What are the Longest Ropes on the Unit Sphere?

Abstract

We consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere. After establishing the existence of solutions for any given thickness between 0 and 1, we explicitly construct for each given thickness $${\Theta_n:= {\rm sin}\, \pi/(2n),}$$ $${n\in\mathbb{N}}$$, exactly $${\varphi(n)}$$ solutions, where $${\varphi}$$ is Euler's totient function from number theory. Then we prove that these solutions are unique, and also provide a complete characterisation of sphere filling curves on the unit sphere; that is of those curves whose spherical tubular neighbourhood completely covers the surface area of the unit sphere exactly once. All of these results carry over to open curves as well, as indicated in the last sectio