A Polyhedral Journey, Beginning With an Expansion of the Rhombic Triacontahedron

The blue figure below is the rhombic triacontahedron. It has thirty identical faces, and is one of the Catalan solids, also known as Archimedean duals. This particular Catalan solid’s dual is the icosidodecahedron.

Rhombic Triaconta

I use a program called Stella 4d (available here) to transform polyhedra, and the next step here was to augment each face of this polyhedron with a prism, keeping all edge lengths the same.

Rhombic Triaconta augmented

After that, I created the convex hull of this prism-augmented rhombic triacontahedron, which is the smallest convex figure which can enclose a given polyhedron.

Convex hull

Another ability of Stella is the “try to make faces regular” function. Throwing this function at this four-color polyhedron above produced the altered version below, in which edge lengths are brought as close together as possible. It isn’t possible to do this perfectly, though, and that is most easily seen in the yellow faces. While close to being squares, they are actually trapedoids.

ch after ttmfr

For the next transformation, I looked at the dual of this polyhedron. If I had to name it, I would call it the trikaipentakis icosidodecahedron. It has two face types: sixty of the larger kites, and sixty of the smaller ones, also.

ch after ttmfr dual

Next, I used prisms, again, to augment each face. The height used for these prisms is the length of the edges where orange kites meet purple kites.

aug ch after ttmfr dual

Lastly, I made the convex hull of the polyhedron above. This convex hull appears below.

Convex hull again

 

One of Many Possible Facetings of the Rhombic Triacontahedron

Faceted Rhombic Triaconta

The simplest way I can explain faceting is that it takes a familiar polyhedron’s vertices, and then connects them in unusual ways, so that you obtain different edges and faces. If you take the convex hull of a faceted polyhedron, it returns you to the original polyhedron.

This was created using Stella 4d, software available (including as a free trial download) right here: http://www.software3d.com/Stella.php.

92 Dodecahedra, Arranged as a Single Rhombic Triacontahedron

With 92 dodecahedra, if you arrange them just right, you can make a model of a rhombic triacontahedron:

RTC Augmented DodecaFor purposes of comparison, here is what the rhombic triacontahedron normally looks like:

Rhombic Triaconta

Also, referring back to the first model shown, here is a picture of just one of the red rhombi-made-of-dodecahedra.

RHOMBUS Dodeca

The first polyhedron shown in this post has an interesting dual, as well. Here it is, colored by face-type (position within the overall shape):

Dual of RTC Augmented Dodeca

Here is another view of the dual, colored by number of edges per face.

Dual of RTC Augmented Dodeca

Here’s one more view of the dual, in “rainbow color mode.”

dual RTC Augmented Dodeca rainbow

Returning to the original model, at the top of this post, here’s what it looks like, if colored by face type:

RTC Augmented Dodeca face type

Here’s one more view, in “rainbow color mode.”

RTC Augmented Dodeca rainbow

All of these images were created using Stella 4d, a program you can buy, or try for free, right here.

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

final stellation of the Rhombic Triaconta

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:

Faceted Icosidodeca dual of final stellation of RTC

A Compound of an Icosahedron and the First Stellation of the Rhombic Triacontahedron

Compound of an icosahedron and a stellation (find out which one) of the RTC

I made this compound using software called Stella 4d: Polyhedron Navigator. This program may be purchased (or a trial download tried for free) at this website.

Two Versions of a Slowly Rotating Rhombic Triacontahedron, Adorned with Spectral Patterns on Each Face

Rhombic Triaconta

It took three programs to make this. First, outlines of the “double rainbow” patterns on each face were constructed using Geometer’s Sketchpad. A screenshot from that program was then pasted into MS-Paint, which was used to add color to the outline of the pattern on each face. Next, the colorized image was projected onto each face of a rhombic triacontahedron, using Stella 4d: Polyhedron Navigator — the program that put this all together, and what I used to generate the rotating .gif above. Stella is available at http://www.software3d.com/Stella.php, with a free trial download available.

Interestingly, while this polyhedron itself is not chiral, the coloring-pattern of it, shown above, is.

With only small modifications, Stella can produce a very different version:

Rhombic Triaconta

Which one do you like better?

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.