I was messing around recently with a unit cube.
If you draw vertices on the midpoint of each edge of the cube, then connect those points by new edges, you will form the wireframe of what I figured out is a cuboctohedron. If you repeat this process on the cuboctohedron (drawing vertices on the midpoints of the edges of the cuboctohedron and connecting them with new edges, you form the wireframe of what I discovered was the rhombicuboctohedron. Repeating this, you end up on the 3rd iteration with another polytope which eludes my identification.
I started drawing net maps of these things and I still don't know what the 3rd+ iteration polytopes actually are. The vertices, edges, and faces are as follows for each iteration (if this helps).
iteration: 1 , 2 , 3 , 4 , 5 ...
vertices: 8, 12, 24, 48, 96, 192 ...
faces: 6, 14, 26, 50, 98, 196 ...
edges: 12, 24, 48, 96, 192, 386 ...
I can't wrap my mind around why this creates these polytopes. Any ideas? Also, what are the polytopes in the 3+ iterations?
Thanks, in advance.