The Pentagonal Hexacontahedron, and Related Polyhedra

The Pentagonal Hexacontahedron

As the dual of the snub dodecahedron, which is chiral, this member of the Catalan Solids is also chiral — in other words, it exists in left- and right-handed versions, known an entantiomers. They are mirror-images of each other, like left and right gloves or shoes. Here’s the other one, by comparison:

Penta Hexeconta enantiomer

It is always possible to make a compound, for a chiral polyhedron, from its two enantiomers. Here’s the one made from the two mirror-image pentagonal hexacontahedra shown above:

Compound of enantiomorphic pair

Stellating this enantiomorphic-pair-compound twenty-one times produces this interesting result:

stellating

And, returning to the unstellated enantiamorphic-pair-compound, here is its convex hull:

Convex hull

This convex hull strikes me as an interesting polyhedron in its own right, so I tried stellating it several times, just to see what would happen. Here’s one result, after seventeen stellations:

Stellation17

Software credit:  I made these rotating images using Stella 4d:  Polyhedron Navigator. That program may be bought at http://www.software3d.com/Stella.php, and has a free “try it before you buy it” trial download available at that site, as well. I also used Geometer’s Sketchpad and MS-Paint to produce the flat purple-and-black image found on faces near the top of this post (and, by itself, in the previous post on this blog), but I know of nowhere to get free trial downloads of these latter two programs.

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