arXiv:1005.4609math.GT
On sphere-filling ropes
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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.
Beyond 100 it becomes a game of avoiding the The Loxodrome of Terror by spiralling into an unescapeable trap. Easiest pattern to get some more length is to do some sort of spherical sinusoid by following the meridian and then some optional final fill loop for the poles. Overall…
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