The Grasshopper Problem

A unit sphere is covered by a lawn such that one of every pair of antipodal points belongs to the lawn. A grasshopper lands on the lawn and then makes a jump with a fixed distance in a random direction. What is the optimal lawn shape such that the grasshopper will have the best chance of landing on the lawn again? This problem has surprising connections to statistical physics and quantum information, specifically Bell’s inequalities. In this setup two parties measure spins about randomly chosen axes and obtain correlations for pairs of axes separated by a fixed angle. By discretizing the lawn to a set of spins it is possible to use numerical methods to optimize the lawn shape. We find that at smaller jump distances the optimal lawns resemble cogwheels, similarly to the planar case. At larger jumps we find shapes such as stripes and increasingly complex labyrinths. We will discuss analytical and numerical results for the spherical grasshopper problem, as well as the connection to Bell’s inequalities involving random measurement choices.