Why is the möbius strip not orientable, in the sense that there cannot exist a globally defined continuous surface normal?

Printable View

- June 10th 2010, 12:10 PMBanachmöbius strip
Why is the möbius strip not orientable, in the sense that there cannot exist a globally defined continuous surface normal?

- June 10th 2010, 12:20 PMBruno J.
This is demonstrated by triangulating the surface, and showing that the triangulation is not orientable.

- June 10th 2010, 12:42 PMBanach
Thanks for your answer! Unfortunately i do not know what a triangulation is. But this should work:

Let us pick a continuous curve c on the strip. If we pick a normal vector at c(0) it will point in the other direction at c(2pi). This is geometrically obvious and we would have a contradiction. But how can i show that formally? - June 10th 2010, 01:01 PMBruno J.
The only formal proof that I know of uses the theory of simplical complexes!