Some geometry with big names

I bought Tony Wright’s GF last fall but didn’t have a chance to use it due to a bit of spousal paranoia (plus I could just use the one at work), so I’m happy to report that it survived the move to Austin and is running great. No smoke in the house at all, wife is happy.

I finally got a chance to try something I’ve been thinking about for a while. Previously I’d made an icosahedron and a dodecahedron (20- and 12-sided polyhedra), and I was curious how they’d work as spheres by rounding the edges. Then it hit me that since they’re duals of each other (one has its vertices where the other’s faces are and vice-versa), the edges are perpendicular. That means we can make a composite of both shapes, with the edges notched and slotted together. After a whole lot of futzing around trying to remember how to Inkscape then only about ten minutes cutting but then a lot of figuring out how to put it together, here it is!

It’s a deltoidal hexecontahedron, says Wikipedia, 60 kite-shaped faces. It uses 30 each of the short and long edge pieces, 20 triangle connectors, and 12 of the pentagonal connectors. I used Woodpeckers 3 mm Baltic Birch, so the slots are tuned to that, 2.9 mm at the mouth and 2.85 mm at the back. I did my previous shapes in Proofgrade Maple and 3.0 mm slots gave a good fit.

sphere

Here are the component shapes built separately:

I originally tried to put this together by building the icosahedron first then the dodecahedron on top of it and snugging it together but I couldn’t get the edge pieces flush with each other. I need some kind of L-shaped thing I could poke through to pull from the other side…? Can’t think of what I could use for that. So instead I paired up the long and short edge pieces first then built up the shape as loosely as possible and snugged it all together once everything was in place, a bit at a time all the way around. If you tighten it up as you build it’s impossible to get the last pieces in.

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Playing around with the pieces I also found you can tile the sphere in a slightly different way, with just short edges (after running off 30 more of them):

This one’s called a rhombic triacontahedron, 30 diamond-shaped faces. The triangular and pentagonal connectors are arranged in the same way and the edges go between them without any crossing. The short edge piece isn’t exactly the right angle–should bend by 36 degrees instead of 44, if I recall?–but it’s close enough it works.

Mathematical!

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There’s a chance you might be totally off your rocker - but you make cool things!!

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Nice work, I like your doubles.

Previously, you might be interested in:

and

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These are awesome. Very cool. Jealous of you guys who can figure these out with only using the laser cutter. All mine use 3D printed internals.

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This is awesome! Also, don’t let @evansd2 fool you; he’s the one who got me started on polyhedra!

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Very neat! Reminds me a little of these little circles with triangular folds inscribed on them that we used to glue together into shapes.

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You could make your connectors longer, and then add a triple of 60 deltoids, and why stop there? Chances are you could keep making shapes that fit the voids and your model to get larger and larger.

If my GF weren’t in Seattle at the moment I’d probably have already tried it.

You might like my edge woven poly:

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That is awesome

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Very cool!

For completeness, here’s an edge piece for the rhombic thingahedron so it’s the same size (~88mm in diameter) as the other one:
rhombedge

It’s not symmetric because the pentagon side doesn’t go as far as the triangle, so the sides are marked.

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