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Message from discussion This Week's Finds in Mathematical Physics (Week 182)

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More options Jul 2 2002, 6:52 pm
Newsgroups: sci.physics.research, sci.physics, sci.math
From: b...@galaxy.ucr.edu (John Baez)
Date: Tue, 2 Jul 2002 22:52:47 +0000 (UTC)
Local: Tues, Jul 2 2002 6:52 pm
Subject: Re: This Week's Finds in Mathematical Physics (Week 182)

Jeffery <jeffery_wink...@hotmail.com> wrote:
>Is the dihedral group D_n the same D_n group as in the ABCDEFGHI
>series?

No; as Chris Hillman explained this is just a notational
coincidence - though "week182" hints at a subtle relation between
the two.

>In this post, you imply that the dodecahedron and icosahedron
>correspond to the E_8 group, although elsewhere you say they
>correspond to the H_3 group.

I should have been a bit clearer.

First of all, there's a more or less straightforward classification
of finite reflection groups in terms of ABCDEFGHI Coxeter diagrams.
Among the simplest examples of finite reflection groups are the
reflection/rotation symmetry groups of the Platonic solids:

The symmetry group of the tetrahedron is called A_3.
The symmetry group of the cube or octahedron is called B_3.
The symmetry group of the dodecahedron or icosahedron is called H_3.

The group A_3 is part of an infinite series of A_n groups
since the tetrahedron has analogues in every dimension n.

The group B_3 is part of an infinite series of B_n groups
since the cube and octahedron have analogues in every dimension n.

The group H_3 is not part of an infinite series of H_n groups
since the dodecahedron and icosahedron do not have analogues
in every dimension n...

... however, they do have analogues in dimension 4, whose symmetry
group is called H_4!  The buck stops in 4 dimensions if you're
looking for regular polytopes with 5-fold symmetry.

For more on the hyperdodecahedron, the hypericosahedron, and
other Platonic delights, try:

Anyway, in "week182" I was describing a *different* and rather
*bizarre* relationship between Platonic solids and the Coxeter-Dynkin
diagrams of type E.  As I mentioned, this extends to a relationship
between finite subgroups of SO(3) and Coxeter-Dynkin diagrams of
types A, D, and E.  This is sometimes called the "McKay correspondence" -
though as Minhyong Kim recently pointed out here on sci.physics.research,
there are aspects involving singularity theory which were understood
before the work of McKay.

In "week182" I was trying to boil this stuff down to its simplest
essence, so that even ancient Egyptians could understand it.
Unfortunately I forgot to say that's what I was doing!  I'll fix that.

Platonic solids here:

and McKay talks about it here:

The most detailed online explanation is probably this:

Joris van Hoboken, Platonic solids, binary polyhedral groups,
Kleinian singularities and Lie algebras of type A,D,E,
Master's Thesis, University of Amsterdam, 2002, available at
http://www.science.uva.nl/research/math/examen/2002/scriptiejoris.ps