The Optimal Twisted Paper Cylinder

Abstract

An embedded twisted paper cylinder of aspect ratio $\hat{I}»$ is a smooth isometric embedding of a flat $\hat{I}»\times 1$ cylinder into ${\mathbb{R}}^3$ such that the images of the boundary components are linked. We prove that for such an object to exist we must have $\hat{I}»>2$ and that this bound is sharp. We also show that any sequence of examples having aspect ratio converging to $2$ must converge to a (non-smooth) $4$-fold wrapping of a right-angled isosceles triangle.

identical to previous version except for one correction on p2 of the intro: the construction of J. Neinhaus only applies to Moebius bands