I made this using Stella 4d. If you’d like to try Stella yourself, the website to visit for a free trial download is http://www.software3d.com/Stella.php.
The Final Stellation of the Great Rhombicosidodecahedron
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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
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
I made this with Stella 4d, which you can try for yourself at this website.
Deliberately 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:
The next one is identical, except I used “rainbow color mode” for it.
Also, just in case you’re curious, here’s the dual of this polyhedron-made-of-polyhedra — this time, colored by face-type.
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:
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.
Another Faceting of the Great Rhombicosidodecahedron
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.
For a different faceting of this polyhedron, just look here: https://robertlovespi.wordpress.com/2013/11/19/a-faceting-of-the-great-rhombicosidodecahedron/
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