Geometric regular polytopes:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2020
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| Ausgabe: | First published |
| Schriftenreihe: | Encyclopedia of Mathematics and its Applications
172 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xi, 603 Seiten |
| ISBN: | 9781108489584 9781108778992 |
Internformat
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Datensatz im Suchindex
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| adam_text | Encyclopedia of Mathematics and its Applications
Geometric Regular Poly topes
PETER MCMULLEN
University College London
Cambridge
UNIVERSITY PRESS
Cambridge
UNIVERSITY PRESS
University Printing House, Cambridge CB2 8BS, United Kingdom
One Liberty Plaza, 20th Floor, New York, NY 10006, USA
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Cambridge University Press is part of the University of Cambridge
It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning, and research at the highest international levels of excellence
www cambridge org
Information on this title: www cambridge org/9781108489584
DOI: 10 1017/9781108778992
© Peter McMullen 2020
This publication is in copyright Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press
Printed in the United Kingdom by TJ International Ltd Padstow Cornwall
A catalogue record for this publication is available from the British Library
ISBN 978-1-108-48958-4 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate
First published 2020
Contents
Foreword
ix
I Regular Polytopes 1
1 Euclidean Space 3
1A Algebraic Properties 4
IB Convexity 10
1C Euclidean Structure 16
ID Isometries 21
IE Reflexion Groups 29
IF Subgroup Relationships 40
1G Angle-Sum Relations 42
1H Group Orders 43
1J Ordinary Space 55
IK Quaternions 58
Abstract Regular Polytopes 63
2A Abstract Polytopes 64
2B Regularity 68
2C Regularity Criteria 73
2D Presentations 78
2E Regular Maps 85
2F Special Polytopes 87
Realizations of Symmetric Sets 94
3A TVansitive Actions 94
3B Realization Cone 98
3C Cosine Vectors 101
3D Examples 106
v
Contents
3E Products of Realizations 111
3F A-Orthogonality 114
3G The Wythoff Space 118
3H A-Orthogonal Bases 123
3J Cosine Matrices 131
3K Cuts and Duality 134
3L Realizations over Subfields 135
3M Realizations and Representations 137
Realizations of Polytopes 139
4A Wythoff’s Construction 139
4B Faithful Realizations 145
4C Degenerate Realizations 148
4D Induced Cosine Vectors 149
4E Alternating Products 153
4F Apeirotopes 154
4G Examples of Realizations 159
Operations and Constructions 169
5A Operations on Polyhedra 170
5B General Mixing 179
5C Twisting 185
5D Modifying Mirrors 190
5E Extensions 195
5F Vertex-Figures 197
Rigidity 200
6A Basic concept 200
6B Fine Schläfli symbols 201
6C Shapes 203
6D Rigidity Criteria 204
II Polytopes of Full Rank 207
Classical Regular Polytopes 209
7A Faces of Full Rank 209
7B Polytopes in All Dimensions 217
7C The 24-Cell 226
7D Pentagonal Polyhedra 234
7E The 600-Cell 240
7F The 120-Cell 248
7G Star-Polytopes 251
7H Honeycombs 262
7J Regular Compounds 266
7K Realizations of {5,3,3} 277
Contents
vii
8 Non-Classical Polytopes 301
8A Polytopes in All Dimensions 301
8B Apeirotopes in All Dimensions 309
8C Apeirohedra and Polyhedra 317
8D Higher-Dimensional Exceptions 320
III Polytopes of Nearly Full Rank 329
9 General Families 331
9A Blends 331
9B Twisting Small Diagrams 339
9C Families of Polytopes 341
9D Families of Apeirotopes 350
10 Three-Dimensional Apeirohedra 361
10A The Classification 361
10B Groups of the Apeirohedra 365
10C Rigidity of the Apeirohedra 373
11 Four-Dimensional Polyhedra 383
11A Mirror Vectors 383
11B Mirror Vector (3,2,3) and Its Relatives 386
11C A Family of Petrials 395
11D Mirror Vector (2,3,2) 411
11E Mirror Vector (2,2,2) 414
11F Further Connexions 426
12 Four-Dimensioned Apeirotopes 431
12A Imprimitive Groups 431
12B Group U5 and Relatives 437
12C Twisting P5 442
13 Higher-Dimensional Cases 450
13A The Gateway 451
13B Rotational Symmetry Groups 452
13C The Gosset-Elte Polytopes 460
13D The First Gosset Class 461
13E The Second Gosset Class 464
13F The Third Gosset Class 470
13G A Degenerate Gosset Class 472
IV Miscellaneous Polytopes 475
14 Gosset-Elte Polytopes
14A Rank 6: {32,32-1}
477
477
Contents
viii
14B Rank 6: {3,32 2} 481
14C Rank 6: {3,32’2}* 483
14D Rank 7: {S3^2 1} 491
14E Rank 7: {32,33,1} 493
14F Rank 8: {34,32,1} 495
14G Rank 8: {32,34,1} 498
15 Locally Toroidal Polytopes 506
15A {{4,4 : 4}, {4,3}} and its Dual 506
15B {{3,4}, {4,4 | 3}} and {{4,4 | 3}, {4,4 | 3}} 508
15C {{4,4 : 4}, {4,4 [ 3}} and its Dual 510
15D {{4,4 : 4}, {4,4 : 6}} and its Dual 514
15E {{4,4 | 4}, {4,4 | 3}} and its Dual 517
15F Polytopes of Type {3m_2,6} 520
15G Polytopes of Type {3m_1,6,3n-1} 526
16 A Family of 4-Polytopes 530
16A The Polyhedron {5,5 : 4} 530
16B A Permutation Representation 537
16C The Polytope {{5,5:4}, {5,3}} 538
16D Layers and Strata of {{5,5:4}, {5,3}} 540
16E The Dual Polytope {{3,5}, {5,5 : 4}} 541
16F The Extended Family 541
16G {3,5,3 :: 4} and {5,3,5 :: 4} 544
17 Two Families of 5-Polytopes 546
17A An Intuitive Approach 547
17B Group and Geometry 549
17C A Quotient of {3,5,3,5} 559
17D The Dual Polytope 562
17E Double Covers 566
17F An Extended Family 571
17G Another Symmetric Set 576
17H Other Combinations 580
Afterword 582
Bibliography 583
Notation Index 591
Author Index 594
Subject Index
|
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| discipline | Mathematik |
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| isbn | 9781108489584 9781108778992 |
| language | English |
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| spelling | McMullen, Peter 1942- (DE-588)1097306666 aut Geometric regular polytopes Peter McMullen: University College London First published Cambridge Cambridge University Press 2020 xi, 603 Seiten txt rdacontent n rdamedia nc rdacarrier Encyclopedia of Mathematics and its Applications 172 Regelmäßiges Polytop (DE-588)4177373-1 gnd rswk-swf Regelmäßiges Polytop (DE-588)4177373-1 s DE-604 Erscheint auch als Online-Ausgabe 978-1-108-77899-2 Encyclopedia of Mathematics and its Applications 172 (DE-604)BV000903719 172 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031738313&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4177373-1 |
| title | Geometric regular polytopes |
| title_auth | Geometric regular polytopes |
| title_exact_search | Geometric regular polytopes |
| title_full | Geometric regular polytopes Peter McMullen: University College London |
| title_fullStr | Geometric regular polytopes Peter McMullen: University College London |
| title_full_unstemmed | Geometric regular polytopes Peter McMullen: University College London |
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