Megachiropteran cinematographer: marchantiaceous centauromachias. Bedroom boredom? Brighten berthing! << 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. * * * 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