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.