This is the truncated cube, which is one of the Archimedean solids.
To make a faceted version of this solid, one must connect at least some of the vertices in different ways. Doing that creates new faces.
This faceted version of the truncated cube includes eight blue equilateral triangles, eight larger, yellow equilateral triangles, and eight irregular, red hexagons. It’s easy to spot the yellow and blue triangles, but seeing the red hexagons is harder. In the final picture here, I have hidden all faces except for three of the hexagons, so that their positions can be more easily seen.
I made all three of these images using Robert Webb’s program called Stella 4d: Polyhedron Navigator. It is available for purchase, or as a free trial download, at http://www.software3d.com/Stella.php.
This is an expansion of the last post here. It may be possible to continue this tiling outward indefnitely, forming an aperioidic tiling — or it may not. I am simply uncertain about this
The 18th stellation of the icosidodecahedron, shown above, is also an interesting compound. The yellow component of this compound is the rhombic triacontahedron, and the blue-and-red component is a “stretched” form of the truncated icosahedron.
This was made using Stella 4d, which you can try for free right here.
The only difference between these two images is that the lower one is in “rainbow color mode.” Both were created using Stella 4d, which you can try for free at this website.
I once made a physical model of this thing, when I was still new to the study of polyhedra. I wish I still had it, but it was lost many years ago.