What a marvellous thing is the stellated dodecahedron!

Miscellaneous and Personal Stuff I Must Share
1

About two months ago my wife and I bought a little decorative lantern, the kind you put a votive candle into, and it’s in the shape of a stellated dodecahedron, that is,
a twelve-sided polyhedron, with pentagonal faces, except that each face is covered by a five-sided pyramid.

So I was looking at it, and figuring out how many separate
faces it has, and I came up with 60. The underlying dodecahedron has 12 faces with 5 sides each. The whole polyhedron could be constructed by an intersection of five five-pointed stars. The outline of the stellated dodecahedron, seen from across the room, looks like a 6-sided Star of David, which itself, is composed of two
triangles.

So in this one object we have all these numbers represented:

2,3,5,6,12,60–all of which are factors of 60. This is in
itself interesting, but I’m thinking that somewhere one ought to be able to find a manifestation of the other two
factors 15 and 30.

Maybe this is a General Question?

2

The dodecahedron (non-stellated) has twenty vertices, giving you another factor.

One that you omitted, I might add.

Fifteen and thirty are going to be hard to find because they would require quadrilateral and bilateral symmetry, respectively, which a dodecahedron has not got.

There are a lot of polyhedra websites out there. Take a minute to search some out, if you haven’t already. And, if you have access to the math program Mathematica, it has some nice polyhedron packages.

3

Do a search on hexahexaflexagons. They’re pretty strange and wonderful. And they’re pretty, strange, and wonderful.

4

Oh, man. I thought this was a thread on D&D dice.