The five regular or “Platonic” solids are abstract shapes, but they sometimes take material forms.

Anthony Barreiro, after seeing the cover picture of *Astronomical Calendar 2015* with its stellated dodecahedrons, happened to revisit his childhood home in Castro Valley, California, and saw hanging on the back porch three lamps his father had bought long ago in Tijuana, Mexico. He sent me this photo of one of them: a stellated dodecahedron.

Dice are cubes, so that when thrown they settle with one of six numbers of spots on top – but they can be other polyhedrons. The more faces they have, the closer to a sphere and the easier they’ll roll. Twice as many numbers are given by a dodecahedral die. (That’s the correct but often forgotten singular of “dice.” It comes down through Old French *dé* from Latin *datum*, “given.”)

The dodecahedron, with its twelve five-edged sides meeting three at a time at twenty vertices, has an overall shape that feels to me more five-ish than twelve-ish (or twenty-ish or three-ish). A contentment resides in it. We are five-fingered and five is at the root of our own being. It has calm mystery; it is a nut, impregnably fortified yet pregnant with growth. It is almost annoyingly subtle and difficult to define. Can you see how to find the “latitude” of one of the points in the row around its middle – the angle to it from the dodecahedron’s “north pole”? It’s nothing simple, like a third or a quarter or a fifth of the way around; you have to travel a mixture of edge-lengths and pentagon-widths. It turns out to be 90 minus arccos (inner radius / outer radius); and the inner radius is (squareroot (75 + 30 * squareroot (5)) / 15) * outer radius; and the outer radius is edgelength * (squareroot (3) / 4) * (1 + squareroot(5)). Maybe you know of an easier way.

This is the template for making a dodecahedron – trace on card, fold along the heavy lines, glue the flaps.

It looks like two humanoid shapes about to become clasped together, reminding us of the comic explanation of love by Aristophanes (in Plato’s *Symposium*): humans are originally spherical beings, who having been chopped in half go forever seeking their other halves.

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All this business of dodecahedrons sent me back to Ian McEwan’s short story ‘Solid Geometry’, in which a brilliant young mathematician invents a ‘plane without a surface’. He folds and cuts a sheet of paper in a certain way, and when he pulls the paper through the incision it disappears. Then, to convince an audience deriding this apparent conjuring trick, he adopts a contorted posture and somehow disappears by crawling through a ‘hoop’ made by his own arms. Perhaps I should let you guys get on with your polywhatnots, as long as you avoid that kind of experiment.

A Borges-like story. A plane without a surface is impossible to conceive, and so, you might think, is a plane with only one surface, but you can make that easily with a strip of paper. You twist one end of it 180 degrees and tape it to the other, and it is called a Mobius strip or cylinder. A line drawn along it arrives on the opposite side. I once wrote a story about that (it too is somewhere in my mislaid “Among the Shapes”) with two two-dimensional beings who live in this Flatland, called Stabilis and Mobilis. Stabilis stays where he is; Mobilis sets off on a journey all the way around the world, and arrives back with his heart seeming (to Stabilis) to be on the other side of his body.

Guy,

I didn’t notice any polyhedrons or orreries, but some of these images provided by Google of what’s being done with 3D printers amaze me and trigger many ideas:

https://www.google.com/search? q=3d+printed+objects&espv=2&biw=1138&bih=511&site=webhp&tbm=isch&tbo=u&source=univ&sa=X&ei=0QMCVbiiLsL5yQTN7IKQDw&sqi=2&ved=0CB0QsAQ

With your expertise in representing 3D objects in 2D, your artistic skills and good judgement, your computer and algorithmic skills, etc, I wonder if this 3D technology might be of interest. I think, for example, of “Kepler’s polyhedral cosmos” at the end of page 34 of Astronomical Calendar 2015. Or maybe there’s a way to produce a better planisphere using this technology (at least as a prototype). (My overall favorite planisphere continues to be the one available from Edmond, but I can think of improvements..)

BTW, the original cover of AC2015 was good enough for me, but thanks for the improved version you blogged to us recently. And thanks for sharing some details of your travails in learning to automate the production of more accurate polyhedron images!

BTW2, the above template you provide for constructing a dodecahedron is very cool, and your related observations about the relevance of the number 5 to humans are worthy of some meditation..

To make James’s link work, delete the space between “search?” and “q”. In other words, copy the whole of his text from “http” to the end of the paragraph, paste it into the browser address box, and delete that space.

I can sort of imagine how 3-D printing works, though I assume it’s done with machinery. I don’t know how 3-D animations are programmed, such as the one you can find somewhere of the two halves of the dodecahedron actually rotating into their clasp. I imagine it could be used to animate the construction of Kepler’s cosmos of polyhedrons enclosing each other. Something else to learn if there were infinite time after doing enough more of the learning of WordPress, Photoshop, Musescore, etc., etc.

I will say more about “five… at the root of our being” if I can find a mass of notes I once made (before computer) for a project called “Among the Shapes”.

I’ve learned so much about polyhedra this year! And the dodecahedron is my favorite. Five is certainly the first number that jumps out at me when looking at a dodecahedron.

By the way, the picture of the lamp should be rotated 90 degrees clockwise. It’s not a great picture to start with, a quick snap with my ipad with the Sun backlighting the lamp (as you said, “without thinking” — no ray tracing needed!).

Rotated!