The Octahedron and the Truncated Octahedron

Octahedron 8 triangular faces 6 vertices

Truncated Octahdedron 8 hexagonal faces, 6 square faces

What's happening?

To cut off the corners of the octahedron, 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 cuboctahedron)

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

In the octahedron, four faces meet at every vertex. Cutting off the original vertices, then, leaves a square shape behind, which is filled in by the purple squares.