Two Polyhedral Compounds Derived From Catalan Solids

The second stellation of the dysdyakis triacontahedron, seen above, is an interesting two-part polyhedral compound. The dysdyakis triacontahedron is one of the Catalan solids, and is the dual of the great rhombicosidodecahedron.

There’s also a “little brother” to this first compound — it’s the second stellation of the dysdyakis dodecahedron, which is the dual of the great rhombicuboctahedron. Like its “big brother,” it’s a two-part compound, and is shown below.

Interestingly, the components of these two compounds are “stretched” versions of two other Catalan solids: the pentagonal hexecontahedron (dual of the snub dodecahedron), and the pentagonal icositetrahedron (dual of the snub cube).

I made these virtual models using Stella 4d, which you can download and try for free at

Two Half-Visible Catalan Solids

The Catalan Solids shown here are the dysdyakis dodecahedron (dual of the great rhombicuboctahedron) and the dysdyakis triacontahedron (dual of the great rhombicosidodecahedron). In each one, all the faces are scalene triangles, and half of them have been rendered invisible, so that you can see the inside view of faces on the far side of each polyhedron. The remaining faces are shown in “rainbow color mode.”

I made these polyhedron models using Stella 4d, which you can try for free right here.

Two Different Double Cuboctahedra, and Their Duals

There are at least two ways to make a double cuboctahedron. One way is to join two cuboctahedra at a square face.

The dual of a single cuboctahedron is a rhombic dodecahedron. The dual of this first double cuboctahedron, however, doesn’t look like a rhombic dodecahedron at all.

Another way to make a double cuboctahedron is to join two cuboctahedra at a triangular face.

Here’s the dual of the second type of double cuboctahedron.

I created these four polyhedra using Stella 4d, a program you can download and try for free, as a trial version, at this website.

Three Different Compounds of the Octahedron and the First Stellation of the Rhombic Dodecahedron

These compounds differ in the relative sizes of their components. I made all three using Stella 4d, which you can try for free right here.

Two Six-Part Polyhedral Compounds

I stumbled across this compound the other day, while playing around with Stella 4d: Polyhedron Navigator (available here).

At first, I thought this was a compound of six tetrahedra, but careful examination reveals that the tetrahedra are missing parts along the middle of some of their edges. I looked up the canonical compound of six tetrahedra in Stella‘s library, and here it is. As you can see, it’s quite similar — but it does have those “missing” pieces added.

The Tetrahemihexahedron, and Its Two Stellations

The tetrahemihexahedron is one of the uniform polyhedra. Its faces are four equilateral triangles, as well as three squares. The squares are shown in yellow in the model above, and pass through the center of the polyhedron. It has tetrahedral symmetry, and seven faces in total. I know of no other polyhedra which have seven faces and any form of polyhedral symmetry.

If you stellate this polyhedron once, you get a tetrahedron.

The tetrahemihexahedron only has two stellations, and I really like the second one, shown below.

I made these models using Stella 4d: Polyhedron Navigator, which you can try for free at this website: