Re: [math-fun] Pentagram snowflakes? In a square grid?

Quoting Dan Asimov <dasimov@earthlink.net>: <snip>

But (as is well-known) a perfectly regular dodecahedron can be inscribed in a perfect cube with 3 pairs of opposite edge passing through the face centers of the 3 pairs of opposite faces, resp., of the cube.

Is there something similar connecting the regular icosahedron and the regular octahedron? It will have to be a little different, since 6|12 but 8~|20. Rich