On Itaumiped, my imaginary planet (see previous post, as well as a few others on this blog), the snow is not crystals of dihydrogen monoxide, but a different chemical — one that shows both 4-fold and 5-fold, but not 6-fold, symmetry.
This may change in the future, however. That’s one of the advantages of having your own imaginary planet.
Itaumiped is my own imaginary planet. I made it up, naming it with an anagram for — what else? — the phrase “I made it up.”
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.
RobertLovesPi.net 