The Great Rhombcuboctahedron As a Building-Block

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The Great Rhombcuboctahedron As a Building-Block

This solid, also known as the great rhombicuboctahedron, and the truncated icosidodecahedron, can be used to build many other things. In addition to the elongated ring of eight above, for example, there’s this octagonal prism.

Augmented Trunc Cubocta2

Augmented Trunc Cubocta 2

Remember the elongated ring at the top of this post? This pic, directly above, is of a ring of four of those rings.

Augmented Trunc Cubocta3

And, yes, that’s a (non-great) rhombcuboctahedron made of great rhombcuboctahedra. Here it is again, with a different color-scheme.

Augmented Trunc Cubocta4

For the last of these constructions, eight more great rhombcuboctahedra are added to the figure in the two posts above, which is also returned to its original color-configuration. These eight new polyhedra have positions which correspond to the corners of a cube.

augmented rhombcuboctahedron made of great rhombcuboctahedra

Manipulating polyhedra in this manner is easy with Stella 4d, the program I used to do all of this. You may buy it, and/or try a free trial version first, at www.software3d.com/Stella.php.

A 182-Faced Convex Hull, with an Explanation of that Term, As It Relates to Polyhedra

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A 182-Faced Convex Hull

I made this polyhedron, using Stella 4d: Polyhedron Navigator, by taking the convex hull of a different polyhedron, one not shown here. To those who don’t already know, though, that just raises a question:  what does it mean to “take the convex hull” of a polyhedron? Precisely-worded mathematical definitions of “convex hull” are easy to find, using Google and/or Wikipedia, but I have a more informal definition — one which matches the way I actually think about this operation one can perform on polyhedra.

Here’s how I picture the process: imagine a thin, spherical rubber sheet is surrounding, but not touching, the starting polyhedron. Next, start shrinking the rubber sheet. It can touch the polyhedron inside it (which might be non-convex), but it cannot penetrate any of its faces. Keep shrinking the sheet until it gets caught at points on the polyhedron inside, and then keep shrinking it further. When it starts to stretch, keep going. Stop just before the rubber sheet starts to burst from being over-stretched — and the shape of the rubber sheet, at that point, is the convex hull of the polyhedron inside it. Stretching the rubber sheet, to the limit, ensures that the convex hull will only have flat, polygonal faces — not any sort of curved surfaces.

Here’s an example — one that will end with a different convex hull than the one found at the top on this post. I’ll start with a great rhombcuboctahedron, which is also known as a great rhombicuboctahedron, as well as a truncated cuboctahedron.

Trunc Cubocta

If I take the convex hull of this great rhombcuboctahedron, the result is simply another great rhombcuboctahedron — the very thing I started with — which doesn’t explain much. Therefore, before taking the convex hull, I’m going to alter it. This can be done in many ways, of course. I’m choosing augmentation of each face with prisms, and setting the prism-height at twice the edge length of these faces. Here’s the result.

Trunc Cubocta

Taking the convex hull of this doesn’t return this same polyhedron, as it would have before the augmentation-with-prisms. Instead, after the “stretching of the imaginary rubber sheet,” this is the result:

Convex hull 1

In this image, the faces that are unmoved still have their original colors. There are also many new faces, of varying types, which were created in the “convex hulling” process. All of these new faces are shown in the same drab-green color.

The next step, changing the color scheme, has little (if any) mathematical significance, but it certainly does increase the attractiveness of the result — and admiration of beauty is, and always has been, a major motivating force in the millenia-old study of polyhedra. I’m choosing a color scheme which gives each face-type a separate color, and also lets the red, yellow, and dark blue keep their same colors.

Convex hull 2

If you’d like to try Stella 4d for yourself, please visit www.software3d.com/Stella.php. A free trial download is available.

A Close-Packing of Space, Using Three Different Polyhedra

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A Close-Packing of Space, Using Three Different Polyhedra

This is like a tessellation, but in three dimensions, rather than two. The pattern can be repeated to fill all of space, using cubes (yellow), truncated octahedra (blue), and great rhombcuboctahedra, also known as truncated cuboctahedra (red).

Software credit: see www.software3d.com/stella.php to try or buy Stella 4d, the software I used to create this image.

Captain Kirk, Mr. Spock, and Dr. McCoy On a Great Rhombcuboctahedron

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Captain Kirk, Mr. Spock, and Dr. McCoy On a Great Rhombcuboctahedron

If any doubt remained about my nerdiness, it’s gone now.

Software credit: see http://www.software3d.com/stella.php.

Great Rhombcuboctahedron with Red and Blue Mandalas

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Great Rhombcuboctahedron with Red and Blue Mandalas

The design on the faces first appeared in the last post, and was made using Geometer’s Sketchpad and MS-Paint.

I used Stella 4d, another program, to put this design (as a colorized version) on the octagonal and hexagonal faces of this great rhombcuboctahedron, and then render the square faces of this polyhedron invisible. This program is available at http://www.software3d.com/stella.php.

Solar Eclipses and Mandalas On a Great Rhombcuboctahedron

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Solar Eclipses and Mandalas On a Great Rhombcuboctahedron

Credit where credit is due:

I found the eclipse picture with a Google-search.

The mandalas are the one in the previous post here (“Seventeen Circles”). I made it using Geometer’s Sketchpad and MS-Paint.

I used software called Stella 4d to assemble this onto the chosen polyhedron, and make the animated .gif file you see here. This software is available at http://www.software3d.com/stella.php, with a free trial download available.

The Great Rhombcuboctahedron

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The Great Rhombcuboctahedron

The images on the faces are colorized versions of my last post here. This transfer was accomplished using software you can find at http://www.software3d.com/stella.php.