San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 Proof That There Are Only Five Platonic Polyhedra

The Platonic polyhedra are the ones made up entirely of one type of polygon and having the same number of edges meeting at each vertex. They are further restricted to be the surfaces of solids which are convex.

Let n be the number of sides of the polygon and m the number of edges meeting at a vertex. Both n and m must be greater than of equal to 3. Let the number of faces, edges and vertices be denoted by f, e and v, respectively.

If the number of edges are counted face-by-face the total count will be nf. But each edge is counted twice so that total is equal to 2e; i.e.,

#### 2e = nf or e = nf/2

A polygon that has n edges also has n vertices. A vertex that has m edges coming together also has m polygons meeting there. A face-by-face counting of vertices gives a total of nv, but each vertex is counted m times. Therefore

#### mv = nf or v = nf/m

The numbers of faces, edges and vertices must satisfy Euler's formula for polyhedra:

#### f -e + v = 2

Substituting the expressions for e and v in terms of f gives

#### f - nf/2 + nf/m = 2 or, equivalently (1 - n/2 + n/m)f = 2 which reduces to f = 4m/(2m - nm + 2n)

For f to be positive (2m-nm + 2n ) must be positive. Since both n and m must be greater than or equal to 3, the conditions to be satified are:

#### 2m -nm + 2n > 0 n ≥ 3 m ≥ 3

Consider the following display of the values of (2m-nm+2n) for various values of n and m.

The Values of (2m-nm+2n)
n\m3456
33210
420-2-4
51-2-5-8
60-4-8-12

The combinations of n and m for which (2m-nm+2n) are positive are shown in red. As can be seen there are only five combinations where this is so. From the relationships previously derived

#### f = 4m/(2m-nm+2n) e = nf/2 v = nf/m

the characteristics of the Platonic polyhedra can be computed and identified.

 (n,m) faces edges vertices name (3,3) 4 6 4 tetrahedron (3,4) 8 12 6 octahedron (3,5) 20 18 12 icosahedron (4,3) 6 12 8 cube (5,3) 12 30 20 dodecahedron

For more on polyhedra and thire Euler-Poincare characteristic see Euler Poincare.