This MathStream post about why an icosahedron inscribes in a cube also shows that a dodecahedron fits into a cube in an analogous way. That raised the prospect that it might also be worth building an “Antidodec” analogous to the Anticos. So I quickly mocked one up in OpenSCAD (here’s the two files you need), producing the following image of what it would look like:
Surprise! it’s just a rectilinear network of (infinite golden) rhombic prisms. While it’s striking and lovely in its own way that this structure turns out to be so simple, it didn’t seem that the result would be visually interesting enough to be worth producing physically. So in the end, this post is really just a footnote to the prior project.