Re: This Week's Finds in Mathematical Physics (Week 241)

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.
In a 2 dimensional ordinary dodecahedron, the Euler characteristic
together with the (5,3) pattern seems to dictate the topology.
Simply by fitting together 3 pentagons at each vertex, you automatically
build a dodecahedron.
(I am not completely sure there is no way out, but I can't think of
one at the moment)
The Euler characteristic of a (5,3) pattern is F(1-5/2+5/3) = F/6

So if F=12, we get Euler=2. If F-24, we have 2 disjoint dodecahedra.
It might be fun to think if we can build anything else, satisfying the
(5,3) restriction.

The generalization of this in 4D is C-F+E-V. This is zero for the
(5,3,3) pattern, regardless of the number of cells. On the one hand,
this suggests that you could topologically stack 3-space with
dodecahedra. But the "wriggle room" is a bit confusing, because in
2D the Euler characteristic corresponds to the total curvature. In
3D, things seem to work differently, but I don't understand how.

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.

Gerard

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