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 http://www.software3d.com/Stella.php.
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.
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.
Both these compounds were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.
The snub cube and its dual make an attractive compound. Since the snub cube is chiral, its chirality is preserved in this compound.
If you examine the convex hull of this compound, you will find it to be chiral as well.
Here is the mirror image of that convex hull:
These two convex hulls, of course, have twin, chiral, duals:
The two chiral convex hulls above (the red, blue, and yellow ones), made an interesting compound, as well.
This is also true of their chiral duals:
I next stellated this last figure numerous times (I stopped counting at ~200), to obtain this polyhedron:
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.
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 http://www.software3d.com/Stella.php.
The decorations on each face were created using the design, made using Geometer’s Sketchpad and MS_Paint, from this post: https://robertlovespi.wordpress.com/2014/05/28/rippling-tessellation-using-squares-regular-octagons-and-octaconcave-equilateral-hexadecagons/. I then used Stella 4d, available at http://www.software3d.com/Stella.php, 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.
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