One of Many Faceted Versions of the Great Rhombicosidodecahedron

The great rhombicosidodecahedron is also known as the truncated icosidodecahedron. I created this faceting of it using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

The Sun, Earth, and Moon Adorning the Faces of a Great Rhombicosidodecahedron

Trunc Icosidodeca

This polyhedral image was created using Stella 4d, a program you can try for yourself, for free, at http://www.software3d.com/Stella.php.

A Great Rhombicosidodecahedron Inspired By David Bowie, As Ziggy Stardust

Ziggy's Trunc Icosidodeca

I made this with Stella 4d, which you can try for yourself at this website.

The Great Rhombicosidodecahedron, Built from Rhombic Triacontahedra, and Its Dual

The great rhombicosidodecahedron is also known as the truncated icosidodecahedron (and, confusingly, several other names). Regardless of what it’s called, these pictures demonstrate that this Archimedean solid can be constructed using rhombic triacontahedra as building-blocks.

First, here’s one in the same color I used for the decagonal ring of rhombic triacontahedra in the last post:

GRID of Rhombic Triaconta

The next one is identical, except I used “rainbow color mode” for it.

GRID of Rhombic Triaconta RB

Also, just in case you’re curious, here’s the dual of this polyhedron-made-of-polyhedra — this time, colored by face-type.

dual of GRID of Rhombic Triaconta

These virtual models were all built using Stella 4d, software you may buy, or try for free, right here.

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.

Selections from the Stellation-Series of the Great Rhombicosidodecahedron

The great rhombicosidodecahedron, also known as the truncated icosidodecahedron, has a long and complex stellation series. Here are some highlights from that series, chosen using aesthetic, rather than mathematical, criteria.

All these virtual models were made using Stella 4d, which you can try and/or buy here.

Nth stellation of the great rhombicosidodecaNt1h stellation of the great rhombicosidodecaN21h stellation of the great rhombicosidodecaN25hg1uyh stellation of the great rhombicosidodecaN25hhgdg1hghjjhfuyh stellation of the great rhombicosidodeca N25hhgdg1hgjhjjhfjhgujhfjhyh stellation of the great rhombicosidodeca N25hhgdg1hgjhjjhfjhgujhjhfjhyh stellation of the great rhombicosidodecaN25hhgdg1uyh stellation of the great rhombicosidodecaN251h stellation of the great rhombicosidodecaN251uyh stellation of the great rhombicosidodecaN25hhgdg1hgjhjjhfjhgujhjjhhfjhyh stellation of the great rhombicosidodecaN25hhgdg1hgjhjjhfjhgujhyh stellation of the great rhombicosidodecaN25hhgdg1hgjhjjhfujhyh stellation of the great rhombicosidodecaN25hhgdg1hgjhjjhfuyh stellation of the great rhombicosidodecaN25hhgdg1jfuyh stellation of the great rhombicosidodecaN25hhgdg1jjhfuyh stellation of the great rhombicosidodeca

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/