On sphere-filling ropes

TL;DR

This paper investigates the problem of determining the longest possible rope on a sphere, constructing and classifying solutions for various thicknesses, revealing patterns similar to natural and physical phenomena.

Contribution

It introduces a comprehensive classification of solution curves for the sphere-filling rope problem across multiple thickness values, connecting geometric solutions to natural patterns.

Findings

Constructed solutions for infinitely many thickness values.

Identified solution patterns resembling tennis ball seams, Turing patterns, and elastic rod arrangements.

Provided a geometric framework linking rope packing to natural pattern formations.

Abstract

What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others exhibit a striking resemblance to Turing patterns in chemistry, or to ordered phases of long elastic rods stuffed into spherical shells.