From: prezky@apple.com (Michael Press) Newsgroups: sci.math Subject: Re: Proof that there are only 5 Platonic solids? Date: Mon, 02 Dec 1996 15:09:30 -0500 >I thought I saw a proof or a proof outline some years back showing that >there are only 5 Platonic solids: 1) tetrahedron, 2) hexahedron (cube), >3) octahedron, 4) dodecahedron, and 5) icosahedron. (Each solid is >assumed to be a regular polyhedron.) > >Does anyone know a proof of this? It would be especially nice to find >one which sheds light on why you can inscribe a regular n-gon in a circle >[the classic way of approximating a circle's area] for arbitrarily large >n, but you evidently *can't* do the same for a regular polyhedron of n faces >inscribed in a sphere for n > 20. > >Any references and/or thoughts are appreciated. The Platonic solids are as you ennumerated, and they are all the regular convex polyhedara. There are nine regular polyhedra. The additional four are the Kepler-Poinsot polyhedra, and they are not convex. First some notation and definitions. The vertex figure for a polyhedral vertex is the polygon whose vertices are the midpoints of the edges incident to the polyhedral vertex. Example: the vertex figures of a cube are all regular triangles. By {n, k} we mean a regular polygon with n sides and exterior angles equal to 2*pi*k/n. {4, 1} is a square, {4, 3} is square oppositely described to {4,1}, {5, 2} is the figure we often call a five pointed star. Note that n and k must be co-prime. A regular polyhedron is one for which all faces are the same kind of regular polygon, and all vertex figures are the same kind of regular polygon. We will now enumerate the nine regular polyhedra by naming the face and vertex figure for each: Name Face Vertex figure Tetrahedron {3,1} {3,1} Hexahedron, Cube {4,1} {3,1} Octahedron {3,1} {4,1} Dodecahedron {5,1} {3,1} Icosahedron {3,1} {5,1} Small stellated dodecahedron {5,2} {5,1} Great stellated dodecahedron {5,2} {3,1} Great dodecahedron {5,1} {5,2} Great icosahedron {3,1} {5,2} Sorry, do not have the full bibliographic entries. Coxeter, Regular Polytopes, Dover. Menninger, Polyhedron Models. -- Michael Press prezky@apple.com