The Geometric Vein: The Coxeter Festschrift

598 Pages·1982·1.7 MB·English

H. S. M. COXETER
The Geometric Vein
The Coxeter Festschrift
Edited by
Chandler Davis Branko Griinbaum
F. A. Sherk
With Contributions by
Patrice Assouad C. M. Campbell Jeffrey Cohen
H. S. M. Coxeter Donald W. Crowe Patrick Du Val
W. L. Edge Erich W. Ellers G. Ewald L. Fejes T6th
J. C. Fisher David Ford Cyril W. L. Gamer
William J. Gilbert J. M. Goethals P. R. Goodey
Branko Griinbaum N. I. Haritonova Howard L. Hiller
S. G. Hoggar Norman W. Johnson William M. Kantor
I. N. Kashirina Ignace I. Kolodner Joseph Malkevitch
John McKay P. McMullen J. C. P. Miller W. O. J. Moser
Stanley E. Payne Jean J. Pedersen J. F. Rigby
E. F. Robertson C. A. Rogers B. A. Rosenfeld D. Ruoff
I. J. Schoenberg J. J. Seidel G. C. Shephard J. Shilleto
J. Tits W. T. Tutte Harold N. Ward Asia Weiss
J. B. Wilker J. M. Wills M. M. Woodcock I. M. Yaglom
With 5 Color Plates,
6 Halftones, and 211 Line Illustrations
Springer-Verlag
New York Heidelberg Berlin
Chandler Davis Branko Griinbaum
Department of Mathematics Department of Mathematics
University of Toronto University of Washington
Toronto M5S lAI Seattle, W A 98195
Canada U.S.A
F. A. Sherk
Department of Mathematics
University of Toronto
Toronto M5S lAI
Canada
AMS Subject Classifications (1980): 51-06,52-06
Library of Congress Cataloging in Publication Data
Main entry under title:
The Geometric vein.
Bibliography: p.
I. Geometry-Addresses, essays, lectures.
2. Coxeter, H. S. M. (Harold Scott Macdonald),
1907- . I. Coxeter, H. S. M. (Harold
Scott Macdonald), 1907- . II. Davis,
Chandler. III. Griinbaum, Branko. IV. Sherk,
F. A.
QA446.G46 516 81-9171
AACR2
© 1981 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1981
All rights reserved. No part of this book may be translated or reproduced in any
form without written permission from Springer-Verlag, 175 Fifth Avenue, New York,
New York 10010, U.S.A.
The use of general descriptive names, trade names, trademarks, etc. in this publication,
even if the former are not especially identified, is not to be taken as a sign that such
names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly
be used freely by anyone.
9 8 7 6 5 432 1
TSBN-13: 978-1-4612-5650-2 e-TSBN-13:978-1-4612-5648-9
DOT: 10.1007/978-1-4612-5648-9
Contents
Introduction. . . . . . . . . I
H. S. M. Coxeter: Published Works . 5
PART I: POLYTOPES AND HONEYCOMBS
Uniform Tilings with Hollow Tiles
BRANKO GRUNBAUM, J. C. P. MILLER,
AND G. C. SHEPHARD. . . . . . . 17
Spherical Tilings with Transitivity Properties
BRANKO GRUNBAUM AND G. C. SHEPHARD . 65
Some Isonemal Fabrics on Polyhedral Surfaces
hmL~~~N ......... . .99
Convex Bodies which Tile Space
P. McMULLEN . . . . . . . . .123
Geometry of Radix Representations
WILLIAM J. GILBERT . . . . . . . . . . . . . . . . . 129
Embeddability of Regular Polytopes and Honeycombs
in Hypercubes
PATRICE ASSOUAD. . . . . . . . . . .. ..... 141
The Derivation of Schoenberg's Star-Polytopes from
Schoute's Simplex Nets
H. S. M. COXETER. . . . . . . . . . . . . . . . . .149
Vl Contents
The Harmonic Analysis of Skew Polygons as a Source
of Outdoor Sculptures
I. J. SCHOENBERG . . . . . . . . . . . . . . . . . . 165
The Geometry of African Art III. The Smoking Pipes
of Begho
DONALD W. CROWE . . . . . . . . . . . 177
Crystallography and Cremona Transformations
PATRICK Du VAL . . . . . . . . . . . . 191
Cubature Formulae, Polytopes, and Spherical Designs
J. M. GOETHALS AND J. J. SEIDEL. . . . . . . . .203
Two Quatemionic 4-Polytopes
S. G. HOGGAR . . . . . . . .219
Span-Symmetric Generalized Quadrangles
STANLEY E. PAYNE. . . . . . . . . .231
On Coxeter's Loxodromic Sequences of Tangent Spheres
ASIA WEISS . . . . . . . . . . . . . . . . .243
PART II: EXTREMAL PROBLEMS
Elementary Geometry, Then and Now
I. M. YA GLOM . . . . . . . . . .253
Some Researches Inspired by H. S. M. Coxeter
L. FEJES T6TH . . . . . . . . . . . . .271
Some Problems in the Geometry of Convex Bodies
C. A. ROGERS . . . . . . . . . . . . . .279
On an Analog to Minkowski's Lattice Point Theorem
J. M. WILLS .............. . .285
Intersections of Convex Bodies with Their Translates
P. R. GOODEY AND M. M. WOODCOCK. . . . .289
An Extremal Property of Plane Convex Curves-
P. Ungar's Conjecture
IGNACE I. KOLODNER. . . . . . . .. ....... 297
PART III: GEOMETRIC TRANSFORMATIONS
Polygons and Polynomials
J. C. FISHER, D. RUOFF, AND J. SHILLETO .321
Algebraic Surfaces with Hyperelliptic Sections
W. L. EDGE ........... . .335
Contents Vll
On the Circular Transformations of Mobius, Laguerre,
and Lie
I. M. Y AGLOM . . . . . . . . . . . .. ..... 345
The Geometry of Cycles, and Generalized Laguerre
Inversion
J. F. RIGBY ............. . .355
Inversive Geometry
J. B. WILKER .379
Absolute Polarities and Central Inversions
NORMAN W. JOHNSON . . . . . . . .443
Products of Axial Affinities and Products of Central
Collineations
ERICH W. ELLERS . . . . .465
Normal Forms of Isometries
G. EWALD ..... . .471
Finite Geometries with Simple, Semisimple, and Quasisimple
Fundamental Groups
B. A. ROSENFELD, N.I. HARITONOVA, AND I. N. KASHIRINA .477
Motions in a Finite Hyperbolic Plane
CYRIL W. L. GARNER. . . . . . . . . . . . . .485
PART IV: GROUPS AND PRESENTATIONS OF GROUPS
Generation of Linear Groups
WILLIAM M. KANTOR. . . . .... 497
On Covering Klein's Curve and Generating Projective
Groups
JEFFREY COHEN. . . . . . . . . . . . . . .511
A Local Approach to Buildings
J. TITs. . . . . . . . . .519
Representations and Coxeter Graphs
DAVID FORD AND JOHN McKAY . . .549
Coinvariant Theory of a Coxeter Group
HOWARD L. HILLER . . . . . . . .555
Two-Generator Two-Relation Presentations for Special
Linear Groups
C. M. CAMPBELL AND E. F. ROBERTSON. . . . . . .561
Groups Related to Fa,b,c Involving Fibonacci Numbers
C. M. CAMPBELL AND E. F. ROBERTSON. . . . . . .569
viii Contents
PART V: THE COMBINATORIAL SIDE
Convex Polyhedra
W. T. TUITE ... .579
Non-Hamilton Fundamental Cycle Graphs
JOSEPH MALKEVITCH . . . . . . . . .583
Some Combinatorial Identities
W. O. J. MOSER . . . . . .585
Binary Views of Ternary Codes
HAROLD N. WARD. . . . . .593
Introduction
Geometry has been defined as that part of mathematics which makes appeal to
the sense of sight; but this definition is thrown in doubt by the existence of great
geometers who were blind or nearly so, such as Leonhard Euler. Sometimes it
seems that geometric methods in analysis, so-called, consist in having recourse to
notions outside those apparently relevant, so that geometry must be the joining
of unlike strands; but then what shall we say of the importance of axiomatic
programmes in geometry, where reference to notions outside a restricted reper-
tory is banned? Whatever its definition, geometry clearly has been more than the
sum of its results, more than the consequences of some few axiom sets. It has
been a major current in mathematics, with a distinctive approach and a distinc-
ti ve spirit.
A current, furthermore, which has not been constant. In the 1930s, after a
period of pervasive prominence, it appeared to be in decline, even passe. These
same years were those in which H.S.M. Coxeter was beginning his scientific
work. Undeterred by the unfashionability of geometry, Coxeter pursued it with
devotion and inspiration. By the 1950s he appeared to the broader mathematical
world as a consummate practitioner of a peculiar, out-of-the-way art. Today
there is no longer anything that out-of-the-way about it. Coxeter has contributed
to, exemplified, we could almost say presided over an unanticipated and dra-
matic revival of geometry.
Coxeter's work, though faithful to the 19th-century traditions and style which
it continues, has gained repeatedly from his openness to other sources: to the
uses of mathematics in physics, and of symmetry in the arts; to the power of
combinatorics, especially its introduction into group theory; to the progress in
recent decades in extremal problems and inequalities, especially those involving
convexity. Thus his work ties together much of the geometry of our times.
This was plain from the success of the Coxeter Symposium held at the
University of Toronto, 21-25 May 1979. Close to a hundred mathematicians,
2 Introduction
from eight countries, gathered for five days of papers. The invited speakers (not
all of whom were able to attend) included, beside Coxeter himself, the following:
John Horton Conway, Erich Ellers, Gunter Ewald, L. Fejes T6th, Branko
Griinbaum, William Kantor, P. McMullen, C. A. Rogers, B. A. Rosenfeld, J.J.
Seidel, G.c. Shephard, J. Tits, W.T. Tutte, and I.M. Yaglom. Rich fare indeed,
enriched further by a roomful of geometrical models (assembled by Barry
Monson) and an evening of films (assembled by Seymour Schuster).
The present volume, growing out of the Symposium, aims to document the
broad range to which some root ideas have ramified. It will be noticed here as it
was at the Symposium, how many of the papers refer to concerns and contribu-
tions of Coxeter directly, and most of the rest do so indirectly. Several of the
papers have the character of surveys, especially those of Kantor, Fejes T6th, and
Tutte, and the first by Yaglom. Most of the papers, though, while tracing a
relationship to a wider area, culminate in a new contribution. (It is characteristic
of the subject that this takes pictorial form almost as often as definition-theorem-
proof.) Despite the closeness of their subject matter, the articles do not refer to
each other. We think it will be helpful, therefore, if we say a few words about the
collection overall.
We have begun it with a listing of the scientific papers of H.S.M Coxeter. This
list, though it is still being added to, surely deserves a place in the literature, and
there could be no more appropriate place.
The papers which follow fall naturally into five divisions, and we have made
this subdivision in presenting them here, though there are manifold interrela-
tionships and overlaps between them. Two fields not recognized in this scheme,
but recurring in the papers, are classical algebraic geometry and the 19th
century's independent synthetic development of projective and other geometries.
The intuitive idea of symmetry, already extended beyond the reach of intu-
ition by the ancients, underlies the extension of their "regular figures" to wider
definitions, to higher dimensions, and to different geometries. This subject was
surveyed in Coxeter's Regular Polytopes (1948), and has proliferated since. We
hope some sense of its achievements and potentialities is conveyed by the papers
in Part I of this book.
The symmetry classes of crystallography are expressed by discrete subgroups
of the Euclidean group; various other sorts of symmetry types by discrete
subgroups of other classical groups. The interplay between the regular figures
and the transformations to which they are subjected is a familiar feature of this
study. If the figures occupy the foreground in the papers of Part I, the transfor-
mation takes precedence in those of Part III, and group in Part IV.
Part II grows out of the study of regular figures in another way: the
minimality and covering properties which they enjoy, and which account for so
many of their applications, are sometimes shared by nonsymmetrical figures.
(Even a tiling of the plane by equal squares illustrates this: it may, but need not,
have translational symmetry in two directions.) Thus there is an area where the
study of figures bound by restrictions of symmetry meets the extremal problem
unbound by any such restriction. Several of the papers in both Parts I and II are
properly in the area of overlap.