Shown above: the compound of the icosahedron and the rhombic dodecahedron. Below is its dual, the compound of the dodecahedron and the cuboctahedron.
Both these compounds were created using the “add/blend polyhedron from memory” function in Stella 4d: Polyhedron Navigator. To check out this program for yourself, just follow this link.
If a zonish icosahedron is created with zones based on the icosahedron’s vertices, here is the result.
If the same thing is done with edges, this is the result.
Another option is faces-only.
The next zonish icosahedron has had zones added based on the icosahedron’s faces and edges, both.
Here’s the one for vertices and edges.
Here’s the one for faces and vertices.
Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges.
All seven of these were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.