On Sphere-Filling Ropes

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 seam lines 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.

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Notes on contributors

Henryk Gerlach

HENRYK GERLACH received his Ph.D. in 2010 from the École Polytechnique Féedérale de Lausanne under the guidance of John H. Maddocks and Peter Buser and was supported by the Swiss National Science Foundation.

Heiko von der Mosel

HEIKO VON DER MOSEL completed his Ph.D. under the supervision of Stefan Hildebrandt at the University of Bonn in 1996. He is a professor of Mathematics at RWTH Aachen University, and his research is devoted to the calculus of variations and geometric analysis.