# Building a Rhombic Enneacontahedron, Using Icosahedra and Elongated Octahedra

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.

Lastly, I made one more image — the same completed shape, but in “rainbow color mode.”

## 6 thoughts on “Building a Rhombic Enneacontahedron, Using Icosahedra and Elongated Octahedra”

1. I am not sure if you are aware of the work of Steve Baer on zonohedra and the 120 cells that compose the enneacontahedron.
There five different cells: 10A, 20B, and 30 each of C,D&E.
I was confounded many years ago by the fact that 4 of the B cells made a regular Rhombic Dodecahedron. What I would associate as a four-fold symmetric form, which could be found within the enneacontahedron.
Many years later I came to understand how those cells were generated and found a myriad of different orientations of those cells can assemble to make the rhombic enneacontahedron.
I did make some videos. I utilize vZome software and credit the late Russel Towle as steering me toward this understanding.

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• Thanks — I’ll have to check this all out!

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