05 | November | 2015 | RobertLovesPi.net

With 92 dodecahedra, if you arrange them just right, you can make a model of a rhombic triacontahedron:

For purposes of comparison, here is what the rhombic triacontahedron normally looks like:

Also, referring back to the first model shown, here is a picture of just one of the red rhombi-made-of-dodecahedra.

The first polyhedron shown in this post has an interesting dual, as well. Here it is, colored by face-type (position within the overall shape):

Here is another view of the dual, colored by number of edges per face.

Here’s one more view of the dual, in “rainbow color mode.”

Returning to the original model, at the top of this post, here’s what it looks like, if colored by face type:

Here’s one more view, in “rainbow color mode.”

All of these images were created using Stella 4d, a program you can buy, or try for free, right here.

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.