A Symmetrohedron Which Is Also a Zonohedron

I made this by zonohedrificaton of an octahedron, using its faces, edges, and vertices, and software called Stella 4d, which you can try for free right here.

This polyhedron has six regular octagons (red), a dozen octagons which are merely equilateral (yellow), eight regular hexagons, and 24 squares.

The Rhombic Octagonoid, a Zonohedron With Ninety Faces

To make this zonohedron with Stella 4d (available as a free trial download here), start with a dodecahedron, and then perform a zonohedrification based on both faces and vertices. It is similar to the rhombic enneacontahedron, with thirty equilateral octagons replacing the thirty narrow rhombic faces of that polyhedron.

I’ve run into this polyhedron from time to time, and I’ve also had students make it. It is the largest zonohedron which can be built using only red and yellow Zome (available here). I thought it needed a name, so I made one up.

A Twice-Zonohedrified Dodecahedron

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.

Two Views of a Zonohedron with 3690 Faces

I used Stella 4d to make these images. You can try this program as a free trial download at http://www.software3d.com/Stella.php.

A Zonohedron Which Is Also a Symmetrohedron

This zonohedron is based on the icosidodecahedron / rhombic triacontahedron compound — more specifically, on its edges. Twelve faces are regular decagons, twenty are regular hexagons, sixty are squares, and the only irregular faces are the thirty equilateral octagons. That’s 122 faces in all.

Four Zonohedra

To build these zonohedra, I used Stella4d, a program you may try for free at http://www.software3d.com/Stella.php.

A Zonohedron with 1382 Faces, Based on the Rhombicosidodecahedron

This zonohedron was formed from zones based on the faces, edges, and vertices of a rhombicosidodecahedron. The first image shows it colored by face type. The second image has the faces colored by number of sides. Finally, here’s one in “rainbow color mode.” These images were all formed using Stella 4d: Polyhedron Navigator, which you can try for free right here. I stumbled upon this zonohedron by adding zones to a truncated octahedron, based on its faces, edges, and vertices. It was created using Stella 4d, which you may try for free at http://www.software3d.com/Stella.php. To the best of my recollection, this is the only zonohedron I have seen which includes rhombi, hexagons, octagons, and, of course, the red hexadecagons.

A Second Type of Double Icosahedron, and Related Polyhedra

After seeing my post about what I called the “double icosahedron,” which is two complete icosahedra joined at one common triangular face, my friend Tom Ruen brought my attention to a similar figure he likes. This second type of double icosahedron is made of two icosahedra which meet at an internal pentagon, rather than a triangular face. Tom jokingly referred to this figure as “a double patty pentagonal antiprism in a pentagonal pyramid bun.” It wasn’t hard to make this figure using Stella 4d, the program I use for polyhedral manipulation and image-creation (you can try it for free here), but I didn’t make it out of icosahedra. It was easier to make this figure from gyroelongated pentagonal pyramids, or “J11s” for short. This polyhedron is one of the 92 Johnson solids. To make the polyhedron Tom had brought to my attention, I simply augmented one J11 with another J11, joining them at their pentagonal faces. Curious about what the dual of this solid would look like, I generated it with Stella. The dual of the double J11 appears to be a modification of a dodecahedron, which is no surprise, for the dodecahedron is the dual of the icosahedron.

I next explored the stellation-series of the double J11, and found several attractive polyhedra there. This one is the double J11’s 4th stellation. The next polyhedron shown is the double J11’s 16th stellation. Here is the 30th stellation: I also liked the 43rd: The next one shown is the double J11’s 55th stellation. Finally, the 56th stellation is shown below. These stellations, as well as the double J11 itself, and its dual, all have five-fold dihedral symmetry. Having “mined” the double J11’s stellation-series for interesting polyhedra, I next turned to zonohedrification of this solid. The next image shows the zonohedron based on the double J11’s faces. It has many rhombic faces in two “hemispheres,” separated by a belt of octagonal zonogons. This zonohedron, as well as the others which follow, all have ten-fold dihedral symmetry. Zonohedrification based on vertices produced this result: The next zonohedron shown was formed based on the edges of the double J11. Next, I tried zonohedrification based on vertices and edges, both. Next, vertices and faces: The next zonohedrification-combination I tried was to add zones based on the double J11’s edges and faces. Finally, I ended this exploration of the double J11’s “family” by adding zones to build a zonohedron based on all three of these polyhedron characteristics: vertices, edges, and faces. 