We can do this because the cube and the octahedron have a very special relationship with each other: They are duals.

Cube: 6 faces, 8 vertices, 12 edges
Octahedron: 8 faces, 6 vertices, 12 edges

They share many of the same properties -- They also share the same polyhedra when they are truncated.  To see this, go back and take another look at the Cubocta Family.