## Slow Dissection of a Loosely-Defined “Faceted” Rhombcuboctahedron

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If you look at the second image from the post two entries ago, and wonder what it would look like without the pink faces, wonder no longer: it’s what you see above.

Next, the red polygons are hidden, and this is what is left (you may click these smaller images if you wish to enlarge them).

The green faces are hidden next.

The next step is to remove the pink faces visible in the interior.

Next, removal of the blue faces leaves only the yellow ones left.

The last step:  change the color scheme, so as to more easily be able to tell one face from another.

All of this polyhedron-manipulation, I did with Stella 4d, software I consider an indispensable research-tool. It is available at http://www.software3d.com/Stella.php.

## The Dual of the Enantiomorphic Pair of Polyhedra from the Last Post

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The last post had two images, and this is the dual of the second one. I was therefore surprised when I ran into this while playing around with Stella 4d, a program which allows easy polyhedron manipulation. (See http://www.software3d.com/stella.php for free trial download.)

Why did it surprise me?

Well, isn’t a polyhedron. for one thing. It is a collection of irregular and concentric polygons which intersect, but they don’t meet at edges. This doesn’t normally happen, so it requires explanation. I figured it out pretty quickly.

I’ve been using the loosest possibly definition for “faceting,” not insisting that faces meet at each edge in pairs, and even making some faces invisible in order to see the interior structure of the “polyhedra.” Since this breaks the faceting-rules, it isn’t surprising that the dual would fail to be a true polyhedron.

That’s my guess, anyway.

## A Chiral Faceting of the Rhombcuboctahedron, and Its Compound with Its Own Mirror-Image

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Once I realized this particular faceting is chiral, I knew I’d want to make a compound of it, and its own mirror-image. As it turns out, that compound is, in my opinion, more attractive:

Both these polyhedra were made with Stella 4d, software you can find at http://www.software3d.com/Stella.php.

## An Excavated Octahedron with Non-Zero Volume, and an Octahedral Nulloid

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This is an octahedron which has had short pyramids excavated from each of its faces. If the pyramids are made taller, their vertices coincide at the octahedron’s center. At that point, unlike in the figure above, the polyhedron’s volume reaches zero — turning it into a special type of polyhedron called a nulloid:

You may click on this second picture to enlarge it, if you wish.

Software credit: I made these images using Stella 4d, which you can find at http://www.software3d.com/Stella.php.

## The Ninth Stellation of the Cube/Octahedron Compound

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Software credit: I made this using Stella 4d, which you can find at http://www.software3d.com/Stella.php.

## The Twelfth Stellation of the Icosahedron/Dodecahedron Compound

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Software credit: I made this using Stella 4d, which you can find at http://www.software3d.com/Stella.php.

## The Icositetrachoron, or 24-Cell: An Oddball In Hyperspace

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In three dimensions, there are five regular, convex polyhedra. Similarly, in five dimensions, there are five regular, convex polytopes. There are also five of them in six, seven, eight dimensions . . . and so on, for as long as care to venture into higher-dimensional realms. However, in hyperspace — that is, four dimensions — there are, strangely, six.

The five Platonic solids have analogs among these six convex polychora, and then there’s one left over — the oddball among the regular, convex polychora. It’s the figure you see above, rotating in hyperspace: the 24-cell, also known as the icositetrachoron. Its twenty-four cells are octahedra.

Like the simplest regular convex polychoron, the 5-cell (analogous to the tetrahedron), the 24-cell is self-dual. No matter how many dimensions you are dealing with, it is always possible to make a compound of any polytope and its dual. Here, then, is the compound of two 24-cells (which may be enlarged by clicking on it):

Both of these moving pictures were generated using software called Stella 4d:  Polyhedron Navigator. You can buy it, or try a free trial version, right here:  www.software3d.com/Stella.php.