The Snub Dodecahedron’s Big Brother

triangles and decagons

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.

Snub Dodeca

The Triangles of a Snub Dodecahedron

snub-dodeca

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.

snub-dodeca

Both of these images were created using Stella 4d, software you may try for free at this website.

Variations of the Snub Dodecahedron

Convex hull of a triangle-expansion 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.

expanded snub truncated dodecahedron

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.

Faceted 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.

nco thing

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.

An Icosahedron, Augmented by Snub Dodecahedra, Plus Two Versions of a Related Polyhedral Cluster

Icosa augmented by snub dodecahedra

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”):

Compound of enantiomorphic pair of snub-dodeca-augemented icosahedra

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:

Compound of enantiomorphic pair of snub-dodeca-augemented icosahedra colored by chirality

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.

exp sn dodeca 2

Exp Sn Dodaca

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.

exp sn dodeca 2 compound of enantiomophic pair

exp sn dodaca Compound of enantiomorphic pair exp snub dodeca

All four of these images were created using Stella 4d: Polyhedron Navigator, software available (including a free trial download) at this website.