The Rhombicuboctahedron

Rhombitruncated Cuboctahedron 8 hexagonal faces 6 octagonal faces 12 square faces

Rhombicuboctahedron 8 triangular faces 18 square faces

What's happening?

If, when truncating the vertices of the cuboctahedron, we move in exactly half way, the result is the rhombicuboctahedron.

In the figure above, we can see that the 8 triangular faces and the 6 purple square faces are what is left of the cubocahedron, after truncating the corners of that solid. The 12 tan square faces arise from the truncation procedure.