Comment on Y.-C. Chen, E. Fried, Möbius bands, unstretchable material sheets and developable surfaces. Proc. R. Soc. A , 20160459 (2016)

Chen & Fried [1] claim that our solution of the shape of a material Möbius strip (taken to be inextensible) in [2] is incorrect as it involves stretching of the material. The purported proof of this claim, the calculation in §7, is in fact the main result of the paper. The claim is elaborated upon in subsequent publications with Fosdick [3–5]. Here, we show that all these papers are based on a misunderstanding of the elastic deformation involved and what it means for such a deformation to be isometric (i.e. to preserve distances). We all agree on the problem at issue, which is to find the equilibrium shape of a Möbius band made of a thin strip of inextensible material. This shape will be given by a developable surface that can be constructed by isometrically deforming a planar rectangular reference strip into a closed structure with half a turn of twist. An arbitrary developable strip of width 2w can be parametrized as