From: edgar@math.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: Platonic Solids in More than Three Dimensions
Date: Mon, 19 Dec 1994 13:54:45 -0500

In article <3d4eduINNd4o@bhars12c.bnr.co.uk>, kevman@bnr.co.uk (Kevin
Mansell) wrote:

> I am interested in Platonic Solids in more than 3 dimensions.
> If you allow 4 or 5 dimensions, are there the equivalents of
> the tetrahedron, cube/octahedron and icosahedron/dodecahedron?

classified by Schl\"afli symbols:

2 dimensions, regular polygons
  {n} for n = 3, 4, 5, ...
3 dimensions, regular polyhedra
  {3,3}  tetrahedron
  {4,3}  cube
  {3,4}  octahedron
  {5,3}  dodecahedron
  {3,5}  icosahedron
4 dimensions
  {3,3,3}  simplex
  {4,3,3}  hypercube
  {3,3,4}  16-cell
  {3,4,3}  24-cell
  {5,3,3}  120-cell
  {3,3,5}  600-cell
higher than 4
  {3,3,3,...,3,3}  simplex
  {4,3,3,...,3,3}  higher cube
  {3,3,3,...,3,4}  octahedron analog

References:  for example:
  INTRODUCTION TO GEOMETRY, H.S.M. Coxeter, 1961.
  AN INTRODUCTION TO THE GEOMETRY OF N DIMENSIONS, D.M.Y. Sommerville, 1929

Regular packings of Euclidean space:
packings of the plane, dimension 2
  {4,4}  packing by squares
  {3,6}  packing by triangles
  {6,3}  packing by hexagons
packings of space, dimension 3
  {4,3,4}  packing by cubes
dimension 4
  {4,3,3,4}  packing by hypercubes
  {3,4,3,3}  packing by 24-cells
  {3,3,4,3}  packing by 16-cells
higher than 4
  {4,3,3,...,3,3,4}  packing by higher cubes

There are no others.
 . . . .
Gerald A. Edgar                  edgar@math.ohio-state.edu
Department of Mathematics
The Ohio State University   telephone: 614-292-0395(Office)
Columbus, OH 43210               614-292-4975 (Math. Dept.)