I'm working on a project about the differences between the original Möbius strip, a strip with an additional even number of half-twists, and a strip with an additional odd number of half-twists. This is a small project and I don't have much knowledge of topology.
Here are some statements about these objects that I would like to prove.
- In every case, the Euler characteristic is 0.
- In the case of the Möbius strip and the odd number of half twists, neither is orientable. But in the even case, since it has 2 sides, it is orientable.
- This one is the most confusing: which surfaces are homeomorphic to the Möbius strip? My teacher said the Möbius strip is a unique surface and it's only homeomorphic to itself, but I read on the internet that it is homeomorphic to a square.
If it really is only homeomorphic to itself, then there is no doubt that a ring with an additional odd number of twists is also homeomorphic to a Möbius strip.
I also think that a ring with an even number of twists is equaled to a torus since they both have 2 edges and 2 surfaces, on the same dimensional.
I would be happy to hear about more interesting mathematical elements that I could use...