The shape of a Möbius strip

The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180 ∘ , 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 in refs 1,2 . 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 3 and paper crumpling 4 , 5 . This could give new insight into energy localization phenomena in unstretchable sheets 6 , 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 7 , 8 , 9 .