The Third Stellation of the Pentagonal Icositetrahedron Is a Compound of Two Irregular Dodecahedra

Here’s the pentagonal icositetrahedron. It is the dual of the snub cube.

And here is its third stellation. As you can see, it’s a compound of two irregular dodecahedra.

I made these images using Stella 4d: Polyhedron Navigator. You can try this program for free at

Compounds of Enantiamorphic Archimedean Solid Duals

An enantiomorphic-pair compound requires a chiral polyhedron, for it is a compound of a polyhedron and its mirror image. Among the Archimedeans, only the snub cube and snub dodecahedron are chiral. For this reason, only threir duals are chiral, among the Archimedean duals, also known as the Catalan solids.

Compound of enantiomorphic pair snub cube duals

That’s a compound of two mirror-image snub cube duals (pentagonal icositetrahedra) above; the similar compound for the snub dodecahedron duals (pentagonal hexacontahedra) is below.

Compound of enantiomorphic pair

Both these compounds were made with Stella 4d, which is available at

A Polyhedral Journey, Beginning with the Snub Cube / Pentagonal Isositetrahedron Base/Dual Compound

The snub cube and its dual make an attractive compound. Since the snub cube is chiral, its chirality is preserved in this compound.

Penta Icositetra & snub cube compound

If you examine the convex hull of this compound, you will find it to be chiral as well.

Convex hull of snub cube& dual compound

Here is the mirror image of that convex hull:

Convex hull mirror image

These two convex hulls, of course, have twin, chiral, duals:

dual of Convex hull of snub cube& dual compound

Dual of Convex hull mirror image

The two chiral convex hulls above (the red, blue, and yellow ones), made an interesting compound, as well.

Compound of enantiomorphic pair not dual

This is also true of their chiral duals:

Compound of enantiomorphic pair

I next stellated this last figure numerous times (I stopped counting at ~200), to obtain this polyhedron:

Stellated Compound of enantiomorphic pair dual

After seeing this, I wanted to know what its dual would look like — and it was a nice polyhedron on which to end this particular polyhedral journey.

dual of Stellated Compound of enantiomorphic pair dual

I  make these transformations of polyhedra, and create these virtual models, using a program called Stella 4d. It may be purchased, or tried for free, at

A Pentagonal Icositetrahedron, Decorated with Rippled Tessellations, Along with Its Compound with Its Own Mirror-Image


A Pentagonal Icositetrahedron, Decorated with Rippled Tessellations

The decorations on each face were created using the design, made using Geometer’s Sketchpad and MS_Paint, from this post: I then used Stella 4d, available at, to project this flat image onto each face of this chiral polyhedron, the dual of the snub cube, and make this rotating image.

Next, I used Stella to add this figure to its own mirror-image, to make a compound — something that is always possible with chiral polyhedra. Here is the result.

Compound of enantiomorphic pair