The last several posts here have been of tessellations I have made using Geometer’s Sketchpad and MS-Paint. To create this rotating polyhedron, I selected one of these tessellations (the one in the last post), and projected it onto each face of a rhombic triacontahedron, using another program called Stella 4d. You can try Stella 4d for yourself, right here, for free: http://www.software3d.com.stella.php.
Hexacontakaihexagons have 36 sides, and dodecagons, of course, have twelve. When a regular hexacontakaihexagon is surrounded by twelve regular dodecagons, in the manner shown here, adjacent dodecagons almost, but not quite, meet at vertices. The gaps between these near-concurrent vertices are so small that they cannot be seen in this diagram — a zoom-in would be required, with thinner line segments used for the sides of the regular polygons.
As a result, the yellow and purple concave polygons aren’t what they appear, at first, to be. They look like triconcave hexagons, but this is an illusion. The yellow ones, in sets of two regions that aren’t quite separate, are actually tetraconcave, equilateral dodecagons with a very narrow “waist” separating the two large halves of each of them. As for the purple ones, they appear to occur in groups of four — but each set of four is actually one polygon, with three such narrow “waists” separating four regions of near-equal area. These purple polygons are, therefore, equilateral and hexaconcave icosikaitetragons — that is, what most people would call 24-gons.
The equiangular hexagons are very nearly regular, with only tiny deviations — probably not visible here — “from equilateralness.”
Little blurbs about posts on this blog get auto-tweeted on my Twitter, @RobertLovesPi. There’s also an A.I. on Twitter, @Hexagonbot, who retweeted my last two tweets about blog-posts here, but will not be retweeting the tweet about this one.
Why is this? Simple: @Hexagonbot is programmed to retweet any tweet which contains the word “hexagon,” which was in the titles of the last two posts here (also tessellations). This tessellation has no hexagons, though, and so the @Hexagonbot will not find it worthy of attention.
I cannot explain why hexagons get their own bot on Twitter, but other polygons do not have such bots. It’s simply one of the mysteries of the Internet.
Software credit: see http://www.software3d.com/stella.php
The faces of this polyhedron include:
12 decagons
30 octagons
60 light-colored hexagons
20 dark-colored hexagons
60 isosceles trapezoids
It was made with Stella 4d, software you can try and/or buy at http://www.software3d.com/stella.php.
This is the shape of the largest zonohedron one can make with red, yellow and blue Zome (see http://www.zometool.com for more on that product for 3-d real-world polyhedron modeling). This image was made using Stella 4d, which you can find at http://www.software3d.com/stella.php.
Created using software you can try at http://www.software3d.com/stella.php.
Later edit: I found this same polyhedron on another website, one that has been online longer than my blog, so I now for, for certain, that this was not an original discovery of my own. At http://www.cgl.uwaterloo.ca/~csk/projects/symmetrohedra/, it is named the “alternate bowtie dodecahedron” by Craig Kaplan and George W. Hart.
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