Building a “Polyhedral Porcupine”

This is the icosahedron, followed by its first stellation.

The first stellation of the icosahedron can be stellated again, and again, and so on. The “final stellation” of the icosahedron is the one right before the stellation-series “wraps around,” back to where it started:

icosa-stellation-final-60-spikes

This final stellation of the icosahedron would serve pretty well as a “polyhedral porcupine,” but I was seeking something even better, so I turned my attention to polyhedral compounds. This is the compound of five icosahedra:

icosahedra-5

The program I use to manipulate these solids is called Stella 4d: Polyhedron Navigator (free trial download available here). My next move, using Stella, was to create the final stellation of this five-icosahedron compound . . . and, when I saw it, I knew I had found my “polyhedral porcupine.”

icosahedra-5-final-stellation

The Final Stellation of the Rhombic Triacontahedron, Together with Its Dual, a Faceting of the Icosidodecahedron

final stellation of the Rhombic Triaconta

Sharp-eyed, regular readers of this blog will notice that this is the same polyhedron shown in the previous post, which was described as the “final stellation of the compound of five cubes,” due to the coloring scheme used in the first image there, which had five colors “inherited” from each of the differently-colored cubes in the five-cube compound. This image, by contrast, is shown in rainbow-color mode.

How can the rhombic triacontahedron and the compound of five cubes have the same final stellation? Simple: the compound of five cubes is, itself, a member of the stellation-series of the rhombic triacontahedron. Because of this, those two solids end up at the same place, after all possible stellations are completed, just as you will reach 1,000, counting by ones, whether you start at one, or start at, say, 170.

I am grateful to Robert Webb for pointing this out to me. He’s the person who wrote Stella 4d, the software I use to make these images of rotating polyhedra. His program may be found at http://www.software3d.com/Stella.php — and there is a free trial version available for download, so you can try Stella before deciding whether or not to purchase the fully-functioning version.

Since faceting is the reciprocal process of stellation, the dual of the polyhedron above is a faceted icosidodecahedron, for the icosidodecahedron is the dual of the rhombic triacontahedron. Here is an image of that particular faceting of the icosidodecahedron, colored, this time, by face-type:

Faceted Icosidodeca dual of final stellation of RTC

The Final Stellation of the Compound of Five Cubes

Stellated 5 Cubes final stellation colors derived from compound

The version of the final stellation of the compound of five cubes shown above has its colors derived from the traditional five-color version of the original compound, itself. The one below, by contrast, has its colors selected by face-type, without regard for the original compound.

Stellated 5 Cubes final stellation colored by face type

Both of these virtual models were created with Stella 4d: Polyhedron Navigator, software available at this website. Also, for more about this particular polyhedron, please see the next post.

The Final Stellation of the Great Rhombicosidodecahedron, Together with Its Dual

In the last post, several selections from the stellation-series of the great rhombicosidodecahedron (which some people call the truncated icosidodecahedron) were shown. It’s a long stellation-series — hundreds, or perhaps thousands, or even millions, of stellations long (I didn’t take the time to count them) — but it isn’t infinitely long. Eventually, if repeatedly stellating this polyhedron, one comes to what is called the “final stellation,” which looks like this:

final valid stellation of the great rhombicosidodeca

Stellation-series “wrap around,” so if this is stellated one more time, the result is the (unstellated) great rhombicosidodecahedron. In other words, the series starts over.

The dual of the great rhombicosidodecahedron is called the disdyakis triacontahedron. The reciprocal function of stellation is faceting, so the dual of the figure above is a faceted disdyakis triacontahedron. Here is this dual:

Faceted Disdyakistriaconta

To complicate matters further, there is more than one set of rules for stellation. For an explanation of this, I refer you to this Wikipedia page. In this post, and the one before, I am using what are known as the “fully supported” rules.

Both these images were made using Stella 4d, software you can buy, or try for free, right here. When stellating polyhedra using this program, it can be set to use different rules for stellation. I usually leave it set for the fully supported stellation criteria, but other polyhedron enthusiasts have other preferences.

The Final Stellation of the Icosahedron

Stellated Icosa

This is what you get if you stellate an icosahedron seventeen times. The eighteenth stellation “loops” back around to the original figure, the icosahedron. For this reason, the figure above is often called “the final stellation of the icosahedron,” as well as “the complete icosahedron.” Its faces are twenty irregular star enneagons, of the type shown below. The red areas are the “facelets” which can be seen, while the other parts of the star enneagon are hidden inside the figure.

Stellated Icosa-StelDiag

Both of these images were made using Stella 4d: Polyhedron Navigator, which you can try for yourself right here. A free trial download is available.