Three Views of a Rotating Cluster of 33 Icosidodecahedra

33-icosidodeca

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.

33-icosidodeca-ft

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.

33-icosidodeca-rc

 

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.

Augmented Icosa Icosidodeca one of each

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.

aug Icosidodeca with excavated icosahedron

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:

Augmented Icosa with icosidodeca excavated from one face

(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.

Augmented Icosidodec surrounded by icosas

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:

Augmented Icosidodeca excavated by icosas

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):

Augmented Icosa

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.

aug tWENTY ICOSIDODECAS EXCAVATED FROM AN ICOSA

[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.

Faceted Icosidodeca 3Faceted IcosidhgfodecaCompound of enantiomorphic pair of facted icosidoecafaceted five octahedra dual of final stellation of 5cubes colored by face typeFaceted DualFaceted IcosidodecaFaceted Icosidodeca xfacetings of the Icosidodecafacetings of the Icosidodeca 2Faceted icosidod and a uniform polyhedron made of 10 hexagons and 12 star pentagons fid out which oneFaceted Icosidodeca six reg decagons going through center twelve star pentagons 30 golden rectangles

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

Faceted Icosidodeca golden rectangles dodecagons and star pentagons

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.

Faceted Icosidodeca compound of 15 cuboids give RW credit.gif

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:

Faceted Icosidodeca compound of 5 cuboids 5 color version

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

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

Four Different Facetings of the Great Rhombcuboctahedron

faceted GRCO

Faceted Trunc Cubocta 2

Faceted Trunc Cubocta 4

Faceted Trunc Cubocta

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.

A Central Icosidodecahedron, Augmented with Twenty Cuboctahedra, and Twelve More Icosidodecahedra

Augmented Icosidodeca aug with 20 cuboctas and 12 icosidodecas color scheme two

Above and below, you will find two different coloring-schemes for this particular cluster of polyhedra. I made both of these rotating images using Stella 4d, software you can buy, or try for free, right here.

Augmented Icosidodeca aug with 20 cuboctas and 12 icosidodecas

Tidally Locked Binary Icosidodecahedra

binary icosidodecahedra

I’ve been trying to figure out for over a year how to make images like the one above, without having holes in the two polyhedra, facing each other. At last, that puzzle of polyhedral manipulation using Stella 4d (software available at this website) has been solved: use augmentation followed by faceting, rather than augmentation followed by simply hiding faces.

Two Polyhedral Compounds: the Icosidodecahedron with the Truncated Cube, and the Rhombic Triacontahedron with the Triakis Octahedron

Compound of Icosidodeca and Trunc Cube

These two compounds, above and below, are duals. Also, in each of them, one polyhedron with icosidodecahedral symmetry is combined with a second polyhedron with cuboctahedral symmetry to form a compound with pyritohedral symmetry: the symmetry of a standard volleyball.

Compound of RTC and Triakis octahedron also pyritohedral

A program called Stella 4d was used to make these compounds, and create these images. It may be purchased, or tried for free, at this website.