ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Abstract Regular Polytopes PETER MMULLEN University College London EGON SCHULTE Northeastern University PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org c Peter McMullen and Egon Schulte 2002 This book 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. First published 2002 Printed in the United States of America Typeface Times New Roman 10/12.5 pt. System LATEX2ε [] A catalog record for this book is available from the British Library. Library of Congress Cataloging in Publication Data McMullen, Peter, 1955– Abstract regular polytopes / Peter McMullen, Egon Schulte. p. cm. – (Encyclopedia of mathematics and its applications) Includes bibliographical references and index. ISBN 0-521-81496-0 1. Polytopes. I. Schulte, Egon, 1955– II. Title. III. Series. QA691 .M395 2002 516.35 – dc21 2002017391 ISBN 0 521 81496 0 hardback Contents Preface page xiii 1 Classical Regular Polytopes 1 1A The Historical Background 1 1B Regular Convex Polytopes 7 1C Extensions of Regularity 15 1D Regular Maps 17 2 Regular Polytopes 21 2A Abstract Polytopes 22 2B Regular Polytopes 31 2C Order Complexes 39 2D Quotients 42 2E C-Groups 49 2F Presentations of Polytopes 60 3 Coxeter Groups 64 3A The Canonical Representation 64 3B Groups of Spherical or Euclidean Type 71 3C Groups of Hyperbolic Type 76 3D The Universal Polytopes {p1,...,pn−1} 78 3E The Order of a Finite Coxeter Group 83 4 Amalgamation 95 4A Amalgamation of Polytopes 96 4B The Classification Problem 101 4C Finite Quotients of Universal Polytopes 103 4D Free Extensions of Regular Polytopes 106 4E Flat Polytopes and the FAP 109 4F Flat Polytopes and Amalgamation 115 ix x Contents 5 Realizations 121 5A Realizations in General 121 5B The Finite Case 127 5C Apeirotopes 140 6 Regular Polytopes on Space-Forms 148 6A Space-Forms 148 6B Locally Spherical Polytopes 152 6C Projective Regular Polytopes 162 6D The Cubic Toroids 165 6E The Other Toroids 170 6F Relationships Among Toroids 172 6G Other Euclidean Space-Forms 175 6H Chiral Toroids 177 6J Hyperbolic Space-Forms 178 7Mixing 183 7A General Mixing 183 7B Operations on Regular Polyhedra 192 7C Cuts 201 7D The Classical Star-Polytopes 206 7E Three-Dimensional Polyhedra 217 7F Three-Dimensional 4-Apeirotopes 236 8 Twisting 244 8A Twisting Operations 244 8B The Polytopes LK,G 247 8C The Polytopes 2K and 2K,G(s) 255 8D Realizations of 2K and 2K,G(s) 259 8E A Universality Property of LK,G 264 8F Polytopes with Small Faces 272 9 Unitary Groups and Hermitian Forms 289 9A Unitary Reflexion Groups 290 9B Hermitian Forms and Reflexions 298 9C General Considerations 305 9D Generalized Triangle Groups 320 9E Tetrahedral Diagrams 332 9F Circuit Diagrams with Tails 347 9G Abstract Groups and Diagrams 355 10 Locally Toroidal 4-Polytopes: I 360 10A Gr¨unbaum’s Problem 360 10B The Type {4,4,3} 363 10C The Type {4,4,4} 369 10D Cuts for the Types {4, 4, r} 378 10E Relationships Among Polytopes of Type {4, 4, r} 383 Contents xi 11 Locally Toroidal 4-Polytopes: II 387 11A The Basic Enumeration Technique 387 T 4 ={{ , } , { , }} 11B The Polytopes p (s,0) : 6 3 (s,0) 3 p 392 11C Polytopes with Facets {6, 3}(s,s) 400 T 4 ={{ , } , { , } } 11D The Polytopes 6 (s,0),(t,0) : 6 3 (s,0) 3 6 (t,0) 410 11E The Type {3, 6, 3} 417 11F Cuts of Polytopes of Type {6, 3, p} or {3, 6, 3} 423 11G Hyperbolic Honeycombs in H3 431 11H Relationships Among Polytopes of Types {6, 3, p} or {3, 6, 3} 437 12 Higher Toroidal Polytopes 445 12A Hyperbolic Honeycombs in H4 and H5 445 12B Polytopes of Rank 5 450 12C Polytopes of Rank 6: Type {3, 3, 3, 4, 3} 459 12D Polytopes of Rank 6: Type {3, 3, 4, 3, 3} 462 12E Polytopes of Rank 6: Type {3, 4, 3, 3, 4} 465 13 Regular Polytopes Related to Linear Groups 471 13A Regular Polyhedra 471 13B Connexions Among the Polyhedra 478 13C Realizations of the Polyhedra 484 13D The 4-Polytopes 490 13E Connexions Among 4-Polytopes 500 14 Miscellaneous Classes of Regular Polytopes 502 14A Locally Projective Regular Polytopes 502 14B Mixed Topological Types 509 Bibliography 519 Indices List of Symbols 539 Author Index 543 Subject Index 544 1 Classical Regular Polytopes Our purpose in this introductory chapter is to set the scene for the rest of the book. We shall do this by briefly tracing the historical development of the subject. There are two main reasons for this. First, we wish to recall the historical traditions which lie behind current research. It is all too easy to lose track of the past, and it is as true in mathematics as in anything else that those who forget history may be compelled to repeat it. But perhaps more important is the need to base what we do firmly in the historical tradition. A tendency in mathematics to greater and greater abstractness should not lead us to abandon our roots. In studying abstract regular polytopes, we shall always bear in mind the geometric origins of the subject. We hope that this introductory survey will help the reader to find a firm basis from which to view the modern subject. The chapter has four sections. In the first, we provide an historical sketch, leading up to the point at which the formal material of Chapter 2 begins. The second is devoted to an outline of the theory of regular convex polytopes, which provide so much of the motivation for the abstract definitions which we subsequently adopt. In the third, we treat various generalizations of regular polytopes, mainly in ordinary euclidean space, including the classical regular star-polytopes. In the fourth, we introduce regular maps, which are the first examples of abstract regular polytopes, although the examples considered here occur before the general theory was formulated. 1A The Historical Background Regular polytopes emerge only gradually out of the mists of history. Apart from certain planar figures, such as squares and triangles, the cube, in the form of a die, was probably the earliest known to man. Gamblers would have used dice from the earliest days, and a labelled example helped linguists to work out the Etruscan words for “one” up to “six”. The Etruscans also had dodecahedral dice; examples date from before 500, and may even be much earlier. The other three “platonic” solids appear not to have been employed in gambling; two out of the three do not roll well in any case. However, it is only somewhat later that the regular solids were studied for their own sakes, and the leap from them to the regular star-polyhedra, analogous to that from 1 2 1 Classical Regular Polytopes pentagon to pentagram, had to await the later middle ages. The nineteenth century first saw regular polytopes of higher dimension, but the real flowering of what is, in origin, one of the oldest branches of mathematics occurred only in the twentieth century. In this section, we shall give a brief outline of the historical background to the theory of regular polytopes. This is not intended to be totally comprehensive, although we have attempted to give the salient features of more than two millenia of investigations in the subject. The Classical Period Before the Greeks As we have already said, the cube was probably the first known regular polyhedron; certainly it was well known before the ideas of geometry and symmetry had themselves been formulated. Curiously, though, stone balls incised in ways that illustrate all the symmetry groups of the regular polyhedra were discovered in Scotland in the nineteenth century; they appear to date from the first half of the third millenium (see [137, Chapter 7]). The Egyptians were also aware of the regular tetrahedron and octahedron. As an aside, we pose the following question. Many attempts have been made to explain why the pyramids are the shape they are, or, more specifically, why the ratio of height to base of a square pyramid is always roughly the same particular number. In particular, such explanations often manage to involve π in some practical way, such as measurements by 256 ≈ . π rolling barrels – the Egyptians’ theoretical value 81 3 16 for was fairly accurate. Is it possible, though, that a pyramid was intended to be half an octahedron? The actual angles of slopes of the four proper pyramids at Giza vary between 50◦47 and ◦ 54 14 ; the√ last is only a little short of half the dihedral angle of the octahedron, namely, arccos(1/ 3) ≈ 54◦44. The Early Greeks Despite various recent claims to the contrary, it seems clear that the Greeks were the first to conceive of mathematics as we now understand it. (The mere listing of, for example, Pythagorean triples does not constitute mathematics; a proof of Pythagoras’s theorem obviously does.) According to Proclus (412–485), the discovery of the five regular solids was attributed by Eudemus to Pythagoras (c582–c500) himself; the fact that a point of the plane is exactly surrounded by six equilateral triangles, four squares or three regular hexagons (these giving rise to the three regular tilings of the plane) was also known to his followers.