# 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.

# Three Views of a Rotating Cluster of 33 Icosidodecahedra

To make these three rotating cluster-polyhedra, I started with one icosidodecahedron in the center, then augmented each of its 32 faces with overlapping, additional icosidodecahedra, for a total of 33 icosidodecahedra per cluster. In the first image, only two colors are used: one for the triangular faces, and another for the pentagons. The second version, however, has the colors assigned by face-type, which is determined by each face’s placement in the overall cluster.

For the third version, I simply put Stella 4d (the program I use to make these images) into “rainbow color mode.” If you’d like to give Stella 4d a try, you can do so for free at this website.

# On Polyhedral Augmentation and Excavation

I have made many posts here using polyhedral augmentation, but what I haven’t done — yet — is explain it. I have also neglected the reciprocal function of augmentation, which is called excavation. It is now time to fix both these problems.

Augmentation is the easier of the two to explain, especially with images. The figure below call be seen as a blue icosahedron augmented, on a single face, by a red-and-yellow icosidodecahedron. It can also be viewed, with equal validity, as the larger figure (the icosidodecahedron) augmented, on a single triangular face, with an icosahedron.

When augmenting an icosidodecahedron with an icosahedron in this manner, one simply attaches the icosahedron to a triangular face of the icosidodecahedron. The reciprocal process, excavation, involves “digging out” one polyhedral shape from the other. Here is what an icosidodecahedron looks like, after having an icosahedron excavated from it, on a single triangular face.

Excavating the smaller polyhedron from the larger one is easier to picture in advance, just as one can imagine what the Earth would look like, if a Moon-sized sphereoid were excavated from it, with a large, round hole making the excavation visible. (This is mathematics, not science, so we’re ignoring the fact that gravity would instantly cause the collapse of such a compound planetary object, with dire consequences for all inhabitants.) What’s more difficult is picturing what would result if this were turned around, and the Earth was used to excavate the Moon.

This “Earth-excavated Moon” idea is analogous to excavating the larger icosidodecahedron from the smaller icosahedron. If one thinks of subtracting the volume of one solid from that of the other, such a creature should have negative volume — except, of course, that this makes no sense, which is consistent with the fact that it would be impossible to do such a thing with physical objects: there isn’t enough matter in the Moon to remove an Earth’s worth of matter from the Moon. Also, moving back to polyhedra, with excavation only into a single face, it turns out that there is no change in appearance when the excavation-order is reversed:

(Well, OK, there was a small change in appearance between the two images, but that’s only because I changed the viewing angle a bit, to give you a better view of the blue faces.)

Things get different — and the augmentation- and excavation-orders begin to matter a lot more — when these operations are performed on all available faces at once, which, in this case, means on all twenty of each polyhedron’s triangular faces. Here is the easiest case to visualize: an icosidodecahedron, augmented by twenty icosahedra.

If you use the reciprocal function, excavation, but leave the order of polyhedra the same, you get a central icosidodecahedron, excavated by twenty smaller, intersecting icosahedra:

It is, of course, possible to have other combinations. The ones I find most interesting, using these two polyhedra, are “global” augmentation and excavation of the smaller figure, the icosahedron, by twenty of the larger ones, the icosidodecahedra. Why? Simple: putting the icosidodecahedra on the outside allows for maximum visibility of both pentagons and triangles. On the other hand, the central icosahedron is completely hidden from view, whether augmentation or excavation is used. Here is the augmentation case, or what I have called a “cluster” polyhedron, many varieties of which can be seen elsewhere on this blog (just search for “cluster,” or “cluster polyhedron,” to find them):

The global-excavation case which has the icosahedron hidden in the middle is similar to the cluster immediately above, in that all that can be seen are twenty intersecting icosidodecahedra. However, it also varies noticeably, because, with excavation, the icosidodecahedra are closer to the center of the entire cluster (the invisible, central icosahedron’s center) than was the case with augmentation. The last image here is of an invisible, central icosahedron, with an icosidodecahedron excavated from all twenty triangular faces. The larger polyhedra “punch through” the smaller one from all sides at once, trapping the central polyhedron — the blue icosahedron — from view. The remaining object looks, to me at least, more like a faceted icosidodecahedron than a cluster-polyhedron. I am of the opinion, but have not verified, that this resemblance to a faceting of the icosidodecahedron is illusory.

[Image credits: all images in this post were made using Stella 4d: Polyhedron Navigator. This program may be purchased, or tried as a free trial download, at http://www.software3d.com/Stella.php.]

# Multiple Facetings of the Icosidodecahedron

To create a faceted icosidodecahedron, you simply start with an icosidodecahedron, then remove its edges and faces. Next, you connect the vertices differently, thus creating new faces and edges. As you can see below, this can be done in many different ways.

All these were created using Stella 4d, software you may try for free at http://www.software3d.com/Stella.php.

# A Faceted Icosidodecahedron with 48 Faces

The 48 faces of this faceted icosidodecahedron are:

• 6 green regular decagons (difficult to see, because they pass through the figure’s center)
• 30 golden rectangles, shown in yellow, and each interpenetrating other faces
• 12 blue star pentagons

Software called Stella 4d was used to make this, and it is available (with a free trial download available) at this website.

# A Compound of Fifteen Cuboids — Which Is Also a Particular Faceting of the Icosidodecahedron

The creator of Stella 4d, the program I used to make these rotating polyhedral images, is Robert Webb (and the software itself may be tried for free here). Recently, on Facebook, he displayed a paper model of this compound of fifteen cuboids, pointed out that it is a faceting of the icosidodecahedron, and I (being me) took that as a challenge to make it myself. Here is my first result, in which all fifteen cuboids have different colors.

I then realized that RW had rendered his in only five colors, so I studied his post more carefully, and made the appropriate adjustments to do the same:

If you’d like to find the Stella page on Facebook, here is a link to it.

# 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:

# Four Different Facetings of the Great Rhombcuboctahedron

All four of these rotating images were created using software called Stella 4d: Polyhedron Navigator. You can buy this program, or try it for free, at this website. Faceting is the inverse function of stellation, and involves connecting the vertices of an already-established polyhedron in new ways, to create different polyhedra from the one with which one started. For each of these, the convex hull is the great rhombcuboctahedron, itself.