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

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

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

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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An Excavated Octahedron, and an Octahedral Nulloid

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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The Ninth Stellation of the Cube/Octahedron Compound

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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The Twelfth Stellation of the Icosahedron/Dodecahedron Compound

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

In three dimensions, there are five regular, convex polyhedra. 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: http://www.software3d.com/Stella.php.

Three Unusual Polyhedral Compounds

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Three Unusual Polyhedral Compounds

The other two appear smaller here, but can be enlarged with a single click.

All three were created using software called Stella 4d, which you may find at http://www.software3d.com/Stella.php.

Two Versions of a Stellated Polyhedron with Icosidodecahedral Symmetry

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Two Versions of a Stellated Polyhedron with Icosidodecahedral Symmetry

The next image was produced by rendering all the blue faces above invisible, and then changing the color-scheme to “rainbow color mode.” For both of these stellated polyhedra, the convex hull would simply be an icosidodecahedron (not shown here).

These images were both created using Stella 4d, which you can try and/or buy at www.software3d.com/Stella.php.

Faced-Based Zonish Versions of the Icosahedron and the Icosidodecahedron

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Faced-Based Zonish Versions of the Icosahedron and the Icosidodecahedron

I’ve had some success lately finding near-misses to the Johnson solids by making face-based zonish versions of various polyhedra. These were found during that search, and are certainly not near-misses, but I still find them interesting, primarily due to their symmetry. Like the others, they were found using Stella 4d, which you can try or buy at http://www.software3d.com/Stella.php.

The top image was formed by making this modification to the icosahedron, and the one below (which you can enlarge with a click) was created by doing the same thing to an icosidodecahedron.