elongated | RobertLovesPi.net

Thinking about the post immediately before this one led me to see if I could connect opposite triangular faces of a rhombicosidodecahedron to form a ten-part compound — and it worked with Stella 4d just as it had when I “previewed” it in my head.

The interesting dual of the above polyhedral compound, also a ten-part compound, I was not able to preview in my head (although that would be a nice ability to have), but creating it was easy with Stella:

It is difficult, in the dual, to tell what the ten components are. To help with this, in the next image, all but one component has been removed. This reveals the components of the dual to be rhombus-faced parallelopipeds which are quite flattened, compared to most parallelopipeds I have seen before. This polyhedron is isomorphic to the cube, just as the elongated octahedra in the first compound were each isomorphic to the Platonic octahedron. Given that the cube and octahedron are duals, this is no surprise.

Stella 4d may be tried for free, as a trial download, at this website: http://www.software3d.com/Stella.php.

With four icosahedra, and four octahedra, it is possible to attach them to form this figure:

This figure is actually a rhombus, but the gap between the two central icosahedra is so small that this is hard to see. To remedy this problem, I elongated the octahedra, thereby creating this narrow rhombus:

It is also possible to use the same collection of polyhedra to make a wider rhombus, as seen below.

These aren’t just any rhombi, either, but the exact rhombi found in the polyhedron below, the rhombic enneacontahedron. It has ninety rhombi as faces: sixty wide ones, and thirty narrow ones.

As a result, it is possible to use the icosahedra-and-elongated-octahedra rhombi, above, to construct a rhombic enneacontahedron made of these other two polyhedra. The next several images show it under construction (I “built” it using Stella 4d, available at this website), culminating with the complete figure.