Abstract Regular Polytopes

by Peter McMullen , Egon Schulte

• ISBN13: 9780521814966
• ISBN10: 0521814960
• Format: Hardcover
• Copyright: 2003-01-20
• Publisher: Cambridge University Press

Summary

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).

Table of Contents

Preface	xiii
Classical Regular Polytopes	1	(20)
The Historical Background	1	(6)
Regular Convex Polytopes	7	(8)
Extensions of Regularity	15	(2)
Regular Maps	17	(4)
Regular Polytopes	21	(43)
Abstract Polytopes	22	(9)
Regular Polytopes	31	(8)
Order Complexes	39	(3)
Quotients	42	(7)
C-Groups	49	(11)
Presentations of Polytopes	60	(4)
Coxeter Groups	64	(31)
The Canonical Representation	64	(7)
Groups of Spherical or Euclidean Type	71	(5)
Groups of Hyperbolic Type	76	(2)
The Universal Polytopes {p1, . . . , pn--1}	78	(5)
The Order of a Finite Coxeter Group	83	(12)
Amalgamation	95	(26)
Amalgamation of Polytopes	96	(5)
The Classification Problem	101	(2)
Finite Quotients of Universal Polytopes	103	(3)
Free Extensions of Regular Polytopes	106	(3)
Flat Polytopes and the FAP	109	(6)
Flat Polytopes and Amalgamation	115	(6)
Realizations	121	(27)
Realizations in General	121	(6)
The Finite Case	127	(13)
Apeirotopes	140	(8)
Regular Polytopes on Space-Forms	148	(35)
Space-Forms	148	(4)
Locally Spherical Polytopes	152	(10)
Projective Regular Polytopes	162	(3)
The Cubic Toroids	165	(5)
The Other Toroids	170	(2)
Relationships Among Toroids	172	(3)
Other Euclidean Space-Forms	175	(2)
Chiral Toroids	177	(1)
Hyperbolic Space-Forms	178	(5)
Mixing	183	(61)
General Mixing	183	(9)
Operations on Regular Polyhedra	192	(9)
Cuts	201	(5)
The Classical Star-Polytopes	206	(11)
Three-Dimensional Polyhedra	217	(19)
Three-Dimensional 4-Apeirotopes	236	(8)
Twisting	244	(45)
Twisting Operations	244	(3)
The Polytopes LK.G	247	(8)
The Polytopes 2K and 2K.G(s)	255	(4)
Realizations of 2K and 2K.G(s)	259	(5)
A Universality Property of LK.G	264	(8)
Polytopes with Small Faces	272	(17)
Unitary Groups and Hermitian Forms	289	(71)
Unitary Reflexion Groups	290	(8)
Hermitian Forms and Reflexions	298	(7)
General Considerations	305	(15)
Generalized Triangle Groups	320	(12)
Tetrahedral Diagrams	332	(15)
Circuit Diagrams with Tails	347	(8)
Abstract Groups and Diagrams	355	(5)
Locally Toroidal 4-Polytopes: I	360	(27)
Grunbaum's Problem	360	(3)
The Type {4,4,3}	363	(6)
The Type {4,4,4}	369	(9)
Cuts for the Types {4,4,r}	378	(5)
Relationships Among Polytopes of Type {4,4,r}	383	(4)
Locally Toroidal 4-Polytopes: II	387	(58)
The Basic Enumeration Technique	387	(5)
The Polytopes pT4(s,0) := {{6, 3}(s,0), {3, p}}	392	(8)
Polytopes with Facets {6, 3}(s,s)	400	(10)
The Polytopes 6T4(s,0),(t,0) := {{6, 3}(s,0), {3, 6}(t,0)}	410	(7)
The Type {3,6,3}	417	(6)
Cuts of Polytopes of Type {6, 3, p} or {3, 6, 3}	423	(8)
Hyperbolic Honeycombs in H3	431	(6)
Relationships Among Polytopes of Types {6, 3, p} or {3, 6, 3}	437	(8)
Higher Toroidal Polytopes	445	(26)
Hyperbolic Honeycombs in H4 and H5	445	(5)
Polytopes of Rank 5	450	(9)
Polytopes of Rank 6: Type {3, 3, 3, 4, 3}	459	(3)
Polytopes of Rank 6: Type {3, 3, 4, 3, 3}	462	(3)
Polytopes of Rank 6: Type {3, 4, 3, 3, 4}	465	(6)
Regular Polytopes Related to Linear Groups	471	(31)
Regular Polyhedra	471	(7)
Connexions Among the Polyhedra	478	(6)
Realizations of the Polyhedra	484	(6)
The 4-Polytopes	490	(10)
Connexions Among 4-Polytopes	500	(2)
Miscellaneous Classes of Regular Polytopes	502	(17)
Locally Projective Regular Polytopes	502	(7)
Mixed Topological Types	509	(10)
Bibliography	519
Indices
List of Symbols	539
Author Index	543
Subject Index	544