The shape of a M¨obius strip E.L. Starostin 1 , G.H.M. van der Heijden 1 The M¨obius 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 character- istic developable shape has been an open problem ever since its first formulation in [22, 23]. 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 M¨obius 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 [6] and paper crumpling [28, 17]. This could give new insight into energy localisation phenomena in unstretchable sheets [5], which might help to predict points of onset of tearing. It could also aid our understanding of the rela- tionship between geometry and physical properties of nano- and microscopic M¨obius strip structures [26, 27, 11]. It is fair to say that the M¨ obius strip is one of the few icons of mathematics that have been absorbed into wider culture. It has mathematical beauty and inspired artists such as M.C. Escher [8]. In engineering, pulley belts are often used in the form of M¨ obius strips in order to wear ‘both’ sides equally. At a much smaller scale, M¨ obius strips have recently been formed in ribbon-shaped NbSe 3 crystals under certain growth 1 Centre for Nonlinear Dynamics, Department of Civil and Environmental Engineering, University College London, London WC1E 6BT, UK Figure 1: Photo of a paper M¨ obius strip of aspect ratio 2π. The strip adopts a characteristic shape. Inextensibility of the material causes the surface to be developable. Its straight generators are drawn and the colouring varies according to the bending energy density. 1