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

[wildeabandon]

This is a lovely post :)

I find the self-dual nature of the tetrahedron to be rather charming myself, but I always have had a soft spot for narcissistic types.

[drdoug]

Thank you! I enjoyed writing it.

I see what you mean about the tetrahedron's self-duality. I'd been seeing it as a bit unsubtle and lacking in interesting complexity. But looked at from a different angle (sorry), it's straightforward and clear, and also admirably self-contained and independent.

I think I would struggle to defend my ratings on a strictly objective scale ... in fact I think I would struggle to reproduce them myself on a separate occasion.

[wildeabandon]

Heh, yes, I'm actually slightly surprised to discover that my preference for the tetrahedron has been sustained since 2010, rather than being a passing whim.

[thekumquat]

The Dodecahedron in the Phantom Tollbooth is one of my favourite literary characters - none of the other solids have that. Possibly cubes as the "square sweets that look round" in Charlie and the Chocolate Factory.

[drdoug]

Oh yes! "My faces are many, My sides are not few, I'm the Dodecahedron, Who are you?" Or is it his angles that are many? Either way, he's great, thanks for the reminder.

Are there no other talking cubes in literature (broadly construed)? There must surely be, but none are leaping to mind.

I'm pretty sure all the Platonic solids have non-speaking roles in fiction - cubes and cuboids all over the places, but the others crop up in SF(&F) from time to time. Although they may sometimes serve as MacGuffins, I can't think of any that would count as characters.

[wildeabandon]

Do any polyhedra show up in Flatland? It's so long since I read it.

[simont]

Searching the ebook: just about. The main three-dimensional character is a Sphere, but a Cube (referred to as a Being and as 'he', so presumably sentient rather than architectural) makes a cameo appearance at one point:He [the Sphere] then introduced me to the Cube, and I found that this marvellous Being was indeed no Plane, but a Solid; and that he was endowed with six plane sides and eight terminal points called solid angles; and I remembered the saying of the Sphere that just such a Creature as this would be formed by a Square moving, in Space, parallel to himself: and I rejoiced to think that so insignificant a Creature as I could in some sense be called the Progenitor of so illustrious an offspring.
But then the narrator apparently just wanders off, so the Cube never quite gets a speaking part.

[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

[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.

[drdoug]

Interesting to look at Google Trends

[babysimon]

[drdoug]

Got to love the teapotahedron

[babysimon]

Gosh, you actually had something more intelligent to say than I did on the subject. Thanks for sharing!