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