The Icosahedron and the Truncated Icosahedron

Icosahedron 20 triangular faces 12 vertices

Truncated Icosahedron 20 hexagonal faces 12 pentagonal faces

What's happening?

To cut off the corners of the icosahedron, we move in the same distance from each corner along the edges. The distance we move is less than half of the side length, because it is impossible to move in more. (If we move in exactly half way, the result is the icosidodecahedron)

This process shortens the side lengths of the edges in the original icosahedron by the same amount. Notice that it also doubles the number of edges -- changing the green triangular faces of the icosahedron (left) into green hexagonal faces in the truncated icosahedron (right).

In the icosahedron, five faces meet at every vertex. Cutting off the original vertices, then, leaves a pentagonal shape behind, which is filled in by the purple pentagons.