A soccer ball has faces which are pentagons and hexagons wit

A soccer ball has faces which are pentagons and hexagons, with each petagon surrounded by hexagons. Assuming that every vertex has degree 3, determine the number of vertices, edges, and faces on the soccer ball

Solution

Compute the number of vertices, edges, and faces of the soccer ball and verify that they satisfy Euler’s formula V + F – E = 2.

Slicing off each corner of the icosahedron replaces each vertex with a pengtagon with 5 new vertices of degree 3. An icosahedron has 12 vertices, so there are 5*12 = 60 vertices in the truncated icosahedron.

2E = dV = 3 * 60, so the there are 90 edges. The 12 new faces are all pentagons, and the 20 original triangular faces become 20 hexagons giving a total of 32 faces. 60 + 32 – 90 = 2.