Articles of mobius band

In how many dimensions is the full-twisted “Mobius” band isotopic to the cylinder?

There is a question on this site about the distinctions between the full-twisted Mobius band and the cylinder, but I would like to ask something different, so I start a new question. Let us call $C$ the standard cylinder embedded in $\mathbb{R}^3$, and call $F$ the full-twisted Mobius band (aka. the Mobius band with two […]

Is it possible to determine if you were on a Möbius strip?

I understand that if you were to walk on the surface of a Möbius strip you would have the same perspective as if you walked on the outer surface of a cylinder. However, would it be possible for someone to determine whether they were on a Möbius strip or cylinder.

Topologically distinguishing Mobius Strips based on the number of half-twists

We can distinguish between a (closed) Mobius strip and ‘regular’ (untwisted) strip by examining the set of points which have no neighborhood homeomorphic to a disk (intuitively, the ‘boundary’ of the strip). For a Mobius strip, this set is connected, while for an untwisted strip the set has two connected components. However, the above criteria […]

Klein-bottle and Möbius-strip together with a homeomorphism

Consider the Klein bottle (this can be done by making a quotient space). I want to give a proof of the following statement: The Klein Bottle is homeomorphic to the union of two copies of a Möbius strip joined by a homeomorphism along their boundaries. I know what such a Möbius band looks like and […]

Drawing a thickened Möbius strip in Mathematica

I would like to have Mathematica plot a “thickened Möbius strip”, i.e. a torus with square cross section that is given a one-half twist. Ideally, I would like this thickened Möbius strip to be transparent with a (non-thickened) solid Möbius strip sitting at its center; here is the best approximation I could draw by hand […]

Is there a Möbius torus?

Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in any relevant respect? [This equivalent to the well-known Möbius strip should […]

Cutting a Möbius strip down the middle

Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist–the identification of the top left corner with the bottom right is a half twist; similarly, the top right corner and bottom left identification contributes another half twist. But where […]

Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ \phi_{\beta}^{-1}))> 0$ (where defined). My question is: Using this definition of orientation, how can one prove […]