muzik wrote:I'd like to see a compound fo two excavated dodecahedra as the duals of each other, so I can actually visualise it.

It wouldn't be very impressive, as they would either overlap entirely or one would be hidden inside the other as a smaller version of it. This is because each vertex is along the same ray, starting from the center, as the middle of one of the faces. Since taking the dual swaps faces and vertices, it will replace each vertex with a face whose center is directly below or above that vertex. Therefore, the original faces and the dual faces must be along the same rays from the center of the figure. The same argument can be made about the vertices.

I think there is a more interesting way to prove the the dual is a non-rotated version of the original. The excavated dodecahedron has reflexive dodecahedral symmetry. If we look at

a spherical tiling where each triangle corresponds to the smallest asymmetric building block of an object with dodecahedral symmetry, we see that each triangle still looks asymmetric. This means that if you were to superimpose two copies of the same object with dodecahedral symmetry, in exactly the same position, there is no way to rotate/reflect one of the two shapes to form a compound, where they are in different positions, that retains the dodecahedral symmetry.

If such a task were possible, it would require the symmetry modules -- but not the asymmetric "building blocks" of each shape -- to overlap. This means that the symmetry modules (the triangles in the spherical tiling) must be symmetric; they would need to carry the same asymmetric block in multiple orientations. But as those triangles are asymmetric, these is no way to preserve the dodecahedral symmetry when taking a compound of two copies of the same shape, except to have them in exactly the same orientation.

All self-dual shapes with non-doublable symmetry (octahedral and dodecahedral symmetry with reflection) have this property. Interestingly,

the modules of tetrahedral symmetry are symmetric triangles, and it is indeed possible to have multiple tetrahedra in different orientations share the same symmetry axes, as in the stella octangula.