Right! Great! I’ll say something about this on part 4, where we are talking about a formula that classifies these planar tilings.

By the way, it’s ‘Coxeter’, not ‘Coexter’. Harold Scott MacDonald Coxeter, who discovered these diagrams, was called ‘the king of geometry’. His book *Regular Polytopes* is really great.

I saw that you posted part 4 of the Symmetry and The Fourth Dimension series and read it already and hopefully will get to it shortly. Nevertheless, I wanted to answer your second puzzle as well, for the coexter diagrams of the square tiling and hexagonal tiling.

For the square tiling, the coexter diagram is: V-4-E-4-F, whereas the coexter diagram for the hexagonal tiling is: V-6-E-3-F.

]]>Yes, great! The Coxeter diagram for the octahedron is

**V—3—E—4—F**

because it has 3 vertices and edges around each face, and 4 edges and faces around each vertex:

You write:

Also, the Coxeter diagrams of the cube and octahedron seem to be the “reverse” or “inverse” of each other with regard to the ordering between V, E, and F, whereas the dodecahedron and icosahedron Coxeter diagrams indicate that they are the “reverse” or “inverse” of each other, which makes sense.

That’s right, and that’s one of the most exciting patterns in the Platonic solids, made vividly visible by their Coxeter diagrams.

The usual term is ‘dual’: if we draw a dot in the center of each face of a Platonic solid, these dots are the vertices of another Platonic solid, its **dual**. The dual of the dual is the original solid (though smaller, the way I’ve described it now).

So:

• The dual of the tetrahedron is itself.

• The dual of the cube is the octahedron.

• The dual of the dodecahedron is the icosahedron.

It’ll be fun to exploit this fact in future posts, and also to see how it generalizes to 4 dimensions.

]]>Thank you for posting such interesting things on your blog. I started following your blog a little while ago (about a couple of months, I think) and really find the topics covered very fascinating–thanks for your great work!

I did find my mistake, indeed. It is for the Coxeter diagram of the octahedron, which has a Coxeter diagram of V-3-E-4-F.

All the other Coxeter diagrams for the other Platonic solids remain the same as in my previous comment, after a second look at them.

I noticed that for each of the Platonic solids, the relation

holds for each of them.

Also, the Coxeter diagrams of the cube and octahedron seem to be the ”reverse” or ”inverse” of each other with regard to the ordering between V, E, and F, whereas the dodecahedron and icosahedron Coxeter diagrams indicate that they are the “reverse” or ”inverse” of each other, which makes sense. For the cube and octahedron cases, the ”base” that unites the two pyramids of the octahedron is a square, which constitutes the base of a cube.

Hopefully, this is more complete and correct in evaluation than my previous attempt. I still have yet to figure out the second puzzle, but look forward to it!

]]>Hint: the way the spinning solids are arranged into rows is actually significant!

]]>Hi! Good work! I’m glad you enjoyed this challenge. You made one little mistake, which affects some of your conclusions. Maybe you can spot it, or someone else here can. The picture at the very top of this post is a clue.

]]>John Baez wrote:

Getting nice non-rotating pictures of these solids from Wikipedia might help.

That might be more effective than the method I used, which was to look at the spinning polyhedra and rotate my chair.

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