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Various proofs: pairs of points in a circle = Möbius strip But removing a disk from RP2 yields the Möbius strip, as desired (See Sarkaria's note for a nice illustration of this fact ) Map a pair of points {p, q} on the unit circle to their midpoint (p + q) 2 This is a bijection between the punctured disk D2 −0 and pairs {p, q} of non-antipodal points, but every pair of antipodal points gets mapped
general topology - Why is the Möbius strip not orientable . . . The Möbius strip M^ M ^ inherits the differentiable structure from R2 R 2 We have to prove that M^ M ^ does not admit an atlas of the described kind which is compatible with the differentiable structure on M^ M ^
Understanding the Equation of a Möbius Strip Is there a reason you need an equation for the Mobius band? In terms of topology, there are much much easier ways of defining the Mobius band, in terms of gluing ends of a strip together
differential geometry - A proof for the Mobius Strip parametrization . . . According to Elementary Differential Geometry by A N Pressley, a parameterization for Mobius strip is : Example 4 9 The Möbius band is the surface obtained by rotating a straight line segment L around its midpoint P at the same time as P moves around a circle C, in such a way that as P moves once around C, L makes a half-turn about P If we take C to be the circle x2 + y2 = 1 in the xy -plane
How does one parameterize the surface formed by a *real paper* Möbius . . . The Möbius strip you show is a developable surface No one, as far as I know, has been able to create a parametrization of it Since 1858, when the Möbius strip was discovered, mathematicians have been looking for a way to model it The problem was finally solved in 2007 by E L Starostin and G H M van der Heijden
Real projective plane and Möbius strip - Mathematics Stack Exchange The natural way to show this would be to explicitly write down the embedding For example, the Möbius strip can map to a thin belt around the equator of the sphere Then the relation between this belt (in spherical coordinates!) and your defining parameterization of the Möbius strip is a simple affine transformation of each coordinate separately Remember that [0, 1] should map to only one
Möbius strip covering space - Mathematics Stack Exchange how can we describe the universal covering space of the Möbius strip? the Möbius strip is a square $[0,1]\\times [0,1]$ with identifications $(0,y)\\sim (1,1-y)$ So my guess is that the universal