Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. The rapid development of the subject in the past twenty years has resulted in a rich new theory featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. This is the first comprehensive, up-to-date account of the subject and its ramifications. It meets a critical need for such a text, because no book has been published in this area since Coxeter's "Regular Polytopes" (1948) and "Regular Complex Polytopes" (1974).
Author(s): Peter McMullen, Egon Schulte
Series: Encyclopedia of Mathematics and its Applications 92
Edition: 1st
Publisher: Cambridge University Press
Year: 2002
Language: English
Pages: 566
Cover......Page 1
About......Page 2
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS 92......Page 4
Abstract Regular Polytopes......Page 6
Copyright - ISBN: 0521814960......Page 7
Contents......Page 10
Preface......Page 14
1A The Historical Background......Page 16
1B Regular Convex Polytopes......Page 22
1C Extensions of Regularity......Page 30
1D Regular Maps......Page 32
2 Regular Polytopes......Page 36
2A Abstract Polytopes......Page 37
2B Regular Polytopes......Page 46
2C Order Complexes......Page 54
2D Quotients......Page 57
2E C-Groups......Page 64
2F Presentations of Polytopes......Page 75
3A The Canonical Representation......Page 79
3B Groups of Spherical or Euclidean Type......Page 86
3C Groups of Hyperbolic Type......Page 91
3D The Universal Polytopes {p_1, . . . , p_{n-1}}......Page 93
3E The Order of a Finite Coxeter Group......Page 98
4 Amalgamation......Page 110
4A Amalgamation of Polytopes......Page 111
4B The Classification Problem......Page 116
4C Finite Quotients of Universal Polytopes......Page 118
4D Free Extensions of Regular Polytopes......Page 121
4E Flat Polytopes and the FAP......Page 124
4F Flat Polytopes and Amalgamation......Page 130
5A Realizations in General......Page 136
5B The Finite Case......Page 142
5C Apeirotopes......Page 155
6A Space-Forms......Page 163
6B Locally Spherical Polytopes......Page 167
6C Projective Regular Polytopes......Page 177
6D The Cubic Toroids......Page 180
6E The Other Toroids......Page 185
6F Relationships Among Toroids......Page 187
6G Other Euclidean Space-Forms......Page 190
6H Chiral Toroids......Page 192
6J Hyperbolic Space-Forms......Page 193
7A General Mixing......Page 198
7B Operations on Regular Polyhedra......Page 207
7C Cuts......Page 216
7D The Classical Star-Polytopes......Page 221
7E Three-Dimensional Polyhedra......Page 232
7F Three-Dimensional 4-Apeirotopes......Page 251
8A Twisting Operations......Page 259
8B The Polytopes L^{K,G}......Page 262
8C The Polytopes 2^K and 2^{K,G(s)}......Page 270
8D Realizations of 2^K and 2^{K,G(s)}......Page 274
8E A Universality Property of L^{K,G}......Page 279
8F Polytopes with Small Faces......Page 287
9 Unitary Groups and Hermitian Forms......Page 304
9A Unitary Reflexion Groups......Page 305
9B Hermitian Forms and Reflexions......Page 313
9C General Considerations......Page 320
9D Generalized Triangle Groups......Page 335
9E Tetrahedral Diagrams......Page 347
9F Circuit Diagrams with Tails......Page 362
9G Abstract Groups and Diagrams......Page 370
10A Grünbaum’s Problem......Page 375
10B The Type {4,4,3}......Page 378
10C The Type {4,4,4}......Page 384
10D Cuts for the Types {4, 4, r}......Page 393
10E Relationships Among Polytopes of Type {4, 4, r}......Page 398
11A The Basic Enumeration Technique......Page 402
11B The Polytopes _pT^4_{(s,0)} := {{6, 3}_{(s,0)}, {3, p}}......Page 407
11C Polytopes with Facets {6, 3}_{(s,s)}......Page 415
11D The Polytopes _6T_{(s,0),(t,0)} := {{6, 3}_{(s,0)}, {3, 6}_{(t,0)}}......Page 425
11E The Type {3, 6, 3}......Page 432
11F Cuts of Polytopes of Type {6, 3, p} or {3, 6, 3}......Page 438
11G Hyperbolic Honeycombs in H^3......Page 446
11H Relationships Among Polytopes of Types {6, 3, p} or {3, 6, 3}......Page 452
12A Hyperbolic Honeycombs in H^4 and H^5......Page 460
12B Polytopes of Rank 5......Page 465
12C Polytopes of Rank 6: Type {3, 3, 3, 4, 3}......Page 474
12D Polytopes of Rank 6: Type {3, 3, 4, 3, 3}......Page 477
12E Polytopes of Rank 6: Type {3, 4, 3, 3, 4}......Page 480
13A Regular Polyhedra......Page 486
13B Connexions Among the Polyhedra......Page 493
13C Realizations of the Polyhedra......Page 499
13D The 4-Polytopes......Page 505
13E Connexions Among 4-Polytopes......Page 515
14A Locally Projective Regular Polytopes......Page 517
14B Mixed Topological Types......Page 524
Bibliography......Page 534
List of Symbols......Page 554
Author Index......Page 558
Subject Index......Page 559
