This is a continuation of the process shown in the last post here, but with a different coloring-scheme.

I made this using *Stella 4d*, which you can try for free at this website.

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This is a continuation of the process shown in the last post here, but with a different coloring-scheme.

I made this using *Stella 4d*, which you can try for free at this website.

…And so on….

[Software credit: I made these images using *Stella 4d*, which you can try for free right here.]

To start building this space-filling honeycomb of three Archimedean solids, I begin with a great rhombcuboctahedron. This polyhedron is also called the great rhombicuboctahedron, as well as the truncated cuboctahedron.

Next, I augment the hexagonal faces with truncated tetrahedra.

The next polyhedra to be added are truncated cubes.

Now it’s time for another layer of great rhombcuboctahedra.

Now more truncated tetrahedra are added.

Now it’s time for a few more great rhombcuboctahedra.

Next come more truncated cubes.

More great rhombcuboctahedra come next.

More augmentations using these three Archimedean solids can be continued, in this manner, indefinitely. The images above were created with *Stella 4d: Polyhedron Navigator*, a program you may try for yourself at http://www.software3d.com/Stella.php.

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

The design on each face of these great rhombcuboctahedra is made from 19 circles, and was created using both *Geometer’s Sketchpad* and *MS-Paint*. I then used a third program, *Stella 4d* (available here), to project this image on each of a great rhombcuboctahedron’s 26 faces, creating the image above.

If you watch carefully, you should notice an odd “jumping” effect on the red, octagonal faces in the polyhedron above, almost as if this polyhedron is suffering from an anxiety disorder, but trying to conceal it. Since I like that effect, I’m leaving it in the picture above, and then creating a new image, below, with no “jumpiness.” Bragging rights go to the first person who, in a comment to this post, figures out how I eliminated this anxiety-mimicking effect, and what caused it in the first place.

Your first hint is that no anti-anxiety medications were used. After all, these polyhedra do not have prescriptions for anything. How does one “calm down” an “anxious” great rhombcuboctahedron, then?

On a related note, it is amazing, to me, that simply *writing* about anxiety serves the purpose of reducing my own anxiety-levels. It is an effect I’ve noticed before, so I call it “therapeutic writing.” That helped me, as it has helped me before. (It is, of course, no substitute for getting therapy from a licensed therapist, and following that therapist.) However, therapeutic writing can’t explain how this “anxious polyhedron” was helped, for polyhedra can’t write.

For a second hint, see below.

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Second hint: the second image uses approximately twice as much memory-storage space as the first image used.

I made this with *Stella 4d*, software you can try here.

The images on the faces of this polyhedron were created with *Geometer’s Sketchpad* and *MS-Paint*. Projecting these images onto these faces, and then creating this rotating image, was accomplished using *Stella 4d: Polyhedron Navigator*— a program you can try for yourself, for free, at http://www.software3d.com/Stella.php.

This polyhedral monster has 578 faces of 26 types. In the image above, hexagons of any type are red, rhombi of any type (including squares) are yellow, and the blue faces are octagons. If each face-type is given a different color, though, this zonohedron looks like this:

Another coloring-scheme — the best one, in my opinion — is like the first one here, except that regular hexagons are given their own color (purple), and squares are given their own as well (black):

All three images were created with *Stella 4d*, software available at http://www.software3d.com/Stella.php.

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.

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.

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:

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.

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

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.

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