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.

Flying Kites into the Snub Dodecahedron, a Dozen at a Time, Using Tetrahedral Stellation

I’ve been shown, by the program’s creator, a function of Stella 4d which was previously unknown to me, and I’ve been having fun playing around with it. It works like this: you start with a polyhedron with, say, icosidodecahedral symmetry, set the program to view it as a figure with only tetrahedral symmetry (that’s the part which is new to me), and then stellate the polyhedron repeatedly. (Note: you can try a free trial download of this program here.) Several recent posts here have featured polyhedra created using this method. For this one, I started with the snub dodecahedron, one of two Archimedean solids which is chiral.

Snub Dodeca

Using typical stellation (as opposed to this new variety), stellating the snub dodecahedron once turns all of the yellow triangles in the figure above into kites, covering each of the red triangles in the process. With “tetrahedral stellation,” though, this can be done in stages, producing a greater variety of snub-dodecahedron variants which feature kites. As it turns out, the kites appear twelve at a time, in four sets of three, with positions corresponding to the vertices (or the faces) of a tetrahedron. Here’s the first one, featuring one dozen kites.

Snub Dodeca variant with kites

Having done this once (and also changing the colors, just for fun), I did it again, resulting in a snub-dodecahedron-variant featuring two dozen kites. At this level, the positions of the kite-triads correspond to those of the vertices of a cube.

Snub Dodeca variant with kites 1

You probably know what’s coming next: adding another dozen kites, for a total of 36, in twelve sets of three kites each. At this point, it is the remaining, non-stellated four-triangle panels, not the kite triads, which have positions corresponding to those of the vertices of a cube (or the faces of an octahedron, if you prefer).

Snub Dodeca variant with kites 2

Incoming next: another dozen kites, for a total of 48 kites, or 16 kite-triads. The four remaining non-stellated panels of four triangles each are now arranged tetrahedrally, just as the kite-triads were, when the first dozen kites were added.

Snub Dodeca variant with kites 3

With one more iteration of this process, no triangles remain, for all have been replaced by kites — sixty (five dozen) in all. This is also the first “normal” stellation of the snub dodecahedron, as mentioned near the beginning of this post.

Snub Dodeca variant with kites 4

From beginning to end, these polyhedra never lost their chirality, nor had it reversed.

An Attempt to Blend Five Snub Cubes with One Snub Dodecahedron


Viewers will be the judges of how successful this attempt to blend these polyhedra actually is. I made it using Stella 4d, software you can try right here.

One Faceting, Each, of the Snub Cube and Snub Dodecahedron

Faceted snub cube

These are facetings of the snub cube (above) and snub dodecahedron (below). I made both using Stella 4d, software you can try for yourself right here.

faceted Snub Dodeca

Faceted Snub Dodecahedron

Faceted Snub Dodeca

Facetings are created by joining vertices to other vertices, but not choosing the vertices in the usual manner, which results in new positions for edges and faces. Faceting is also the reciprocal-function for polyhedral stellation. This is one of many possible facetings of the snub dodecahedron, and I created it using Stella 4d, which 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

72-Faced Snub Dodecahedron Variant, and Related Polyhedra

72 faced snub dodecahedron variant mirror image

Like the snub dodecahedron itself, which this resembles, this polyhedron is chiral, meaning it exists in left- and right-handed forms. One version is shown above, and its mirror-image is shown below.

72-faced snub dodecahedron variant

With any chiral polyhedron, it is possible to make a compound out of the two mirror-images. Here is the enantiomorphic-pair compound for this polyhedron.

Compound of enantiomorphic pair

After making this compound, I was curious about what sort of convex hull it would have, so I used the program I employ for these polyhedral investigations, Stella 4d (available at, to find out:

Convex hull of compound on enantiomorphic pair

This polyhedron contains irregular icosagons, which are twenty-sided polygons. After playing around with this for a while, I was able to construct a related polyhedron in which the icosagons were regular — and that was one of the polyhedra seen on the post immediately before this one, which I then altered to form the others there. Had I not actually seen it happen myself, I would not have suspected there would be any connection between the snub dodecahedron, and polyhedra containing regular icosagons.