A Symmetrohedron Featuring Eight Regular Hexagons, Six Squares, and 24 Isosceles Triangles

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.

The Compound of the Truncated Cube and the Rhombic Dodecahedron

This was created using Stella 4d: Polyhedron Navigator. You may try this program for free at this website.

The Tetrahedral Icosahedron

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.

Filling Space With Rhombic Dodecahedra

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.