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PREFACE
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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 |
366
367
368
369
370
371 |
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