I used three programs to make this: *Stella 4d*, *Geometer’s Sketchpad*, and *MS-Paint*. You can try *Stella* for free at http://www.software3d.com/Stella.php.

# Tag Archives: rhombic triacontahedron

# The Eighteenth Stellation of the Rhombicosidodecahedron Is an Interesting Polyhedral Compound

The 18th stellation of the rhombicosidodecahedron, shown above, is also an interesting compound. The yellow component of this compound is the rhombic triacontahedron, and the blue-and-red component is a “stretched” form of the truncated icosahedron.

This was made using *Stella 4d*, which you can try for free right here.

# Using Rhombic Triacontahedra to Build an Icosidodecahedron

These two polyhedra are the icosidodecahedron (left), and its dual, the rhombic triacontahedron (right).

One nice thing about these two polyhedra is that one of them, the rhombic triacontahedron, can be used repeatedly, as a building-block, to build the other one, the icosidodecahedron. To get this started, I first constructed one edge of the icosidodecahedron, simply by lining up four rhombic triacontahedra.

Three of these lines of rhombic triacontahedra make one of the icosidodecahedron’s triangular faces.

Next, a pentagon is attached to this triangle.

Next, the pentagonal ring is surrounded by triangles.

More triangles and pentagons bring this process to the half-way point. If we were building a pentagonal rotunda (one of the Johnson solids), this would be the finished product.

Adding the other half completes the icosidodecahedron.

All of these images were created using *Stella 4d: Polyhedron Navigator*. You may try this program yourself, for free, at http://www.software3d.com/Stella.php. The last thing I did with *Stella*, for this post, was to put the finished model into rainbow color mode.

# A Transparent Rhombic Triacontahedron

I made this using *Stella 4d: Polyhedron Navigator*. You may try this software, for free, at http://www.software3d.com/Stella.php.

# 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.

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.

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.

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.

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.

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.

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

# One of Many Possible Facetings of the Rhombic Triacontahedron

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:

For purposes of comparison, here is what the rhombic triacontahedron normally looks like:

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

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):

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

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

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

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

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

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:

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

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

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:

Which one do you like better?