[xbipeice] Concentric Platonic solids

Start with an icosahedron.  Inscribe within it a dodecahedron with a vertex centered on each face (dual polyhedron).

The "compound of five cubes" shows one can inscribe a cube within a dodecahedron by selecting 8 vertices.  The remaining 12 dodecahedron vertices sit 2 per cube face, roughly evenly spaced on an orthogonal line through the center of each square.  Build a d12 cubical die which has two pegs per face instead of pips. Each cube edge corresponds to a dodecahedron face.

A tetrahedron may be inscribed within a cube by selecting face diagonals.  There are two ways: the union is a stella octangula; the intersection, a octahedron.

Inscribe an octahedron inside a tetrahedron by joining midpoints of the edges. 

The octahedron has square cross sections, so this is how the Pyramorphix is equivalent to the Pocket Cube.  Build a twisty puzzle (probably virtual) which twists both like a Pyramorphix and Pyraminx.  Or a both face turning and vertex turning octahedron -- could be done with LEDs.  Text and and static images do a poor job of describing twisty puzzles on Wikipedia; video is great.  Slow motion, fast forward, zoom, brighten, loop segment are needed.

From a cube, one can get to all regular polyhedra.  Isn't this beautiful?