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.)