# A Dozen Snub Dodecahedra, Surrounding a Dodecahedral Core Software used — Stella 4d: Polyhedral Navigator, available as a free trial download at http://www.software3d.com/Stella.php.

# The Snub Dodecahedron’s Big Brother The polyhedron above reminded me of the snub dodecahedron, which is shown below. Both rotating images were made using Stella 4d, which you can try for yourself — with a free trial download available — right here. # The Triangles of a Snub Dodecahedron The snub dodecahedron, one of the Archimedean solids, has eighty faces which are triangles, and twelve pentagonal faces as well. In the view above, the pentagons are rendered invisible, allowing the interior to be viewed as the solid rotates.

The eighty triangles are of two types: the sixty yellow ones share an edge with a pentagon, and the twenty blue ones do not. If the blue triangles are also hidden, the “transparency” of this solid becomes even greater, as seen below. Both of these images were created using Stella 4d, software you may try for free at this website.

# Variations of the Snub Dodecahedron To make the first of these variations, above, I augmented each triangular face of a snub dodecahedron with an antiprism 2.618 times as tall as the triangles’ edge length, and then took the convex hull of the result. The other polyhedra shown, below, were obtained by various other manipulations of the snub dodecahedron, all performed using a program called Stella 4d: Polyhedron Navigator, which you can try right here. The variant above looked like it needed a name, so I called it an expanded snub truncated dodecahedron. As for the one below, it is one of many facetings of the snub dodecahedron. Finally, the last figure shown (stumbled upon during a “random walk” with Stella) is one of many possible figures which are non-convex relatives of the snub dodecahedron. # The Snub Dodecahedron and Related Polyhedra, Including Compounds 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. This polyhedral manipulation and .gif-making was performed using Stella 4d, a program you can find here.

# An Icosahedron, Augmented by Snub Dodecahedra, Plus Two Versions of a Related Polyhedral Cluster Because the snub dodecahedron is chiral, the polyhedral cluster, above, is also chiral, as only one enantiomer of the snub dodecahedron was used when augmenting the single icosahedron, which is hidden at the center of the cluster.

As is the case with all chiral polyhedra, this cluster can be used to make a compound of itself, and its own enantiomer (or “mirror-image”): The image above uses the same coloring-scheme as the first image shown in this post. If, however, the two enantiomorphic components are each given a different overall color, this second cluster looks quite different: All three of these virtual models were created using Stella 4d, software available at this website.

# Two Different Versions of an Expanded Snub Dodecahedron, Both of Which Feature Regular Decagons

The snub dodecahedron, one of the Archimedean solids, can be expanded in multiple ways, two of which are shown below. In each of these expanded versions, regular decagons replace each of the twelve regular pentagons of a snub dodecahedron.  Like the snub dodecahedron itself, both of these polyhedra are chiral, and any chiral polyhedron can be used to create a compound of itself and its own mirror-image, Below, you’ll find these enantiomorphic-pair compounds, each made from one of the two polyhedra above, together with its own reflection.  All four of these images were created using Stella 4d: Polyhedron Navigator, software available (including a free trial download) at this website.