In the compound above, the yellow hexagons are not quite regular, which is why I’m calling the yellow-and-orange polyhedron a truncation of the icosahedron, rather than simply the truncated icosahedron. I stumbled upon it while playing with Stella 4d, which you may try for free at http://www.software3d.com/Stella.php.
This is the compound of five rhombic dodecahedra, with each component shown in a different color. This is one of the few well-known polyhedral compounds which is actually more attractive with the faces hidden, and that’s what’s shown in the next image.
The first image shows a central yellow rhombic triacontahedron, with smaller, blue rhombic triacontahedra attached to each of its thirty-two vertices. The second polyhedron shown is the dual of the first one, with colors chosen by the number of sides per face in the second image — pentagons red, and triangles yellow. The convex hull of this second polyhedral complex shown would be an icosidodecahedron, itself the dual of the rhombic triacontahedron.
I use software called Stella 4d: Polyhedron Navigator to make the rotating polyhedral images on this blog. You can try Stella for yourself, for free, at http://www.software3d.com/Stella.php.
This chiral polyhedral compound was generated from a partial faceting of the polyhedron shown in the last post here, using Stella 4d‘s faceting function, plus its “try to make faces regular” operation afterwards. Making the six-prism compound in the first place was suggested by Tony Hartley, on Facebook, where I posted a link to that last post in a mathematical group for discussion. Thanks, Tony!
I found this by further stellation of the polyhedra shown in the two posts right before this one, using a program called Stella 4d: Polyhedron Navigator. You can try Stella for free at http://www.software3d.com/Stella.php.