Sharp-eyed, regular readers of this blog will notice that this is the same polyhedron shown in the previous post, which was described as the “final stellation of the compound of five cubes,” due to the coloring scheme used in the first image there, which had five colors “inherited” from each of the differently-colored cubes in the five-cube compound. This image, by contrast, is shown in rainbow-color mode.

How can the rhombic triacontahedron and the compound of five cubes have the same final stellation? Simple: the compound of five cubes is, itself, a member of the stellation-series of the rhombic triacontahedron. Because of this, those two solids end up at the same place, after all possible stellations are completed, just as you will reach 1,000, counting by ones, whether you start at one, or start at, say, 170.

I am grateful to Robert Webb for pointing this out to me. He’s the person who wrote Stella 4d, the software I use to make these images of rotating polyhedra. His program may be found at http://www.software3d.com/Stella.php — and there is a free trial version available for download, so you can try Stella before deciding whether or not to purchase the fully-functioning version.

Since faceting is the reciprocal process of stellation, the dual of the polyhedron above is a faceted icosidodecahedron, for the icosidodecahedron is the dual of the rhombic triacontahedron. Here is an image of that particular faceting of the icosidodecahedron, colored, this time, by face-type: