This is the icosahedron, one of the Platonic solids. It has twenty faces.
The polyhedron below is the rhombic enneacontahedron, a well-known zonohedron with ninety faces.
Finally, here is a polyhedron which blends these two. It has 20 + 90 = 110 faces.
I used Stella 4d: Polyhedron Navigator to make these images. You can try this program for free at http://www.software3d.com/Stella.php.
If one starts with a dodecahedron, and then creates a zonohedron based on that solid’s vertices, the result is a rhombic enneacontahedron.
If, in turn, one then creates a new zonohedron based on the vertices of this rhombic enneacontahedron, the result is this 1230-faced polyhedron — a twice-zonohedrified dodecahedron. Included in its faces are thirty dodecagons, sixty hexagons, and sixty octagons, all of them equilateral.
Stella 4d: Polyhedron Navigator was used to perform these transformations, and to create the rotating images above. You can try this program for yourself, free, at http://www.software3d.com/Stella.php.
Zome is a ball-and-stick modeling system which can be used to make millions of different polyhedra. If you’d like to get some Zome for yourself, just visit http://www.zometool.com.
The polyhedron above is the rhombic enneacontahedron. Sixty of its faces are wide (yellow) rhombi, while the other thirty are narrow (red) rhombi. The wider rhombi are arranged in twelve panels of five rhombi each. If those panels are moved outward from the center by just the right amount, the narrower rhombi have room to expand, becoming equilateral octagons:
Both of these rotating images were created using Stella 4d, a program you can try, for free, at http://www.software3d.com/Stella.php.
Software credit: I made this using Stella 4d, available 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.
Lastly, I made one more image — the same completed shape, but in “rainbow color mode.”
The rhombic enneacontahedron has thirty faces which are narrow rhombi, and sixty faces which are wider rhombi. It is also known as a vertex-based zonohedrified dodecahedron. To create this cluster-polyhedron, I started with one rhombic enneacontahedron in the center, and then augmented its thirty red faces (the narrow rhombi) with additional rhombic enneacontahedra. In the image above, I kept the yellow color for all the wide rhombi, and red for all the narrow ones. In the next image, however, the rhombi are colored by face type, referring to their position in the entire cluster-polyhedron.
Software credit: I created this using Stella 4d, software you can buy, or try for free, at this website.
