The Theory of Uniform Polytopes and Honeycombs

Thesis for the Degree of Ph.D.
UNIVERSITY OF TORONTO
Norman W. Johnson
1966
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CONTENTS (vi-vii)

  • PREFACE (i-iv)

  • 1. POLYTOPES AND HONEYCOMBS IN N-SPACE (1-32)
    • 1.1 Euclidean and non-Euclidean spaces … 1-5
    • 1.2 Hyperspheres and horospheres … 6-8
    • 1.3 Convexity … 9-11
    • 1.4 Polytopes and honeycombs … 12-19
    • 1.5 Combinatorial properties … 20-22
    • 1.6 Nonsimple figures … 23-26
    • 1.7 Orientability and density … 27-30
    • 1.8 Regular and uniform figures … 31-32

  • 2. SIMPLEXES (33-49)

    • 2.1 Properties of simplexes … 33-39
    • 2.2 The Schläfli criterion … 40-45
    • 2.3 The graphical notation … 46-49

  • 3. GROUPS GENERATED BY REFLECTIONS (50-90)

    • 3.1 Groups of isometries … 50-52
    • 3.2 The product of reflections … 53-58
    • 3.3 Groups embeddable in hyperbolic n-space … 59-69
    • 3.4 Crystallographic groups … 70-71
    • 3.5 Enumeration of admissible groups … 72-81
    • 3.6 Nonembeddable groups … 82-90

  • 4. SUBGROUPS AND AUTOMORPHISMS (91-138)

    • 4.1 Alternating and semi-rotational subgroups … 91-97
    • 4.2 Commutator and other subgroups … 98-107
    • 4.3 Central quotient groups … 108-112
    • 4.4 Augmented spherical groups … 113-120
    • 4,5 Augmented Euclidean group… 121-138

  • 5. THE SCHWARZ POLYGONS (139-161)

    • 5.1 The Kepler dyads … 139-143
    • 5.2 Amalgamations … 144-148
    • 5.3 The Schwarz-Mobius triangles … 149-152
    • 5.4 The Schwarz-Cartan polygons … 153-155
    • 5.5 The Schwarz-Lanner triangles … 156-161

  • 6. THE GOURSAT POLYTOPES (162-182)

    • 6.1 Admissible amalgamations … 162-165
    • 6.2 The Goursat tetrahedra … 166-173
    • 6.3 Prisms, spikes, and wedges … 174-175
    • 6.4 Inadmissible tetrahedra … 176-179
    • 6.5 Other investigations … 180-182

  • 7. REGULAR FIGURES IN N-SPACE (183-216)

    • 7.1 Symmetry and regularity … 183-188
    • 7.2 Reciprocation … 189-192
    • 7.3 Regular polytopes and honeycombs … 193-199
    • 7.4 Regular compounds … 200-202
    • 7.5 Fully reflexible compounds … 203-207
    • 7.6 Compounds with irreflexible components … 208-216

  • 8. UNIFORM POLYTOPES AND HONEYCOMBS (217-296)

    • 8.1 Uniformity and quasi-uniformity … 217-225
    • 8.2 Wythoff's construction … 226-236
    • 8.3 Interpretation of subgraphs … 237-241
    • 8.4 The figures kij and 0[n] … 242-248
    • 8.5 Alternation … 249-256
    • 8.6 Antiprisms and snub polytopes … 257-270
    • 8.7 Partial alternation … 271-279
    • 8.8 Magnification … 280-290
    • 8.9 Historical remarks … 291-296

  • Tables (297-365)

    • Notes on the tables … 297-365
    • Table 1. Irreducible spherical groups generated by reflections … 301-303
    • Table 2. Irreducible Euclidean groups generated by reflections … 304-307
    • Table 3. Embeddable hyperbolic groups generated by reflections … 308-317
    • Table 4. The Schwarz-Mobius triangles … 318
    • Table 5. The Schwarz-Cartan polygons … 319-320
    • Table 6. The Schwarz-Lanner triangles … 321
    • Table 7. The Goursat-Mobius tetrahedra … 322-336
    • Table 8. The Goursat-Cartan polyhedra … 337-345
    • Table 9. The Goursat-Lanner tetrahedra … 346-351
    • Table 10. Regular polytopes … 352-353
    • Table 11. Regular spherical honeycombs … 354-355
    • Table 12. Regular elliptic honeycombs … 356-357
    • Table 13. Regular Euclidean honeycombs … 358
    • Table 14. Regular hyperbolic honeycombs … 359

  • Bibliography (366-371)

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PREFACE (i-iv)

A POLYTOPE in n-dimensional space, Euclidean or non-Euclidean, may be defined recursively, starting with the zero-dimensional case, a single point, as a collection of lower-dimensional polytopes (its elements) and a region (the interior of the polytope) bounded thereby, subject to certain conditions that exclude degenerate and pathological cases. The interior of a polytope need not be simply connected, or even connected, permitting the consideration of star (i.e., self-intersecting) polytopes. An n-dimensional honeycomb is defined analogously as a collection of polytopes, or polytopes and lower-dimensional honeycombs, that essentially "fill" n-space.

Polytopes or honeycombs that have a special kind of symmetry are said to be regular. The five Platonic solids and the plane tessellations of squares, equilateral triangles, or regular hexagons are familiar examples. An n-dimensional polytope or (n-1)-dimensional honeycomb is uniform if it is regular or if, for n > 3, its (n-1)-dimensional elements are uniform and its vertices are all alike. The well-known Archimedean polyhedra are instances of figures that are uniform but not regular.

Beginning with the discovery by J. Kepler in 1619 and Poinsot in 1809 of the four regular star polyhedra, the investigation of uniform figures both in three dimensions and in higher space has been a continuing subject of interest. All the regular polytopes and honeycombs, both Euclidean and non-Euclidean, have been known for some time, and a general theory for them has been worked out, most notably by H.S.M Coxeter in Regular Polytopes [20].* The theory of convex uniform polytopes and simple uniform honeycombs of Euclidean space is also more or less well established, but little has been done with honeycombs of hyperbolic space or with higher-dimensional uniform star polytopes and honeycombs. A thorough investigation, along the lines of Coxeter, M.S. Longuet-Higgins, and J.C.P. Miller's treatment of uniform polyhedra [27], depends on the solution of certain preliminary problems.

*Numbers in brackets refer to the Bibliography on pages 366-371.

The basis of a general theory of uniform polytopes and honeycombs is the fact, observed by W.A. Wythoff, that, with only trivial exceptions, the vertices of each uniform figure are the transforms of a suitably chosen point in the fundamental region for a discrete group generated by reflections, either under the whole group or some proper subgroup. In the case of star polytopes and honeycombs, it is more useful to associate the chosen point not with the fundamental region itself but with a region that is an amalgamation of adjacent replicas of the fundamental region.

A discrete group generated by reflections is said to be embeddable in hyperbolic n-space if it can be generated by reflections in n linearly independent hyperplanes therein such that every n-1 hyperplanes have either a common point or a common direction. Every such group can be realized in spherical, Euclidean, or hyperbolic (n-1)-space; in the non-Euclidean cases and the irreducible Euclidean cases the fundamental region is the closure of a simplex. I describe all the embeddable groups, most of the hyperbolic ones for the first time. Using some of the symbolism and results of Coxeter and Moser [28] and with the aid of an appropriate extension of a graphical notation invented by Coxeter, I obtain various subgroups of each embeddable group, as well as larger groups that arise when the generators of a group are permuted by an automorphism.

The spherical triangles that occur as amalgamations of the fundamental region for a finite group generated by three reflections were enumerated by H. Schwarz in 1873, and A. Goursat proposed, but did not solve, the corresponding problem for spherical tetrahedra in 1889. The solution is given here as part of a determination of all admissible amalgamations for embeddable groups generated by not more than four reflections.

Following a summary of the more important properties of regular polytopes and honeycombs, including the derivation of each regular figure from its characteristic simplex, I extend the definition of regularity to compounds of two or more polytopes or honeycombs. Making use of previously established relationships between groups, I give a complete enumeration of such regular compounds.

The final chapter explains the details of Wythoff's construction for uniform polytopes and honeycombs and introduces some useful variations of it. I show how the operation of alternation can be applied to certain "quasi-uniform" figures obtained by the basic method to produce the interesting snub polytopes and honeycombs. The new operations of partial alternation and magnification account for some uniform figures previously regarded as anomalous and lead to others that are entirely new.

These results provide most of the information needed to carry the investigation of uniform figures at least as far as four-dimensional polytopes and three-dimensional honeycombs. The tables list all the irreducible, embeddable groups generated by reflections, the Schwarz polygons and the Goursat polyhedra, and all the regular polytopes, honeycombs, and compounds.

I should like to acknowledge the support given me at a preliminary stage of my work by a grant from the National Research Council of Canada. I also owe many thanks to my adviser, H.S.M. Coxeter, for his inspiration and guidance and most of all for his patience, to F. A. Scherk and Branko Grünbaum for reading and criticizing the manuscript, to G. de B. Robinson for his appraisal of the final version, and to Ann Brown and Carolyn Piersma for their fine handling of a most difficult job of preparing the typescript.

NORMAN W. JOHNSON

East Lansing, Michigan

March 1966
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BIBLIOGRAPHY (366-371)


(Created August 16, 2024)