If we truncate either the cube or the octahedron in this way, we get exactly the same thing! The result is called a cuboctahedron. Similarly, we get the same thing if we perform this procedure to the icosahedron and the dodecahedron. In this case, the resulting polyhedron is the icosidodecahedron. The fact that truncating both of the polyhedra in these pairs gives us the same semi-regular solid is related to the fact that they are dual to each other, a concept that we may get a chance to cover later. Finally, if we slice the corners off of the tetrahedron in this way, we get another tetrahedron!

Pictures of the cuboctahedron and icosidodecahedron can be found at http://home.teleport.com/~tpgettys/dualpair.shtml