The Rhombitruncated Icosidodecahedron

Icosidodecahedron 20 triangular faces 12 pentagonal faces 30 vertices

Rhombitruncated Icosidodecahedron 20 hexagonal faces 12 10-sided faces 30 square faces

What's happening?

To cut off the corners of the icosidodecahedron, 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 rhombicosidodecahedron)

This process shortens the side lengths of the edges in the original icosidodecahedron by the same amount. Notice that it also doubles the number of edges -- changing the purple pentagonal faces of the icosidodecahedron (left) into purple 10-sided faces in the truncated icosidodecahedron; changing the green triangular faces of the icosidodecahedron into green hexagonal faces in the truncated icosidodecahedron. (right).

In the icosidodecahedron, four faces meet at every vertex. Cutting off the original vertices, then, leaves a square shape behind, which is filled in by the tan squares. The prefix "rhombi" indicates that the truncation results in four-sided faces.