Tag Archives: cuboctahedral

Combining Octahedral and Icosahedral Symmetry to Form Pyritohedral Symmetry

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Compound of Octa and Icosa

Pyritohedral symmetry, seen by example both above and below, is most often described at the symmetry of a volleyball:

volleyball-306791_640

[Image of volleyball found here.]

To make the rotating polyhedral compound at the top, from an octahedron and an icosahedron, I simply combined these two polyhedra, using Stella 4d, which may be purchased (or tried for free) here.

In the process, I demonstrated that it is possible to combine a figure with octahedral (sometimes called cuboctahedral) symmetry, with a figure with icosahedral (sometimes called icosidodecahedral) symmetry, to produce a figure with pyritohedral symmetry.

Now I can continue with the rest of my day. No matter what happens, I’ll at least know I accomplished something.

A Non-Convex Polyhedron with Cuboctahedral Symmetry

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co symm and nc

I used Stella 4d to make the polyhedron above, and you can try this software for yourself here.

A Collection of Rotating Polyhedra with Cuboctahedral Symmetry

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I have received a request to slow down the rotational speed of the polyhedral models I make and post here, and am going to try to do exactly that. First, though, I need to empty my collection of already-made image files which haven’t yet been posted, so that I can start again, with models which rotate more slowly, after deleting all the “speedy” ones. From my backlog of polyhedral images to post, then, here are some which have cuboctahedral symmetry.

Convex hull

Convexhgdfb hull

Compound of jhfjhdgdtuare-based pyramids

stellated Compound of Cubocta and dual

Unnamed Duhgddal

This last one actually has the symmetry of a snub cube — a chiral variant of “normal” cuboctahedral symmetry.

Convsdgtagaqex hull

All these images were created using Stella 4d:  Polyhedron Navigator. If you’d like to try this program for yourself, the website to visit for a free trial download is www.software3d.com/Stella.php.

A Gallery of 27 Polyhedra with Cuboctahedral Symmetry

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Some of these polyhedra have “normal” cuboctahedral symmetry, while others have the chiral variant of that symmetry-type — in other words, the same type of symmetry found in the snub cube.

Some Polyhedra with Cuboctahedral Symmetry

CO1Convex hullCO2 Convex hullCconvex hullDual of Convex hullhexagons and squaresStellated Convex hullStellated Convex hulllAugmented Stellated Convex hullCoonvex hullConvex hull of prism-augmented snub cubedual of Coonvex hulllreaugmented dual of Convex hull of prism-augmented snub cubeStellated Polydual of Convex hull of prism-augmented snub cubeanother augmentation of dual of Convex hull of prism-augmented snub cubedual of Coonvex hullFaceted Convex hullStellated Poly2Stellated Poly3Faceted Convex hulllStellated Poly4Dual of Convex hulglDual of Convex hugFaceted Convex hulfgdsgthxlFaceted Convex hulfgdsgthgfsxlFaceted Convex hufdlfgdsgthgfsxl

I used Stella 4d:  Polyhedron Navigator to make these images, and you can find that program at http://www.software3d.com/Stella.php.