## The shape of a Möbius strip

E.L. Starostin & G.H.M. van der Heijden

The Möbius strip, obtained by taking a rectangular strip of plastic or paper,
twisting one end through 180 degrees, and then joining the ends, is the
canonical example of a one-sided surface. Finding its characteristic
developable shape has been an open problem ever since its first formulation
by Sadowksy in 1930. Here we use the invariant variational bicomplex
formalism to derive the first equilibrium equations for a wide developable
strip undergoing large deformations, thereby giving the first non-trivial
demonstration of the potential of this approach. We then formulate the
boundary-value problem for the Möbius strip and solve it numerically.
Solutions for increasing width show the formation of creases bounding nearly
flat triangular regions, a feature also familiar from fabric draping and
paper crumpling. This could give new insight into energy localisation
phenomena in unstretchable sheets, which might help to predict points of
onset of tearing. It could also aid our understanding of the relationship
between geometry and physical properties of nano- and microscopic Möbius
strip structures.

*Nature Materials* **6**, 563-567 (2007)