The Spherical Grasshopper Problem

TL;DR

This paper investigates the spherical Grasshopper Problem, using discretisation and simulated annealing to find optimal colourings on a sphere that maximize the probability of a grasshopper landing on the same colour, revealing solutions similar to planar cases.

Contribution

It extends the analysis of the Grasshopper Problem to the sphere with antipodality constraints and employs computational methods to identify optimal solutions across different jump distances.

Findings

Cogwheel solutions are optimal for certain angles.

Critical solutions emerge near the midpoint angle.

Stripe patterns dominate at larger jump distances.

Abstract

The aim of this essay is to better understand the Grasshopper Problem on the surface of the unit sphere. The problem is motivated by analysing Bell inequalities, but can be formulated as a geometric puzzle as follows. Given a white sphere and a bucket of black paint, one is asked to paint half of the sphere, such that antipodal pairs of points are oppositely coloured. A grasshopper lands on the sphere, and jumps a fixed distance in a random direction. How should the sphere be coloured such that the probability of the grasshopper landing on the same colour is maximized? Goulko and Kent have explored this problem on the plane without an antipodality constraint. This essay gives clear indication that the spherical problem with the antipodality constraint yields colourings with similar shapes as the planar problem does. This research has discretised the problem and used a simulated annealing algorithm to search for the optimal solution. Results are consistent with the planar results of \cite{goulkokent}. For $0.10\pi\leq\theta\leq0.44\pi$ cogwheel solutions are found to be optimal, with odd integer of cogs $n_o$ such that $n_o$ is close to $\frac{2\pi}{\theta}$. For $0.45\pi\leq\theta\leq 0.55\pi$ critical solutions are found, in which domains of identical colour decrease in size towards $0.5\pi$ (moving from either side). For $\theta\geq 0.55$ colourings are found consisting of stripes with cogs. Towards $\theta=\pi$ colourings are generated that display just stripes that scale in width with $\pi-\theta$.