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Re: This Week's Finds in Mathematical Physics (Week 241)
Posted:
Nov 24, 2006 8:13 AM
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A reply to Gerard Westendorp, but first some errata: I misspelled
Joe Giaime's name, and the final E7 should have been an E6:
the Dynkin diagram of E6:
o----o----o----o----o
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o
In article <4563AA9F.8080905@xs4all.nl>,
Gerard Westendorp <westy31@xs4all.nl> wrote:
>John Baez wrote:
>> I then went on to discuss the 120-cell, which gives a way of chopping
>> a spherical universe into 120 dodecahedra. This leads naturally to
>> the Poincare homology sphere, a closely related 3-dimensional manifold
>> made by gluing together opposite sides of *one* dodecahedron.
>I am a bit puzzled by the topology of this.
To get the Poincare homology sphere, take a dodecahedron, and
identify each point on any face with a point on the opposite face,
in the simplest possible way. More precisely, identify each face
with the opposite face after giving it a clockwise 1/10 turn!
(Or, if you prefer, a counterclockwise 1/10 turn - but be consistent.)
If you look, you'll see that a 1/10 turn (36 degrees) is the
smallest amount of turning that can work.
When you're done, you'll see that four edges and four faces meet
at each vertex.
As for the 120-cell:
>Anyway, if I just imagine gluing together dodecahedra, I get a
>sphere that has an outer shell that is composed of an ever-increasing
>number of dodecahedra. They don't seem to come together to a close,
>like the pentagons do in a dodecahedron.
Well, they don't close until you "fold it up" into the fourth
dimension. Did you look at these pictures?
http://www.weimholt.com/andrew/120_stage1.html
You might also like these:
http://www.ams.org/featurecolumn/archive/boole.html
which show the successive layers more systematically:
1 + 12 + 20 + 12 + 30 + 12 + 20 + 12 + 1 = 120
although they actually just go to the halfway-point:
1 + 12 + 20 + 12 + 30
which gives approximately the "top half" of the 120-cell.
Also look at this:
http://www.georgehart.com/hyperspace/hart-120-cell.html
Since I'm posting to sci.physics.research, I should also recommend
Brett McInnes' paper on the instability of the Poincare
3-sphere in the context of brane-world cosmology:
http://arxiv.org/abs/hep-th/0401035
As is well known, classical General Relativity does not constrain the
topology of the spatial sections of our Universe. However, the Brane-
World approach to cosmology might be expected to do so, since in general
any modification of the topology of the brane must be reflected in some
modification of that of the bulk. Assuming the truth of the Adams-
Polchinski-Silverstein conjecture on the instability of non-supersymmetric
AdS orbifolds, evidence for which has recently been accumulating, we
argue that indeed many possible topologies for accelerating universes
can be ruled out because they lead to non-perturbative instabilities.
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