Here is a picture of a Möbius strip, made out of some thick green paper:
I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as it appears in the picture. Now before you jump up and declare, "That's easy! It's just $\left(\left[1+u \cos \frac \theta2\right]\cos \theta,\left[1+u\cos\frac \theta2\right]\sin \theta,u\sin \frac \theta2\right)$ for $u\in\left[-\frac12\!,\frac12\right]$, $\theta\in[0,2\pi)$!" Understand that this parametrization misses some features of the picture; specifically, if you draw a line down the center of the strip, you get a circle, but the one in the picture is a kidney-bean shape, and non-planar. What equations would I need to solve to get a "minimum-energy" curve of a piece of planar paper which is being topologically constrained like this? Is it even true that the surface has zero curvature? (When I "reasonably" bend a piece of paper into a smooth shape, will it have zero curvature across the entire surface, or does some of paper's resistance correspond to my imparting non-zero curvature to the surface?)
This question is thus primarily concerned with the equilibrium shapes formed by paper and paper-like objects (analogous to minimal surface theory in relation to soap-bubble models). Anyone know references for this topic?
Best Answer
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.
You might want to read their paper "The equilibrium shape of an elastic developable Möbius strip" by going to this site - http://www.ucl.ac.uk/~ucesgvd/pamm.pdf