The Optimal Twisted Paper Cylinder

TL;DR

This paper investigates the geometric conditions for embedding a twisted paper cylinder into three-dimensional space, establishing a sharp aspect ratio bound and describing the limiting behavior of near-bound examples.

Contribution

It proves the necessary aspect ratio condition for the existence of embedded twisted paper cylinders and characterizes the limit of sequences approaching this bound.

Findings

Embedded twisted paper cylinders exist only if aspect ratio > 2

The aspect ratio bound of 2 is sharp

Sequences approaching the bound converge to a 4-fold wrapping of a triangle

Abstract

An embedded twisted paper cylinder of aspect ratio $\lambda$ is a smooth isometric embedding of a flat $\lambda \times 1$ cylinder into $\R^3$ such that the images of the boundary components are linked. We prove that for such an object to exist we must have $\lambda>2$ 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.