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.