Platonic Solid - Combinatorial Properties

Combinatorial Properties

A convex polyhedron is a Platonic solid if and only if

1. all its faces are congruent convex regular polygons,
2. none of its faces intersect except at their edges, and
3. the same number of faces meet at each of its vertices.

Each Platonic solid can therefore be denoted by a symbol {p, q} where

p = the number of edges of each face (or the number of vertices of each face) and

q = the number of faces meeting at each vertex (or the number of edges meeting at each vertex).

The symbol {p, q}, called the Schläfli symbol, gives a combinatorial description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.

All other combinatorial information about these solids, such as total number of vertices (V), edges (E), and faces (F), can be determined from p and q. Since any edge joins two vertices and has two adjacent faces we must have:

The other relationship between these values is given by Euler's formula:

This nontrivial fact can be proved in a great variety of ways (in algebraic topology it follows from the fact that the Euler characteristic of the sphere is 2). Together these three relationships completely determine V, E, and F:

Note that swapping p and q interchanges F and V while leaving E unchanged (for a geometric interpretation of this fact, see the section on dual polyhedra below).