Megachiropteran cinematographer: marchantiaceous centauromachias.

Bedroom boredom?  Brighten berthing!

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Oh!  -- but which direction you get rotated in invariant!  That is,
the Poincare Dodecahedron is not its own mirror image.  So such
a universe would have a large-scale handedness.  Wouldn't that
be neat...
>>

I have the impression that the Poincaré dodecahedral manifold D created by a 1/10 rotation identifying opposite faces, and the one using a -1/10 rotation, are isometric to each other -- much as a right- and a left-handed H
opf fibration of the 3-sphere are isometric.

What I'm wondering now is whether D double-covers a non-orientable manifold.

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By the way, a cool thing about D is that it was the first counterexample to something Poincaré thought he had proved -- that any 3-manifold M with trivial (i.e., zero) first homology group H_1(M) must be homeomorphic to the 3-sphere.

He discovered that the fundamental group pi_1(D), when abelianized (to obtain H_1(D)) becomes the zero group.  So Poincaré, chastened, then suggested that *maybe* if the fundamental group of a 3manifold is trivial, *then* it must be homeomorphic to the 3-sphere.  (He didn't actually couch this as a conjecture, but only as a possibility.)

--Dan