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.
I stumbled across this while playing around with Stella 4d, a program you can try for free right here. The red component is the rhombic triacontahedron, while the yellow component is a slightly-stretched version of the strombic hexecontahedron. The dual of this compound is shown below.
The blue figure in the center of this model is the compound of five cubes. If you take a cube, and build pyramids of just the right height on each of that cube’s faces, those pyramids form a rhombic dodecahedron, as seen below.
In the model at the top of this post, yellow rhombic dodecahedra have been built around each cube in the compound of five cubes. The yellow figure in the top is, therefore, the compound of five rhombic dodecahedra.
I made these models out of Zome. If you’d like to try Zome for yourself, the place to go to buy it is http://www.zometool.com.
The polyhedral compound above contains an icosidodecahedron (blue) and a rhombic triacontahedron (red). In this compound, the icosidodecahedron’s edges are bisected, while the rhombic triacontahedron’s edges are split into segments with lengths in the square of the golden ratio (~2.618 to 1).
“Morph Duals By Tilting To Duals” is a Stella 4d feature that I haven’t used much. Here’s what happens if you apply it to an icosahedron, at the 50% morphing level: you get the compound of the icosahedron and its dual, the dodecahedron,
If you apply this same operation a second time, here’s what you get.
This appears to be a three-part compound, with two familiar components: the icosahedron (red) and the dodecahedron (orange). Remove those two components, and you get this:
Since this reminds me of an icosidodecahedron, I colored its faces to better suit that identity.
Little peeks at the edges of the solid above made me suspicious, so I hid these purple and green faces, to see the inner structure. Here’s the result.
RobertLovesPi.net 