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.

Two Compounds with Pyritohedral Symmetry: the Icosidodecahedron / Truncated Octahedron Compound, and the Rhombic Triacontahedron / Tetrakis Cube Compound

Compound of Icosidodeca and Trunc Octa its pyritohedralCompound of RTC and tetrakis cube its pyritohedral

Stella 4d, a program you can try here, was used to create these two compounds. Both have pyritohedral symmetry: the symmetry of a standard volleyball. The two compounds are also duals.

Selections from the Stellation-Series of the Icosidodecahedron

The icosidodecahedron has a long and interesting stellation-series, and you can see the whole thing using Stella 4d, the program I used to make the rotating .gifs here. Rather than keep the scale the same in each frame, I set the program to make the polyhedron as large as possible, while still fitting in the image-box. This creates the illusion that the polyhedra below are “breathing.”

Glimpses of the invisible visible version 20th stellation of the icosidodecahedron

The polyhedron above is the 20th stellation of the icosidodecahedron — the one that appeared as the sole image in the last post here, but with completely different colors. The next one shown is the 31st stellation.

Glimpses of the invisible visible version 31st stellation of the icosidodecahedron

Glimpses of the invisible visible version 55th stellation of the icosidodecahedron

The 55th stellation is immediately above, while the next one is the 69th.

Glimpses of the invisible visible version 69th stellation of the icosidodecahedron

Glimpses of the invisible visible version 84th stellation of the icosidodecahedron

The 84th stellation is immediately above, while the next one is the 89th.

Glimpses of the invisible visible version 89th stellation of the icosidodecahedron

Glimpses of the invisible visible version 106th stellation of the icosidodecahedron

The 106th stellation is immediately above, while the next one is the the 110th.

Glimpses of the invisible visible version 110th stellation of the icosidodecahedron

Glimpses of the invisible visible version 135th stellation of the icosidodecahedron

The 135th stellation is immediately above, while the next one, which is chiral, is the 157th.

Glimpses of the invisible visible version 157th stellation of the icosidodecahedron

Glimpses of the Invisible

Glimpses of the invisible

Created using Stella 4d, available here, by multiple stellations of a black icosidodecahedron, rendered as a rotating figure, against a black background.

32 Octagonal Mandalas, Rotating in the Dark

Icosidodeca

To create the octagonal mandalas, I used Geometer’s Sketchpad and MS-Paint. I then projected them onto the faces of an all-but invisible icosidodecahedron, and created this rotating .gif image of it, using Stella 4d: Polyhedron Navigator, software you can try for free, right here.