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I have shown that the Gaussian curvature of the Möbius band $$\sigma (t, \theta )=\left (\left (1-t\sin \frac{\theta}{2}\right)\cos \theta, \left (1-t\sin \frac{\theta}{2}\right )\sin \theta, t\cos \frac{\theta}{2}\right )$$ is equal to $-\frac{1}{4}$ everywhere along its median circle.

How can we deduce that this Möbius band cannot be constructed by taking a strip of paper and joining the ends together with a half-twist?

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A strip of paper has Gaussian curvature $0$. Since twisting a surface doesn't change its Gaussian curvature, you can't twist a strip of paper into a mobius band with Gaussian curvature $-1/4$.

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  • $\begingroup$ Is it known that a strip of paper has Gaussian curvature $0$? Or do we have to prove it? $\endgroup$
    – Mary Star
    Jan 11 '16 at 0:49
  • $\begingroup$ It's a flat strip of paper. $\endgroup$ Jan 11 '16 at 0:53
  • $\begingroup$ What is flat surface? Why is a strip of paper flat? $\endgroup$
    – Mary Star
    Jan 11 '16 at 10:14
  • $\begingroup$ The fact that twisting a surface doesn't change its Gaussian curvature follows from the Gauss’ Theorema Egregium, according to which the Gaussian curvature of a surface is preserved by local isometries, right? $\endgroup$
    – Mary Star
    Jan 11 '16 at 10:23
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To expand on the answers provided, this Möbius band cannot be made from a rectangular (or any shaped) strip of paper since paper is a (roughly) unstretchable material. Thus any Möbius band you could make from a strip of paper would have to be an isometry a plane strip (a developable surface) since it would preserve distances. A plane strip has Gaussian curvature zero since it is a plane, and by Gauss's Theorema Egregium, isometric mappings preserve Gaussian curvature.

It is in fact not true that a parametrization of a flat Möbius band is unknown. There is the parametrization due to Sadowsky which proved the existence of such a parametrization as early as 1930, and Wunderlich found a closed form parametrization in 1962 (the texts of both authors are in German, but there are translations in the Journal of Elasticity if you care to take a look). The shape that is unknown is the Möbius band of minimum energy i.e., the shape that is adopted if you make a paper strip and leave it to adopt its own shape. There have however been significant numerical investigations of such a shape (see Starostin and van der Heijden for example).

Here are the references for the translations:

Hinz, D. F. & Fried, E. Translation of Michael Sadowsky’s paper “An elementary proof for the existence of a developable Möbius band and the attribution of the geometric problem to a variational problem”. J. Elasticity 119, 3–6 (2015).

Hinz, D. F. & Fried, E. Translation and interpretation of Michael Sadowsky’s paper “Theory of elastically bendable inextensible bands with applications to the Möbius band”. J. Elasticity 119, 7–17 (2015).

Hinz, D. F. & Fried, E. Translation of Michael Sadowsky’s paper “The differential equations of the Möbius band”. J. Elasticity 119, 19–22 (2015).

Todres, R. Translation of W. Wunderlich’s “On a Developable Möbius Band”. J. Elasticity 119, 23–34 (2015).

And the Starostin and van der Heijden articles:

Starostin, E. L. & van der Heijden, G. H. M. The equilibrium shape of an elastic developable Möbius strip. PAMM Proc. Appl. Math. Mech. 7, 2020115–2020116 (2007).

Starostin, E. L. & van der Heijden, G. H. M. The shape of a Möbius strip. Nat. Mater. 6, 563–567 (2007).

There are several other more recent works on this topic, but these give a good background to the problem.

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The doubly curved Möbius band has Gauss curvature K negative and varies point to point. A strip of flat paper has K=0.

By virtue of Gauss Egregium theorem isometric mapping aka bending ( bendings and twistings leave K unchanged) distortion transformation is impossible between the two due to K mismatch. There is no local isometry. Match of K is a necessary condition. (Such a two-way transformation is possible between catenoid and helicoid given usually as an example with demo.)

Parametrization of flat Möbius strip leading to zero K in this manner of mapping is so far unknown.

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