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.

Another Faceting of the Great Rhombicosidodecahedron

Faceted Trunc Icosidodeca

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.

Trunc Icosidodeca

For a different faceting of this polyhedron, just look here: https://robertlovespi.wordpress.com/2013/11/19/a-faceting-of-the-great-rhombicosidodecahedron/

A Rotating Great Rhombicosidodecahedron, with Spinning Mandalas On Its Faces

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A Rotating Great Rhombicosidodecahedron, with Spinning Mandalas On Its Faces

This polyhedron is also known as the truncated icosidodecahedron. However, I prefer the name which appears in the title of this post.

I made the image which appears on each face with Geometer’s Sketchpad and MS-Paint, and then used Stella 4d to project this image onto each face of this polyhedron, and create this rotating .gif image.

If you’d like to try Stella 4d for free, just visit this site: http://www.software3d.com/Stella.php. To my knowledge, a free trial download is only available for Stella 4d, but not for the other programs mentioned above.

An Icosidodecahedral Cluster of Great Rhombicosidodecahedra

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An Icosidodecahedral Cluster of Great Rhombicosidodecahedra

I made this, using Stella 4d, by augmenting the thirty square faces of a great rhombicosidodecahedron with additional great rhombicosidodecahedra. The result has one of these polyhedra located in each position which corresponds to a vertex of an icosidodecahedron.

You can give this program a try yourself, for free, at http://www.software3d.com/stella.php.

Great Rhombicosidodecahedron with Interference Patterns

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Great Rhombicosidodecahedron with Interference Patterns

Programs used: Geometer’s Sketchpad and MS-Paint, to make the image seen in the previous post; and Stella 4d, to place that image of each face of this polyhedron, and then make this rotating .gif file.

Stella 4d may be tried for free at http://www.software3d.com/stella.php.