To create this, I started with a rhombic triacontahedron, and then used software called Stella 4d to zonohedrify it, adding zones to the existing faces along x, y, and z axes meeting at the polyhedron’s center.
I find its dual even more interesting:
The software used to create these rotating images may be tried for free at http://www.software3d.com/Stella.php.
Here is the dual of the solid above, also.
These images were created with Stella 4d, which you can find at http://www.software3d.com/Stella.php.
This is the dual of the polyhedron seen in the last post. It appears to be an interesting blend of the snub cube and an icosidodecahedron.
(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
Once I completed the polyhedral cluster seen in the last post, I became curious about its dual, which is what you see here. The overall shape here resembles an icosidodecahedron — the Archimedean solid which is, itself, the dual of the rhombic triacontahedron.
To try the software I use to generate these images, simply visit http://www.software3d.com/stella.php — where a free trial download is available.
As the dual of the snub dodecahedron, which is chiral, this member of the Catalan Solids is also chiral — in other words, it exists in left- and right-handed versions, known an entantiomers. They are mirror-images of each other, like left and right gloves or shoes. Here’s the other one, by comparison:
It is always possible to make a compound, for a chiral polyhedron, from its two enantiomers. Here’s the one made from the two mirror-image pentagonal hexacontahedra shown above:
Stellating this enantiomorphic-pair-compound twenty-one times produces this interesting result:
And, returning to the unstellated enantiamorphic-pair-compound, here is its convex hull:
This convex hull strikes me as an interesting polyhedron in its own right, so I tried stellating it several times, just to see what would happen. Here’s one result, after seventeen stellations:
Software credit: I made these rotating images using Stella 4d: Polyhedron Navigator. That program may be bought at http://www.software3d.com/Stella.php, and has a free “try it before you buy it” trial download available at that site, as well. I also used Geometer’s Sketchpad and MS-Paint to produce the flat purple-and-black image found on faces near the top of this post (and, by itself, in the previous post on this blog), but I know of nowhere to get free trial downloads of these latter two programs.
Software credit: see http://www.software3d.com/stella.php
This is the dual of the figure shown in the last post.
Software used: see http://www.software3d.com/Stella.php.
Alternate names for components of this compound are the rhombicuboctahedron and the kited icositetrahedron.
For more information on the software used to produce this image, please visit http://www.software3d.com/stella.php.
Created with Stella 4d (site to try it: http://www.software3d.com/stella.php).
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