Decagonal Ring of Rhombic Triacontahedra

ring of ten Rhombic Triaconta

Ten rhombic triacontahedra fit perfectly into a decagonal ring. It’s not a “near-miss” — the fit is exact.

I made this with Stella 4d, software you can try for free, or purchase, at

The Compound of the Truncated Icosahedron and the Rhombic Triacontahedron

Compound of Rhombic Triaconta and Trunc Icosa

I put these two polyhedra together using Stella 4d: Polyhedron Navigator. If you’d like to try this program yourself, for free, this website is the one to visit:

Two Polyhedral Compounds: the Icosidodecahedron with the Truncated Cube, and the Rhombic Triacontahedron with the Triakis Octahedron

Compound of Icosidodeca and Trunc Cube

These two compounds, above and below, are duals. Also, in each of them, one polyhedron with icosidodecahedral symmetry is combined with a second polyhedron with cuboctahedral symmetry to form a compound with pyritohedral symmetry: the symmetry of a standard volleyball.

Compound of RTC and Triakis octahedron also pyritohedral

A program called Stella 4d was used to make these compounds, and create these images. It may be purchased, or tried for free, at this website.

Two Compounds with Pyritohedral Symmetry: the Icosidodecahedron / Truncated Octahedron Compound, and the Rhombic Triacontahedron / Tetrakis Cube Compound

Compound of Icosidodeca and Trunc Octa its pyritohedralCompound of RTC and tetrakis cube its pyritohedral

Stella 4d, a program you can try here, was used to create these two compounds. Both have pyritohedral symmetry: the symmetry of a standard volleyball. The two compounds are also duals.

At 47, My Age Is a Prime Number Again =D

For some reason, I like having my age be a prime number of years. Today, I turn 47, so I get to have a prime-number-age for a whole year now. This hasn’t happened since I was 43, so I made this 47-pointed star to celebrate:


I also make birthday-stars for composite-number ages as well, just because it’s fun, and you can find at least two others on this blog, on January 12, in past years. Also, I wouldn’t want to have to wait until I’m 53 (my next prime age) to make another one of these.

At the moment, I certainly don’t feel 47. There are times when I feel twenty-two . . .

There are also times when I feel six.


At the moment, however, I feel about thirty. For that reason, I put the 47-pointed stars on the thirty faces of a rotating rhombic triacontahedron, because (a) it’s my birthday, (b) I want to, and (c) I can.

Rhombic Triaconta

Image/music credits:

  1. I created this using Geometer’s Sketchpad and MS-Paint.
  2. “When Yer Twenty-Two,” by The Flaming Lips, via a YouTube posting.
  3. Two panels from a Calvin and Hobbes cartoon, by Bill Watterson. (Calvin is perpetually six years old.)
  4. Created using the image at the top of this post, and the program Stella 4d: Polyhedron Navigator, which is available here.

My First Solution to the Zome Cryptocube Puzzle, with Special Guest Appearances by Jynx the Kitten

Last month, in a special Christmas promotion, the Zometool company ( briefly sold a new kit (which will return later) — a fascinating game, or puzzle, called the “Cryptocube.” Zome usually comes in a variety of colors, with each color having mathematical significance, but the Cryptocube is produced in black and white, which actually (in my opinion) makes it a better puzzle. Here’s how the Crypocube challenge works:  you use the black parts to make a simple cube, and then use the smaller white parts to invent a structure which incorporates the cube, is symmetrical, is attractive, and can survive having the twelve black cube-edges removed, leaving only the cube’s eight black vertices in place. I had a lot of fun making my first Cryptocube, and photographed it from several angles.

imageIf this was built using standard Zome colors, the round white figure inside the cube, a rhombic triacontahedron, would be red, and the pieces outside the cube, as well as those joining the rhombic triacontahedron to the cube (from inside the cube), would be yellow.

It isn’t only humans who like Zome, by the way. Jynx the Kitten had to get in on this!

image (1)

Jynx quickly became distracted from the Cryptocube by another puzzle, though: he wanted to figure out how to pull down the red sheet I had attached to the wall, as a photographic backdrop for the Cryptocube. Jynx takes his feline duties as an agent of entropy quite seriously.

image (2)As usually happens, Jynx won (in his never-ending struggle to interfere with whatever I’m doing, in this case by pulling the sheet down) and it took me quite a while to get the red sheet back up, in order to take kitten-free pictures of my Cryptocube solution, after removal of the black cube’s edges.

image (3)

Here’s the view from another angle.

image (4)

The Cryptocube will be back, available on the Zometool website, later in 2015. In the meantime, I have advice for anyone not yet familiar with Zome, but who wants to try the Cryptocube when it returns: go ahead and get some Zome now, at the link above, in the standard colors (red, blue, and yellow, plus green in advanced kits), and have fun building things with it over the next few months. The reason to do this, before attempting to solve the Crypocube, is simple: the colors help you learn how the Zome system works, which is important before trying to solve a Zome puzzle without these colors visible. After gaining some familiarity with the differing shapes of the red, blue, yellow, and green pieces, working with them in white becomes much easier.

On a related note, Zome was recommended by Time magazine, using the words “Zometool will make your kids smarter,” as one of the 14 best toys of 2013. I give Zome my own strong, personal recommendation as well, and, as a teacher who uses my own Zome collection in class, for instructional purposes, I can attest that Time‘s 2013 statement about Zome is absolutely correct. Zome is definitely a winner!

If You Have Enough Platonic Dodecahedra Around, and Glue Them Together Just Right, You Can Make a Rhombic Triacontahedron.

Aren’t you glad to know that? As soon as I found out isocahedra can form a rhombic dodecahedron (see last post), I knew this would be true as well. Why? Zome explains why, actually. It’s at Anything buildable with yellow Zome can be built out of icosahedra. Dodecahedra con form anything buildable with red Zome. Finally, if you can make it with blue Zome, it can be built out of rhombic triacontahedra. It follows that rhombicosidoecahedra can build anything Zome-constructable — but one look at a Zomeball makes that easy to believe, since Zomeballs are modified rhombicosidodecahedra.

Anyway, here’s the rhombic triacontahedron, made of dodecahedra:

Augmented Dodeca

[Image created with Stella 4d; see for more info re: this program.]