The Platonic solids, rated

[drdoug]

Platonic solids are always presented in order of number of faces, which seems terribly unfair - it’s the mathematical equivalent of going in alphabetical order of surname - so I randomly permuted them to give the order here. Which isn’t addressing millennia of discrimination, of course, but doing them in reverse order seemed wrong too

Comments

[simont]

I ran a poll on this very subject a few years back. The audience agreed with you on the overall winner, but I think they were more influenced than you were by the cube's overexposure, and the tetrahedron got surprisingly little love too.

A couple of people took the interesting approach of also considering the aesthetic appeal of the solid's unfolded net, which generally caused them to give extra points to the dodecahedron. (They didn't say which of the possible net layouts they thought was exceptionally lovely, but I'm guessing it was the fairly standard one of 'pentagon, with a pentagon on each edge, now do that again and join the two pieces side by side'.)

And drswirly argued persuasively in favour of the octahedron deserving a higher score, on the basis that it looks so startlingly different from multiple points of view - with a vertex facing you it's wide and chunky, and yet with an edge facing you it becomes surprisingly thin - and also because it can be regarded as a triangular antiprism, which is perhaps surprising if you'd previously been starting from a 'dual of the cube' sort of headspace and hence given mental primacy to the vertices. But switch to a face-oriented point of view, designating one face as the bottom and one as the top, and suddenly it's a member of an apparently unrelated family of solids.

[drswirly]

I'm glad you thought it was persuasive. My actual argument from 2010 was the following.

The tetrahedron is too dull. I mean, self-dual? What's the point of that?
The cube is very functional and useful, but not really exciting.
The icosahedron is a bit too busy, and like a sphere having a bad day.
The dodecahedron is just showing off. Really, pentagons? Smug thing.

The octahedron is pleasantly surprising. It looks all wide and square when you look at it with a vertex towards you, then tall and thin with an edge towards you. And when placed flat on a table, it taunts and teases with its antiprismness, the cheeky little thing.

[drdoug]

Good points. If I were running the ratings again I'd probably bump it up to 9/10. But the referee's decision has to be final. It's also often overlooked - even in that 'your D12 cries itself to sleep' cartoon, the octahedron is missing. This, I think, is part of my criticism of 'lack of vavoom'. Empirically the octahedron is not as memorable as the others.

[drdoug]

Oh, that's a great post, thanks.

I like the idea of thinking about the net, but I really don't care much for the dodecahedron's nets. Even the one you mention. They look like an unpromising mess of pentagons. It's only when the thing is assembled that the lovely symmetry is there for my money.

Now, the icosahedron's nets, on the other hand, are fantastic.