The stellation-series of the truncated dodecahedron contains some interesting polyhedra. Selections from this series appear below.
The polyhedron above is the 24th stellation of the truncated dodecahedron, while the one below is the 25th stellation.
The polyhedron immediately above is the truncated dodecahedron’s 27th stellation. The one shown below is the 29th stellation.
The last two polyhedra in this post are the truncated dodecahedron’s 36th stellation (above), and its 70th stellation (below).
These images were created using Stella 4d, software available here.
First, this is the 11th stellation.
Next, the 13th:
And, finally, the 15th stellation of the icosahedron:
I used Stella 4d, which you can find here, to make these.
An icosahedron is hidden from view in the center of this cluster-polyhedron. To create the cluster, each of the icosahedron’s triangles was augmented with a rhombicosidodecahedron. The resulting cluster has the overall shape of a dodecahedron.
To create the next cluster-polyhedron, I started with the one above, and then augmented each of its triangular faces with icosidodecahedra.
I used a program named Stella 4d: Polyhedron Navigator to create these cluster-polyhedra. This software may be bought (or tried for free) at this website.
The polyhedron above had 422 faces and 360 vertices. In dual polyhedra, these numbers are reversed, so the next polyhedra (the dual of the first one) has 360 faces and 422 vertices. Both were created using Stella 4d, available here.