The polyhedron above is a zonish icosahedron, with zones added to that Platonic solid based on its faces and vertices. Its faces are twenty equilateral triangles, thirty equilateral decagons, and sixty rhombi. After making it, I used faceting to truncate the vertices where sets of five rhombi met, creating the polyhedron below. It has twelve regular pentagons as faces, with the sixty rhombi of the polyhedron above turned into sixty isosceles triangles, along with the thirty decagons and twenty triangles from the first of these two polyhedra. This second one could be called either a faceted zonish icosahedron, or a truncated zonish icosahedron.
Both of these polyhedra were created using Stella 4d: Polyhedron Navigator, software you can try for free at http://www.software3d.com/Stella.php.
The isosceles triangles in this polyhedron have legs which are each 22.475% longer than their bases. I made this by creating the dual of the convex hull of the base/dual compound of the truncated octahedron, using a program called Stella 4d, which you can try for free right here.
This was created using Stella 4d: Polyhedron Navigator. You may try this program for free at this website.
This is an icosahedron because it has twenty faces, and it is of tetrahedral symmetry. There are four regular hexagons and four equilateral triangles among these faces, along with twelve isosceles triangles with a leg:base length ratio of 1.73205:1. I found it while investigating variations to the base/dual compound of the truncated tetrahedron using Stella 4d, which you can try for free at this website. This polyhedron is the dual of the convex hull of the base/dual compound of the truncated tetrahedron and the triakis tetrahedron.
This is the cuboctahedron, one of the Archimedean solids. Its dual, shown below, is the rhombic dodecahedron.
The rhombic dodecahedron has a property which sets it apart from most other polyhedra: it can fill space with copies of itself, leaving no gaps. The next stage of such growth is shown below.
The next step is to add more rhombic dodecahedra on each face.
One more set added, and the edge-length of the cluster reaches four rhombic dodecahedra.
This could be continued without limit. As is does, the overall shape of the cluster becomes more and more shaped like a cuboctahedron, which is back where we started. You can easily see this in the convex hull of the last cluster.
All of these rotating images were created using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.