# Möbius Band non-orientable? (demonstrative)

I heard an easy way to see if a connected 2-dimensional manifold is non-orientable is applying local rotations (clockwise) in a consistant way. And if you encounter an anti-clockwise rotation the manifold can't be orientable. I tried to apply this on the Möbius Band but i haven't encountered ani anti-clockwise rotations, everything is consistent. This shouldn't be the case. And i'm 99% sure i have a proper Möbius Band. Is this method wrong in this case or am i doing something wrong here?

Edit: I guess i found my mistake, i locally treated the paper's front- and backside different. If i just follow the band and apply local rotations - of course i'll met the same rotation at some point. I just had to compare the consistency of a rotation on the frontside with the rotaton on the backside. Well...

-
You must be doing something wrong. What, we can't tell without seeing what you did and have. –  Daniel Fischer Sep 26 '13 at 15:45
Your edit explains things correctly. I've seen the following idea mentioned as well. Imagine you have a hole puncher that punches out an asymmetric shape of some kind. If you keep applying this hole punch as you work your way around the Möbius strip, eventually the hole you cut out meets its mirror image. My wife made a Möbius quilt that illustrates this idea. –  Grumpy Parsnip Sep 26 '13 at 19:14

Draw the Möbius band as a rectangle with the usual identifications on the top and bottom edge and triangulate the space by first splitting the rectangle in to two squares, one on top of the other, and then splitting these squares diagonally in to right angled triangles.

Put an orientation on one of the 2-simplices. It will induce an orientation on all of the other simplices except for one, as the last simplex will have two simplices sharing an edge with it, and their induced orientations will not coincide. (You should draw this to check).

-