Geometry revealed: a Jacob's Ladder to modern higher geometry
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2010
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 831 S. Ill., graph. Darst. |
| ISBN: | 9783540709961 |
Internformat
MARC
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Datensatz im Suchindex
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|---|---|
| adam_text | TABLE OF CONTENTS ABOUT THE AUTHOR V INTRODUCTION VII CHAPTER I. POINTS
AND LINES IN THE PLANE 1 1.1. IN WHICH SETTING AND IN WHICH PLANE ARE WE
WORKING? AND RIGHT AWAY AN UTTERLY SIMPLE PROBLEM OF SYLVESTER ABOUT THE
COLLINEARITY OF POINTS 1 1.2. ANOTHER NAIVE PROBLEM OF SYLVESTER, THIS
TIME ON THE GEOMETRIC PROBABILITIES OF FOUR POINTS 6 1.3. THE ESSENCE OF
AFFINE GEOMETRY AND THE FUNDAMENTAL THEOREM 12 1.4. THREE CONFIGURATIONS
OF THE AFFINE PLANE AND WHAT HAS HAPPENED TO THEM: PAPPUS, DESARGUES AND
PERLES 17 1.5. THE IRRESISTIBLE NECESSITY OF PROJECTIVE GEOMETRY AND THE
CONSTRUCTION OF THE PROJECTIVE PLANE 23 1.6. INTERMEZZO: THE PROJECTIVE
LINE AND THE CROSS RATIO 28 1.7. RETURN TO THE PROJECTIVE PLANE:
CONTINUATION AND CONCLUSION 31 1.8. THE COMPLEX CASE AND, BETTER STILL,
SYLVESTER IN THE COMPLEX CASE: SERRE S CONJECTURE 40 1.9. THREE
CONFIGURATIONS OF SPACE (OF THREE DIMENSIONS): REYE, MOEBIUS AND SCHIARII
43 1.10. ARRANGEMENTS OF HYPERPLANES 47 I. XYZ 48 BIBLIOGRAPHY 57
CHAPTER II. CIRCLES AND SPHERES 61 II. 1. INTRODUCTION AND BORSUK S
CONJECTURE 61 11.2. A CHOICE OF CIRCLE CONFIGURATIONS AND A CRITICAL
VIEW OF THEM 66 11.3. A SOLITARY INVERSION AND WHAT CAN BE DONE WITH IT
78 11.4. HOW DO WE COMPOSE INVERSIONS? FIRST SOLUTION: THE CONFORMAI
GROUP ON THE DISK AND THE GEOMETRY OF THE HYPERBOLIC PLANE 82 U.S
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/98389647X DIGITALISIERT
DURCH XII TABLE OF CONTENTS II.9. CIRCLES OF APOLLONIUS 113 II. XYZ 116
BIBLIOGRAPHY 137 CHAPTER III. THE SPHERE BY ITSELF: CAN WE DISTRIBUTE
POINTS ON IT EVENLY? 141 III. 1. THE METRIC OF THE SPHERE AND SPHERICAL
TRIGONOMETRY 141 111.2. THE MOEBIUS GROUP: APPLICATIONS 147 111.3.
MISSION IMPOSSIBLE: TO UNIFORMLY DISTRIBUTE POINTS ON THE SPHERE S 2 :
OZONE, ELECTRONS, ENEMY DICTATORS, GOLF BALLS, VIROLOGY, PHYSICS OF
CONDENSED MATTER 149 111.4. THE KISSING NUMBER OF S 2 , ALIAS THE HARD
PROBLEM OF THE THIRTEENTH SPHERE 170 111.5. FOUR OPEN PROBLEMS FOR THE
SPHERE S 3 172 111.6. A PROBLEM OF BANACH-RUZIEWICZ: THE UNIQUENESS OF
CANONICAL MEASURE 174 111.7. A CONCEPTUAL APPROACH FOR THE KISSING
NUMBER IN ARBITRARY DIMENSION 175 III. XYZ 177 BIBLIOGRAPHY 178 CHAPTER
IV. CONIES AND QUADRICS 181 IV. 1 . MOTIVATIONS, A DEFINITION PARACHUTED
FROM THE LADDER, AND WHY ... 181 IV.2. BEFORE DESCARTES: THE REAL
EUCLIDEAN CONIES. DEFINITION AND SOME CLASSICAL PROPERTIES 183 IV.3. THE
COMING OF DESCARTES AND THE BIRTH OF ALGEBRAIC GEOMETRY . . . . 198
IV.4. REAL PROJECTIVE THEORY OF CONIES; DUALITY 200 IV.5. KLEIN S
PHILOSOPHY COMES QUITE NATURALLY 205 IV.6. PLAYING WITH TWO CONIES,
NECESSITATING ONCE AGAIN COMPLEXIFICATION . 208 IV.7 TABLE OF CONTENTS
XIII V.7. THE ALGEBRAIC CURVATURE IS A CHARACTERISTIC INVARIANT:
MANUFACTURE OF RULERS, CONTROL BY THE CURVATURE 269 V.8. THE FOUR VERTEX
THEOREM AND ITS CONVERSE; AN APPLICATION TO PHYSICS . 271 V.9.
GENERALIZATIONS OF THE FOUR VERTEX THEOREM: ARNOLD I 278 V.10. TOWARD A
CLASSIFICATION OF CLOSED CURVES: WHITNEY AND ARNOLD II ... 281 V.U.
ISOPERIMETRIC INEQUALITY: STEINER S ATTEMPTS 295 V.I2. THE ISOPERIMETRIC
INEQUALITY: PROOFS ON ALL RUNGS 298 V.13. PLANE ALGEBRAIC CURVES:
GENERALITIES 305 V.I4. THE CUBICS, THEIR ADDITION LAW AND ABSTRACT
ELLIPTIC CURVES 308 V.I5. REAL AND EUCLIDEAN ALGEBRAIC CURVES 320 V.16.
FINITE ORDER GEOMETRY 328 V. XYZ 331 BIBLIOGRAPHY 336 CHAPTER VI. SMOOTH
SURFACES 341 VI. 1. WHICH OBJECTS ARE INVOLVED AND WHY? CLASSIFICATION
OF COMPACT SURFACES 341 VI.2. THE INTRINSIC METRIC AND THE PROBLEM OF
THE SHORTEST PATH 345 VI.3. THE GEODESIES, THE CUT LOCUS AND THE
RECALCITRANT ELLIPSOIDS 347 VI.4. AN INDISPENSABLE ABSTRACT CONCEPT:
RIEMANNIAN SURFACES 357 VI.5. PROBLEMS OF ISOMETRIES: ABSTRACT SURFACES
VERSUS SURFACES OF E 3 . . . 361 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 364 VI.7. WHAT IS KNOWN ABOUT THE TOTAL CURVATURE (OF
GAUSS) 373 VI.8 XIV TABLE OF CONTENTS VII.8. VOLUME AND MINKOWSKI
ADDITION: THE BRUNN-MINKOWSKI THEOREM AND A SECOND PROOF OF THE
ISOPERIMETRIC INEQUALITY 439 VII.9. VOLUME AND POLARITY 444 VII.10. THE
APPEARANCE OF CONVEX SETS, THEIR DEGREE OF BADNESS 446 VU. 11. VOLUMES
OF SLICES OF CONVEX SETS 459 VII. 12. SECTIONS OF LOW DIMENSION: THE
CONCENTRATION PHENOMENON AND THE DVORETSKY THEOREM ON THE EXISTENCE OF
ALMOST SPHERICAL SECTIONS 470 VII.13. MISCELLANY 477 VII. 14.
INTERMEZZO: CAN WE DISPOSE OF THE ISOPERIMETRIC INEQUALITY? . . . . 493
BIBLIOGRAPHY 499 CHAPTER VIII. POLYGONS, POLYHEDRA, POLYTOPES 505 VIII.
1. INTRODUCTION 505 VIII.2. BASIC NOTIONS 506 VIII.3. POLYGONS 508
VIII.4. POLYHEDRA: COMBINATORICS 513 VIII.5. REGULAR EUCLIDEAN POLYHEDRA
518 VIII.6. EUCLIDEAN POLYHEDRA: CAUCHY RIGIDITY AND ALEXANDROV
EXISTENCE . 524 VIII.7. ISOPERIMETRY FOR EUCLIDEAN POLYHEDRA 530 VIII.8.
INSCRIBABILITY PROPERTIES OF EUCLIDEAN POLYHEDRA; HOW TO ENCAGE A SPHERE
(AN EGG) AND THE CONNECTION WITH PACKINGS OF CIRCLES . . . 532 VIII.9.
POLYHEDRA: RATIONALITY 537 VIII. 10. POLYTOPES (D 3; 4): COMBINATORICS I
539 VIII.LL. REGULAR POLYTOPES (D ^4) 544 VIII. 12. POLYTOPES (D ^ 4):
RATIONALITY, COMBINATORICS II 550 VIII TABLE OF CONTENTS XV IX.9.
ALGORITHMICS AND PLANE TILINGS: APERIODIC TILINGS AND DECIDABILITY,
CLASSIFICATION OF PENROSE TILINGS 607 IX. 10. HYPERBOLIC TILINGS AND
RIEMANN SURFACES 617 BIBLIOGRAPHY 620 CHAPTER X. LATTICES AND PACKINGS
IN HIGHER DIMENSIONS 623 X.I. LATTICES AND PACKINGS ASSOCIATED WITH
DIMENSION 3 623 X.2. OPTIMAL PACKING OF BALLS IN DIMENSION 3, KEPLER S
CONJECTURE AT LAST RESOLVED 629 X.3. A BIT OF RISKY EPISTEMOLOGY: THE
FOUR COLOR PROBLEM AND THE KEPLER CONJECTURE 639 X.4. LATTICES IN
ARBITRARY DIMENSION: EXAMPLES 641 X.5. LATTICES IN ARBITRARY DIMENSION:
DENSITY, LAMINATIONS 648 X.6. PACKINGS IN ARBITRARY DIMENSION: VARIOUS
OPTIONS FOR OPTIMALITY . . . . 654 X.7. ERROR CORRECTING CODES 659 X.8.
DUALITY, THETA FUNCTIONS, SPECTRA AND ISOSPECTRALITY IN LATTICES 667
BIBLIOGRAPHY 673 CHAPTER XL GEOMETRY AND DYNAMICS I: BILLIARDS 675 XI.
1. INTRODUCTION AND MOTIVATION: DESCRIPTION OF THE MOTION OF TWO
PARTICLES OF EQUAL MASS ON THE INTERIOR OF AN INTERVAL 675 XI.2. PLAYING
BILLIARDS IN A SQUARE 679 XI.3. PARTICLES WITH DIFFERENT MASSES:
RATIONAL AND IRRATIONAL POLYGONS . . . 689 XI.4. RESULTS IN THE CASE OF
RATIONAL POLYGONS: FIRST RUNG 692 XI.5. RESULTS IN THE RATIONAL CASE:
SEVERAL RUNGS HIGHER ON THE LADDER . . . 696 XI.6. RESULTS IN THE CASE
OF IRRATIONAL POLYGONS 705 XI.7. RETURN TO THE CASE OF TWO MASSES:
SUMMARY 710 XI.8 XVI TABLE OF CONTENTS XII.7. DO THE MECHANICS DETERMINE
THE METRIC? 779 XII.8. RECAPITULATION AND OPEN QUESTIONS 781 XII.9.
HIGHER DIMENSIONS 781 BIBLIOGRAPHY 782 SELECTED ABBREVIATIONS FOR
JOURNAL TITLES 785 NAME INDEX 789 SUBJECT INDEX 795 SYMBOL INDEX 827
|
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| author | Berger, Marcel 1927-2016 |
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| dewey-ones | 516 - Geometry |
| dewey-raw | 516 |
| dewey-search | 516 |
| dewey-sort | 3516 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV025599853 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:37:15Z |
| institution | BVB |
| isbn | 9783540709961 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020195402 |
| oclc_num | 663496387 |
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| physical | XVI, 831 S. Ill., graph. Darst. |
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| spelling | Berger, Marcel 1927-2016 Verfasser (DE-588)12047672X aut Geometry revealed a Jacob's Ladder to modern higher geometry Marcel Berger Berlin [u.a.] Springer 2010 XVI, 831 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Geometrie (DE-588)4020236-7 gnd rswk-swf Geometrie (DE-588)4020236-7 s DE-604 Erscheint auch als Online-Ausgabe 978-3-540-70997-8 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020195402&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Berger, Marcel 1927-2016 Geometry revealed a Jacob's Ladder to modern higher geometry Geometrie (DE-588)4020236-7 gnd |
| subject_GND | (DE-588)4020236-7 |
| title | Geometry revealed a Jacob's Ladder to modern higher geometry |
| title_auth | Geometry revealed a Jacob's Ladder to modern higher geometry |
| title_exact_search | Geometry revealed a Jacob's Ladder to modern higher geometry |
| title_full | Geometry revealed a Jacob's Ladder to modern higher geometry Marcel Berger |
| title_fullStr | Geometry revealed a Jacob's Ladder to modern higher geometry Marcel Berger |
| title_full_unstemmed | Geometry revealed a Jacob's Ladder to modern higher geometry Marcel Berger |
| title_short | Geometry revealed |
| title_sort | geometry revealed a jacob s ladder to modern higher geometry |
| title_sub | a Jacob's Ladder to modern higher geometry |
| topic | Geometrie (DE-588)4020236-7 gnd |
| topic_facet | Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020195402&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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