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.)