Möbius bands, unstretchable material sheets and developable surfaces

A Möbius band can be formed by bending a sufficiently long rectangular unstretchable material sheet and joining the two short ends after twisting by 180°. This process can be modelled by an isometric mapping from a rectangular region to a developable surface in three-dimensional Euclidean space. Attempts have been made to determine the equilibrium shape of a Möbius band by minimizing the bending energy in the class of mappings from the rectangular region to the collection of developable surfaces. In this work, we show that, although a surface obtained from an isometric mapping of a prescribed planar region must be developable, a mapping from a prescribed planar region to a developable surface is not necessarily isometric. Based on this, we demonstrate that the notion of a rectifying developable cannot be used to describe a pure bending of a rectangular region into a Möbius band or a generic ribbon, as has been erroneously done in many publications. Specifically, our analysis shows that the mapping from a prescribed planar region to a rectifying developable surface is isometric only if that surface is cylindrical with the midline being the generator. Towards providing solutions to this issue, we discuss several alternative modelling strategies that respect the distinction between the physical constraint of unstretchability and the geometrical notion of developability.