Five Variants of the Compound of Two Tetrahedra

This is the compound of two tetrahedra, also known as Johannes Kepler’s Stella Octangula.

I found the five variations of this polyhedral compound shown below, located deep within the stellation-series of the great rhombicuboctahedron.

These .gif images were all made using Stella 4d, a program you can try for free at

Expanding the Icosidodecahedron

This is the icosidodecahedron. It’s one of the thirteen Archimedean solids. To make an expanded version of it, I first augmented each of its faces with a prism.

Next, I formed the augmented icosidodecahedron’s convex hull.

This expanded icosidodecahedron has the twelve pentagonal faces (shown in red) and twenty triangular faces (shown in blue) of the original icosidodechedron. It also has sixty rectangular faces (yellow), and sixty isosceles triangles (shown in green). That’s a total of 152 faces.

To do all of this, I used a program called Stella 4d. If you’d like to try Stella for yourself, for free, just visit this website:

One Faceting, Each, of the Snub Cube and Snub Dodecahedron

Faceted snub cube

These are facetings of the snub cube (above) and snub dodecahedron (below). I made both using Stella 4d, software you can try for yourself right here.

faceted Snub Dodeca

On Consistent and Inconsistent Combining of Chiralities, Using Polyhedral Augmentation

For any given chiral polyhedron, a way already exists to combine it with its own mirror-image — by creating a compound. However, using augmentation, rather than compounding, opens up new possibilities.

The most well-known chiral polyhedron is the snub cube. This reflection of it will be referred to here using the letter “A.”

Snub Cube ATo avoid unnecessary confusion, the same direction of rotation is used throughout this post. Apart from that, though, the image below, “Snub Cube B,” is the reflection of the first snub cube shown.

Snub Cube B

There are many ways to modify polyhedra, and one of them is augmentation. One way to augment a snub cube is to attach additional snub cubes to each square face of a central snub cube, creating a cluster of seven snub cubes. In the next image, all seven are of the “A” variety.

Snub Cube seven of them  AA

If one examines the reflection of this cluster of seven “A” snub cubes, all seven, in the reflection, are of the “B” variety, as shown here:

Snub Cube seven of them BB

Even though one is the reflection of the other, both clusters of seven snub cubes above have something in common: consistent chirality. As the next image shows, inconsistent chirality is also possible.

Snub Cube A augmented with B

The cluster shown immediately above has a central snub cube of the “A” variety, but is augmented with six “B”-variety snub cubes. It therefore exhibits inconsistent chirality, as does its reflection, a “B” snub cube augmented with “A” snub cubes:

Snub Cube B augmented with A

With simple seven-part snub-cube  clusters formed by augmentation of a central snub cube’s square faces by six snub cubes of identical chirality to each other, this exhausts the four possibilities. However, multiplying the possibilities would be easy, by adding more components, using other polyhedra, mixing chiralities within the set of polyhedra added during an augmentation, and/or mixing consistent and inconsistent chirality, at different stages of the growth of a polyhedral cluster formed via repeated augmentation.

All the images in this post were created using Stella 4d, which you can try for yourself at this website.

Three Polyhedra, Each Featuring One Dozen Regular Icosagons

Icosagons are polygons with twenty sides, and do not appear in any well-known polyhedra. The first of these three regular-icosagon-based polyhedra has 122 faces.

122 faces including 12 regular icosagons

The second of these polyhedra, each of which bears an overall resemblance to a dodecahedron, has 132 faces.

132 faces including 12 regular icosagons

Finally, the third of these polyhedra has a total of 152 faces.

152 faces ncluding 12 regular icosagonsI used Stella 4d to make each of these virtual polyhedron models — and you may try this program for free at

Some Recommeded Websites for Polyhedral Enthusiasts

There are a lot of websites devoted to polyhedra. Here are some of the best.

This one is run by Jonathan Bowers:

Here’s the portal-page to George Hart’s pages, with links to a LOT of cool stuff he’s made:

I don’t know who runs this one:

This one is Robert Webb’s. He’s the person who wrote Stella 4d, the program I use most often on my own blog to create polyhedral images: (Also, while we do share a first name, he is not the same person as me, as a few readers of my blog have thought in the past.)

Craig Kaplan has a page of links to other pages of his, right here: Of those, my favorites are the sections on John solid near-misses (, as well as on symmetrohedra:

Jim McNeill’s polyhedra site is here:

Here’s a good one, but I don’t know its creator’s full name:

Finally, one by Vladimir Bulatov:

This is definitely not a complete list. If you know of other good polyhedron-oriented websites, please leave links to them in a comment on this post.