From the Great Rhombicosidodecahedron to Something Much Stranger

This is the great rhombicosidodecahedron, one of the thirteen Archimedean solids.

Here’s its dual, the disdyakis triacontahedron.

I use a program called Stella 4d to make these .gifs and manipulate polyhedra, and one of Stella‘s functions is “try to make faces regular.” I performed this function on the disdyakis triacontahedron, which has ten triangles meeting at some vertices — so 600 degrees’ worth of triangle-angles tried to squeeze in around those points when the faces were made to be regular. This forces the polyhedron to become non-convex — to the point of looking wrinkled.

“That’s weird looking,” I thought. “I wonder what its dual looks like?” With Stella, I could find out with one mouse-click, and I was most surprised by the result.

In this polyhedron, there are thirty orange rectangles, twelve light blue 10/4-gons, and twenty violet 6/2-gons. None of them are regular. Here are what the faces look like in isolation, starting with an orange rectangle, then a light blue 10/4-gon, and lastly a violet 6/2-gon.

If you’d like to try Stella for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.

A Compound of the Great Rhombicosidodecahedron and the Small Stellated Dodecahedron

I made this, using Stella 4d (available here), by multiple stellation of the faceted great rhombicosidodecahedron shown in the post immediately before this one.

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.

Deliberately Difficult to Watch

difficult to watch

I’ve never tried this before: create a rotating polyhedral image which is difficult to watch, using disorienting effects, such as the rotation of the images of spirals on the rotating faces. The spiral is made of golden gnomons (obtuse triangles with a base:leg ratio which is the golden ratio). This image, alone and without comment, is shown in the previous post, and was made using Geometer’s Sketchpad and MS-Paint. In the preparation for this post, it was further altered, including the projection of it onto the faces of a great rhombicosidodecahedron, and creating this rotating .gif. This part of the process was performed using a program called Stella 4d: Polyhedron Navigator, available here. You be the judge, please: is it, in fact, difficult to watch? Did I accomplish my (admittedly rather odd) goal?

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.