While examining different facetings of the dodecahedron, I stumbled across one which is also a compound of ten elongated octahedra.

Here’s what this compound looks like with the edges and vertices hidden:

Next, I’ll put the edges and vertices back, but hide nine of the ten components of the compound. This makes it easier to see the single elongated octahedron which is still shown.

Here’s what this elongated octahedron looks like with all those vertices and edges hidden from view.

I made all these polyhedral transformations using Stella 4d, a program you can try for yourself at this website. Stella includes a “measurement mode,” and, using that, I was able to determine that the short edge to long edge ratio in these elongated octahedra is 1:sqrt(2).

The next thing I wanted to try was to make the octahedra regular. Stella has a function for that, too, and here’s the result: a compound of ten regular octahedra.

My last step in this polyhedral exploration was to form the dual of this solid. Since the octahedron’s dual is the cube, this dual is a compound of ten cubes.