Here’s a compound I stumbled across tonight, while playing around with Stella 4d, a program you can try for free at this website. Trapezohedra have kites as faces, and each of the six components of this compound has a different color.
After finding the compound above, I used Stella to create this compound’s dual. Since trapezohedra are the duals of antiprisms, I expected to see a compound of six pentagonal antiprisms — but that’s not what I found. Instead, I saw this:
My initial reaction to this polyhedron was puzzlement. It’s pretty, and it’s interesting, but it’s not a dual of six antiprisms, at least as far as I can tell. I found the first polyhedron by using a lot of stellations, as well as other functions, for a long enough time that I couldn’t even remember what I started with. Faceting is the dual process to stellation, so this second polyhedron should be a faceted polyhedron — which it is.
What about the antiprisms I expected, though? Stella has a large built-in library of polyhedra, including compounds, so I looked up the compound of six regular pentagonal antiprisms, which is the next model shown.
Next, I created the dual of this antiprism-compound, and found myself looking at a compound of six trapezohedra which is quite different from the one at the top of this post.
As the dual of the regular-antiprism compound, this fourth image shows the “canonical” compound of six pentagonal trapezohedra, and it has more elongated kites for faces than the first one has. What I originally found with all of my stellations, etc., shown in the first image above, was a compound of six pentagonal trapezohedra, not the compound of six pentagonal trapezohedra. As for the non-compound dual solid shown in the second image above, it is unusual because it had an unusual origin — my long series of stellations and other transformations of polyhedra. Beyond that, I haven’t yet figured it out.
No matter how much you study geometry, there’s always more to learn.