On Sep 6, 2015, at 11:00 AM, Hilarie Orman <ho@alum.mit.edu> wrote:
Really nice Wikipedia article about orbifolds.
The Moebius band is the underlying 2D-manifold for one of the 17 "parabolic 2D orbifolds". The Moebius band is, of course, one of the 17 wallpaper groups.
I didn't know that about the Moebius band (which is my favorite band), and it was fun figuring out which wallpaper group that is. Some more cheerful facts about the Moebius band are * It is the configuration space of all lines in the plane (not necessarily through the origin). * The open Moebius band can be given a complete metric* of constant curvature 0 or one of constant curvature -1, but not one of constant curvature +1. * There is a cubic polynomial P(x,y,z) such that (a portion of) the locus P(x,y,z) = 0 defines a Moebius band in R^3. (I mentioned this here a while ago.) * The Moebius band is (part of) a minimal surface in the 3-sphere S^3 with its standard "round" metric. —Dan _________________________________________________________________________ * A metric is complete when every geodesic can be extended indefinitely. For example, every compact (smooth) surface with a smooth metric on it.