  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.
