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

Posted on 15 October 2015 by RobertLovesPi

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:

One of Many Possible Facetings of the Rhombicosidodecahedron

Posted on 12 October 2015 by RobertLovesPi

To make a faceted version of the rhombicosidodecahedron, one first (1) starts with a rhombicosidodecahedron, one of the Archimedean solids, then (2) removes the faces and edges of this polyhedron, leaving all the vertices in place, and then (3) connects these vertices in a different way than they were connected in the original polyhedron, forming new edges and faces. Faceting is the reciprocal operation to polyhedral stellation.

This polyhedron was made using Stella 4d, software available here.

Four Different Facetings of the Great Rhombcuboctahedron

Posted on 11 October 2015 by RobertLovesPi

All four of these rotating images were created using software called Stella 4d: Polyhedron Navigator. You can buy this program, or try it for free, at this website. Faceting is the inverse function of stellation, and involves connecting the vertices of an already-established polyhedron in new ways, to create different polyhedra from the one with which one started. For each of these, the convex hull is the great rhombcuboctahedron, itself.

One of Many Faceted Rhombicosidodecahedra

Posted on 16 September 2015 by RobertLovesPi

This was created by making the dual of the 32nd stellation of the strombic hexacontahedron, which is itself the dual of the rhombicosidodecahedron. This technique for finding facetings works because faceting is the reciprocal function of polyhedral stellation.

I did this using Stella 4d, which you can try for yourself, for free, at http://www.software3d.com/Stella.php.

There Are Many Faceted Versions of the Dodecahedron. This One Is the Dual of the Third Stellation of the Icosahedron.

Posted on 6 June 2015 by RobertLovesPi

The twelve purple faces of this faceted dodecahedron show up on Stella 4d‘s control interface as {10/4} star decagons, which would make them each have five pairs of two coincident vertices. I’m informally naming this special decagon-that-looks-like-a-pentagram (or “star pentagon,” if you prefer) the “antipentagram,” for reasons which I hope are clear.

Stella 4d, the program I use to make most of my polyhedral images, may be tried for free at http://www.software3d.com/Stella.php.

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

Posted on 1 June 2015 by RobertLovesPi

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.

Two of Many Possible Facetings of the Truncated Icosahedron

Posted on 31 May 2015 by RobertLovesPi

I made these faceted polyhedra, both facetings of the truncated icosahedron, using Stella 4d, software available here.

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

Posted on 25 May 2015 by RobertLovesPi

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:

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:

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.

A Faceting of the Truncated Dodecahedron, Together with Its Dual

Posted on 28 April 2015 by RobertLovesPi

This faceting of the truncated dodecahedron, one of many, was made with Stella 4d, software you can buy, or try for free, here. Here is its dual, below.

Another Faceting of the Great Rhombicosidodecahedron

Posted on 23 April 2015 by RobertLovesPi

This could also be called one of many possible faceted truncated icosidodecahedra. I made it using Stella 4d, which you can try and/or buy here. Faceting is the reciprocal operation of stellation, and involves connecting the vertices of a polyhedron into faces which are unlike those of the original polyhedron. At least some, and sometimes all, of the faceted faces intersect each other, inside the polyhedron’s convex hull, as is the case here.

For comparison, here is that convex hull: a (non-faceted) great rhombicosidodecahedron, also made using Stella.