Lectures On The Icosahedron

[damigella]

It's all the mods' fault. One of the pics in Camp sick_wilson Pic Tac Toe Challenge, which looked to me like a billiard ball, turned out to be a Magic 8 Ball. Something whose existence I was totally and blissfully unaware of.

Wikipedia informed me that inside the Magic 8 Ball there's just a simple, 20-faced, icosahedral die.

If you throw a usual, cubic die you have six faces, and six possible results. So why twenty? Good number, but could we make it with 15, or 33 faces?

The answer is no, at least if you want a honest die: in other words, if you want a so-called regular polyhedron.

When we draw on a piece of paper we can produce regular polygons with as many sides as we want: an equilateral triangle, a square, a pentagon, an exagon… after that the words become complicated (octagon and decagon I can still spell easily), and the question of how to make the drawing gets more complicated (the answer, as usual, is Google Images) but there's no limit to your fancy, and you can produce a regular polygon with 2000 sides if you want - except of course, it would like like a circle to me, and probably to you as well. Regular, by the way, means "seriously democratic": all sides look precisely the same (yes, there's a technical definition, but I don't want to write it).

Once we go to three dimensions, triangle and square have easy equivalents. The triangle becomes a tetrahedron (tetra means four, four vertices, four faces) which is a pyramid with base an equilateral triangle and whose sides are also equilateral triangles. The square becomes a cube, and I hope you all know what a cube is (although most broth cubes I know aren't cubical). It turns out that these two kinds of democratic arrangements exist in any number of dimensions - whatever that's supposed to mean.

But then things get iffy. There's only three more democratic egalitarian honest dice regular polyhedra, and they have eight, twelve, and twenty faces: they're called octahedron, dodecahedron, and icosahedron, because all of us recognize numbers much better if they're written in (Latin or) ancient Greek.

Not only long and diligent search has failed to produce other examples, but we know that there are none. I learned how to prove this from a book at age 17 (the proof is easy once you know the trick) but the fact in itself was known already in ancient Greece, and influenced Greek philosophers no end: the popular name for all the five regular polyhedra is Platonic solids, because Plato liked geometry and got as much of a kick out of these five guys as I do, and probably more.

So why I am telling you that? Just because an icosahedron is something you can easily build if you have a magnetic construction kit. It's totally easy, and looks cute. You start by assembling a triangle. It becomes equilateral naturally, all the bars have the same length. Then you build another one with a side in common. Then you build a fan, with three equilateral triangle sharing a vertex. Then you add a fourth. And then you make a pyramid with a five-sided basis, just by connecting the free corner of the first triangle in the fan with the free corner of the last. Then you keep going, repeating the same pattern, and lo and behold, when you have used 12 balls and 30 bars, you're done and you get a cute and very stable structure, with no effort on your part.



No thinking required. [If you start with a fan with three triangles, and add a bar to make a pyramid with square basis, you have half an octahedron - to make a full one, add another pyramid on the opposite side of the basis.]

Of course in principle you can build an icosahedron, or any regular polyhedron, with any material you can easily cut: I once figured out a homework assignment in solid geometry by carving a cube of pecorino cheese - once I got the solution I had a very tasty snack. A cool thing you can do is make an icosahedron of some black material, paint it white, and then cut off all the sharp vertices (the tips where the balls would be if it were a magnetic model) so that some black shows up. And you would get…







PS: The title is from a very famous book, Vorlesungen über das Ikosaheder (1884) written by a very, very important German mathematician, Felix Klein (1884). The current school system keeps your mathematical education more than a century behind reality, sorry.