPTER 32. SYMMETRIES AND THE NOTION OF ELEMENTARITY IN HIGH ENERGY PHYSICS 472
1. INTRODUCTION 472
2. SUCCESSFUL METHODS, BROKEN SYMMETRIES AND THE QUESTION OF SIMPLICITY 473
3. THE NOTION OF OBSERVABILITY 475
4. CONCLUSION 478
REFERENCES 479
CHAPTER 33. SYMMETRY-BREAKING PATTERN FORMATION IN SEMICONDUCTOR PHYSICS: SPATIO-TEMPORAL CURRENT STRUCTURES DURING AVALANCHE BREAKDOWN 482
1. INTRODUCTION 482
2. SYSTEM 482
3. RESULTS 483
4. MODEL 486
5. CONCLUDING REMARKS 487
REFERENCES 487
Part 2: SYMMETRY 2 Unifying Human Understanding 490
CHAPTER 34. THE PERCEPTUAL VALUE OF SYMMETRY 494
1. INTRODUCTION 494
2. GLOBAL DETECTION OF SYMMETRY 496
3. VISUAL EXPLORATION OF SYMMETRICAL STIMULI 497
4. AROUSAL POTENTIAL OF SYMMETRICAL COMPOSITIONS 501
5. ENCODING SYMMETRY 502
6. CONCLUSION 502
REFERENCES 503
CHAPTER 35. MIND'S EYE 504
INTRODUCTION 504
VISION AS PROCESS: GENERAL CONSIDERATIONS 505
SYMMETRY: SOME GENERAL OBSERVATIONS 508
ROBOTIC VISION: ANALOGIES AND ANALYSIS 511
GENERAL IMPLICATIONS: RELATION TO SCIENTIFIC PROCESS 519
CONCLUSION: NEW WINE FROM NEW WINESKINS 521
REFERENCES 522
CHAPTER 36. NON-EUCLIDEAN GEOMETRIES AND ALGORITHMS OF LIVING BODIES! 524
1. THE ERLANGER PROGRAM AND GEOMETRIZATION IN BIOLOGY 524
2. THE HISTORY OF RESEARCH IN THE SYMMETRIES AND ALGORITHMS OF THE MORPHOLOGICAL SELF-ORGANIZATION IN BIOLOGICAL OBJECTS 526
3. NON-EUCLIDEAN GEOMETRIES AND ITERATIVE ALGORITHMS IN ORGANIC FORMS 528
4. EUCLIDEAN AND NON-EUCLIDEAN ITERATIVE ALGORITHMS IN THE KINEMATICS OF BIOLOGICAL MOVEMENTS AND PHYSIOLOGICALLY NORMAL POSTURES 533
5. THE KINEMATICS OF THE THREE-DIMENSIONAL GROWTH OF BIOLOGICAL BODIES WHOSE PARTS ARE NOT CYCLOMERICALLY POSITIONED 538
6. AUTONOMOUS AUTOMATA AND GROUP INVARIANT PROPERTIES OF LIVING ORGANISMS 542
7. APERIODIC ITERATIVE BIORHYTHMS 543
8. THE PROTOSYSTEM OF BIOLOGICAL REGULATION AND PSYCHOPHYSICAL MATTERS 544
9. CONCLUSIONS 549
REFERENCES 550
APPENDIX 551
CHAPTER 37. SPIRAL PHYLLOTAXIS 554
REFERENCES 557
CHAPTER 38. A SYMMETRY-ORIENTED MATHEMATICAL MODEL OF CLASSICAL COUNTERPOINT AND RELATED NEUROPHYSIOLOGICAL INVESTIGATIONS BY DEPTH EEG 558
1. INTRODUCTIONTOTHEPROBLEM 558
2. THE GEOMETRY OF INTERVAL DICHOTOMIES 566
3. COUNTERPOINT AS AN INFINITESIMAL VARIATION OF GREGORIAN CHORAL 574
4. LOCAL SYMMETRIES IN COUNTERPOINT 579
5. SOME MUSICOLOGICAL COMMENTS 583
6. METHODS, TEST MATERIAL AND ANALYSIS OF THE NEUROPHYSIOLOGICAL INVESTIGATIONS 585
7. GENERAL OBSERVATIONS OF EVOKED EEG RESPONSES 597
8. DEFECTIVE COUNTERPOINT 603
9. SPECTRAL DATA FROM ISOLATED CONSONANCES AND DISSONANCES 604
10. TESTING THE SYMMETRY OF AUTOCOMPLEMENTARITY ON THE INTERVAL SET 609
11. DISCUSSION 611
REFERENCES 612
CHAPTER 39. SQUARE SPIRALS, DIMENSIONALITY AND BIOPOLYMERS 614
REFERENCES 630
CHAPTER 40. THE GOLDEN SECTION IN THE MEASUREMENT THEORY 632
1. INTRODUCTION 632
2. THE GOLDEN SECTION 632
3. HOW DO RABBITS MULTIPLY? 635
4. SOME FIBONACCI TRIGONOMETRY 637
5. PASCAL'S TRIANGLE OR ANOTHER WAY OF RABBITS MULTIPLYING 638
6. HOW MANY GOLDEN SECTIONS CAN BE FOUND? 641
7. HOW THE FIRST CRISIS IN THE FOUNDATIONS OF MATHEMATICS WAS OVERCOME 643
8. CAN THE MEASUREMENT BE COMPLETED FOR THE INFINITE PERIOD OF TIME OR THE CONSTRUCTIVE APPROACH TO MEASUREMENT? 645
9. GEOMETRIC PROBLEM PUT FORWARD BY N. I. LOBACHEVSKI 645
10. THE PROBLEM OF CHOOSING THE BEST SYSTEM OF SCALE WEIGHTS OR HOW THE BINARY NOTATION WAS INVENTED 647
11. NEUGEBAUER'S HYPOTHESIS 648
12. ASYMMETRY PRINCIPLE OF MEASUREMENT 648
13. THE FUNDAMENTAL RESULTS OF THE ALGORITHMIC MEASUREMENT THEORY 649
14. UNEXPECTED RELATIONS 652
15. FIBONACCI CODES 653
16. THE GOLDEN p-RATIO CODES 655
17. CONCLUSIONS 656
REFERENCES 657
CHAPTER 41. BUCKLING PATTERNS OF SHELLS AND SPHERICAL HONEYCOMB STRUCTURES 658
1. INTRODUCTION 658
2. CIRCLES ON A SPHERE 659
3. BUCKLING OF COMPLETE SPHERICAL SHELLS 663
4. ANALOGIES IN NATURE 664
5. CONCLUSIONS 669
REFERENCES 670
CHAPTER 42. SYMMETRY IN FREE MARKETS 672
INTRODUCTION 672
A GAME OF ROULETTE 673
THE ROULETTE MARKET 673
SYMMETRY IN THE ROULETTE MARKET 674
A SYMMETRY VIOLATION IN THE BLACKJACK MARKET 675
THE PARI MUTUEL MARKET 676
SYMMETRY IN THE PARI MUTUEL MARKET 677
THE STOCK MARKET 679
SYMMETRY IN THE STOCK MARKET 680
SYMMETRY VIOLATION IN THE STOCK MARKET 682
SYMMETRY IN CONSUMER MARKETS 683
THE ROLE OF GOVERNMENT IN SYMMETRIC MARKETS 685
REFERENCES 687
APPENDIX A 687
APPENDIX B 688
APPENDIX C 688
CHAPTER 43. SYMMETRIES IN MUSIC TEACHING 690
SYMMETRIES IN MUSIC TEACHING 690
SYMMETRY OF INTENSITY 690
SYMMETRY OF DURATION —SYMMETRY OF RHYTHM 690
PITCH SYMMETRY —MELODY SYMMETRY 692
SYMMETRY OF HARMONY 697
SYMMETRY OF TONALITY 701
SYMMETRY OF FORM 706
GOLDEN SECTION 709
FCGDAEH 711
CHAPTER 44. UNCERTAINTY PRINCIPLE AND SYMMETRY IN METAPHORS 716
REFERENCES 725
CHAPTER 45. MODERN SYMMETRY 728
REFERENCES 732
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
INTRODUCTION 734
OLD HUNGARIAN ORNAMENTAL ART 735
AVAR-ONOGURIAN ORNAMENTAL ART 740
THE SYMMETRY METHOD IN ARCHAEOLOGY 747
REFERENCES 748
CHAPTER 47. THE GEOMETRY OF DECORATION ON PREHISTORIC PUEBLO POTTERY FROM STARKWEATHER RUIN 750
MATHEMATICAL BACKGROUND 750
LITERATURE SURVEY 754
THE STARKWEATHER SITE 758
COMMENTARY AND ANALYSIS 760
THE VALUE AND MEANING OF SYMMETRY ANALYSIS 765
CONCLUSION 766
REFERENCES 766
CHAPTER 48. IN THE TOWER OF BABEL: BEYOND SYMMETRY IN ISLAMIC DESIGN 770
THEORETIC RELATIONSHIPS AND DESIGN STRUCTURE 783
THE NEED FOR A SCIENTIFIC LANGUAGE AND METHODOLOGY TO UNDERSTAND AND SYSTEMATICALLY CATEGORIZE AND DESCRIBE ISLAMIC GEOMETRIC PATTERNS 790
AN EXAMPLE OF THE PROCESS OF ISLAMIC GEOMETRIC DESIGN 795
CONCLUSION 807
REFERENCES 808
CHAPTER 49. RECONSTRUCTION AND EXTENSION OF LOST SYMMETRIES: EXAMPLES FROM THE TAMIL OF SOUTH INDIA 810
INTRODUCTION: TAMIL THRESHOLD DESIGNS 810
ANALYSIS AND RECONSTRUCTION OF A PA VITRAM-DESIGN 810
A SECOND EXAMPLE 811
TRANSFORMATION RULES 814
RECONSTRUCTION OF THE "ROSEWATER SPRINKLER" "SWINGING BOARD" DESIGNS AND 814
THE PAVITRAM-DESIGN REVISITED 815
EXTENSIONS IN THRESHOLD DESIGNS 817
EXTENSIONS OF THE RECONSTRUCTED P A VITR AM-DESIGN 821
EXAMINATION OF A BRAHMA'S KNOT 823
SOURCE OF INSPIRATION 829
CONCLUDING REMARKS 831
REFERENCES 832
CHAPTER 50. THE ROLE OF SYMMETRY IN JAVANESE BATIK PATTERNS 834
INTRODUCTION 834
SYMBOLIC CONTENT AND SYMMETRY OF TRADITIONAL PATTERNS 835
SYMMETRY ELEMENTS AND TRADITION 842
CONCLUSIONS 844
REFERENCES 845
CHAPTER 51. SYMMETRY IN THE MOVEMENTS OF T'AI CHI CHUAN 846
1. INTRODUCTION 846
2. TYPES OF SYMMETRY 847
3. CONCLUDING REMARKS 852
REFERENCES 852
APPENDIX 853
CHAPTER 52. SYMMETRY ASPECTS OF BOOKBINDINGS 856
INTRODUCTION 856
BOOKBINDING IN THE MIDDLE AGES 858
RENAISSANCE BOOKBINDINGS 873
FROM BAROQUE TO MODERN BINDINGS 893
CONCLUSIONS 902
REFERENCES 903
CHAPTER 53. 515—A SYMMETRIC NUMBER IN DANTE 906
EN GIRO TORTE SOL CICLOS ET ROTOR IGNE, 907
REFERENCES 916
CHAPTER 54. SYMMETRY IN CHRISTIAN TIME AND SPACE 918
REFERENCES 925
CHAPTER 55. URBAN SYMMETRY 926
CHAPTER 56. SYMMETRY AND WAYWARD NATURE 932
INTRODUCTION 932
SYMMETRY AND THE ARTIST 932
ATTEMPTING TO INTEGRATE VARIOUS APPROACHES 937
REFERENCES 938
CHAPTER 57. SYMMETRY IN PICTURES BY YOUNG CHINESE CHILDREN 940
1. INTRODUCTION 940
2. TYPES OF SYMMETRY IN CHINESE CHILDREN'S DRAWINGS 940
3. ORDER OF DEVELOPMENT OF THE VARIOUS KINDS OF SYMMETRY IN CHINESE CHILDREN'S PICTURES 947
4. WHAT CAUSES CHINESE CHILDREN TO DEVELOP THESE FORMS OF SYMMETRY IN THEIR DRAWINGS? 947
CHAPTER 58. FROM GEOMETRICAL RIGOR TO VISUAL EXPERIENCE 950
CHAPTER 59. ORNAMENTAL BRICKWORK THEORETICAL AND APPLIED SYMMETROLOGY AND CLASSIFICATION OF PATTERNS 974
1. INTRODUCTION 974
2. ROSETTES, FRIEZES AND PLANE PATTERNS 974
3. CLASSIFICATION OF PATTERNS 976
4. PLAIN BRICKWORK 985
5. ORNAMENTAL BRICKWORK TECHNIQUES 987
6. SELECTED PATTERN FAMILIES 990
7. COLOURED PATTERNS 999
8. ORDER-DISORDER (OD) PHENOMENA AND TWINNING 1003
9. AFFINE AND OTHER TRANSFORMATIONS 1006
10. CASE STUDY 1 1008
11. CASE STUDY 2 1014
12. EPILOGUE 1016
REFERENCES 1017
CHAPTER 60. INFLUENCES OF THE IDEAS OF JAY HAMBIDGE ON ART AND DESIGN 1020
INTRODUCTION 1020
PRINCIPLES OF DYNAMIC SYMMETRY 1021
THE GOLDEN SECTION AND HUMAN MOVEMENT 1022
THE MODULAR 1023
CONCLUSIONS 1026
REFERENCES 1026
CHAPTER 61. SYMMETRY IN CHINESE ARTS AND CRAFTS 1028
1. INTRODUCTION 1028
2. SYMMETRICAL CHARACTERISTICS OF CHINESE ARTS AND CRAFTS AS SEEN IN THE TAI JI PATTERN 1028
3. SYMMETRY FORMS IN CHINESE ARTS AND CRAFTS 1032
4. HOW ARE THE CHINESE SYMMETRICAL PATTERNS DEVELOPED IN ARTS AND CRAFTS 1036
SPACER COLORS SYMMETRY 1046
STUDY OF A BEAM OF LIGHT 1046
CUTTING OF PLANES 1053
LINEAR PERSPECTIVE—IMPRESSION OF VACANCY 1053
STRIATIONS IN A LOGARITHMIC SCALE 1058
AD INFINITUM 1060
GLOSSARY 1060
SCOPE OF THE SEARCH 1060
PREFACE 1062
FORM IN ARCHITECTURE 1062
METHOD 1062
TECHNIQUES 1063
AD INFINITUM 1068
CONCLUSIONS 1084
REFERENCES 1085
CHAPTER 62. THE JOY OF SYMMETRY 1086

Produkt-Details

• Erscheint lt. Verlag: 25.2.2016
• Sprache: englisch
• Themenwelt: Schulbuch / Wörterbuch ► Lexikon / Chroniken
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Naturwissenschaften ► Chemie ► Physikalische Chemie
• Themenwelt: Technik
• ISBN-10: 1-4832-9949-X / 148329949X
• ISBN-13: 978-1-4832-9949-5 / 9781483299495