A Decorated Pentagonal Hexacontahedron, with Three of Its Stellations


This is a pentagonal hexacontahedron, the dual of the snub dodecahedron. It’s decorated with mandalas of the the type I blogged here, two posts ago. The mandalas do interesting things when this polyhedron is stellated, as you can see below.


That was the first stellation, and here is the second:


The sixth stellation was the last one I found interesting enough to post.


All four polyhedral images above were created using Stella 4d: Polyhedron Navigator, software you may buy, or try for free, at this website.

The Snub Dodecahedron and Related Polyhedra, Including Compounds

Snub Dodeca

The dual of the snub dodecahedron (above) is called the pentagonal hexacontahedron (below, left). The compound of the two is shown below, at right. (Any of the smaller images here may be enlarged with a click.)

Like all chiral polyhedra, both these polyhedra can form compounds with their own mirror-images, as seen below.

Finally, all four polyhedra — two snub dodecahedra, and two pentagonal hexacontahedra — can be combined into a single compound.

Compound of enantiomorphic pair and base-dual compound snub dodeca

This polyhedral manipulation and .gif-making was performed using Stella 4d, a program you can find here.

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 http://www.software3d.com/Stella.php.

A Polyhedron Featuring Sixty Octagons and Sixty Triangles


A Polyhedron Featuring Sixty Octagons and Sixty Triangles

If someone had asked me if it were possible to form a symmetric polyhedra out of irregular triangles and octagons, using exactly sixty of one type each, I would have guessed that it were not possible. Why does it work here? Part of the reason is that each triangle borders three octagons, and each octagon borders three triangles — a necessary, but not sufficient, condition. This is a partial truncation of an isomorph of the pentagonal hexacontahedron, the dual of the snub dodecahedron. As such, no surprise — it’s chiral.

This was made while stumbling about in the wilderness of the infinite number of possible polyhedra using Stella 4d: Polyhedron Navigator. You can get it here: http://www.software3d.com/Stella.php.

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:


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:


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.