Seqram (seqram) wrote,

Tetrakis Hexahedron

So I found myself revisiting some geometry obsessions lately (background: at my brother's house for Passover his children and mine began to be interested on playing Dungeons & Dragons, which led to talk about polyhedral dice, and so forth), including modular origami construction of interesting polyhedra. I dragged out my old copy of Paperfolding Stellated Polyhedra by Jeannine Mosely to look up how to make a rhombic dodecahedron etc. (Jeanine, if you are out there and by some bizarre chance ever read this, a friend gave me a copy of some paper or another you wrote with that title back when I was in high school; I don't know if you ever meant for it to be seen). I thought about making other die-like polyhedra, some of the other Catalan Solids, like, say, a tetrakis hexahedron, sometimes used as a 24-sided die. I've never designed one of these modular shapes before, only followed other peoples' directions for them. But I read and i reckoned and I pondered and actually designed the thing—and it worked! See picture...

(the smaller one is the first one I made; I worked out a simpler way to approximate the ratios needed and made the larger one.)

I record this accomplishment here for posterity so at least I don't forget I did it. I wrote down the instructions and everything too so maybe it could be repeated.  I suppose for some people working this out would be simple, but what I think is cool is how little I started with. I had done none of the calculations to find shortcuts and formulas for all this, and besides, it isn't like my 3D geometry is even all that good. I remember a few simple formulæ from plane geometry and trigonometry and that's about all I have to go on; I re-derive everything when  I need it. I couldn't even find good data on the shapes of the triangles etc (actually, MathWorld did tell me the edge lengths, but I didn't even look at them so I didn't realize how easy it was to derive their very simple 4:3 ratio); I derived everything from only the description of the shape as a cube with each face replaced with a pyramid of height ¼s.  Hey, I thought it was cool.

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