Gallery - icosidodecahedron
The larger the polyhedra get, the more equilateral embeddings. The examples below all correspond to the icosidodecahedron. Note again that embeddings exist with a symmetry that is not present in the classical equihedron.
It can be proved analytically that there are only four examples with (at least) Th symmetry. The classical embedding and its conjugate have symmetry Ih. The other 2 belong to a set of six algebraic conjugates - the remaining 4 have coordinates in complex numbers. If we only assume T symmetry, there are even more solutions.