  The planar grasshopper problem, originally introduced in (Goulko & Kent 2017
Proc. R. Soc. A 473, 20170494), is a striking example of a model with
long-range isotropic interactions whose ground states break rotational
symmetry. In this work we analyze and explain the nature of this symmetry
breaking with emphasis on the importance of dimensionality. Interestingly,
rotational symmetry is recovered in three dimensions for small jumps, which
correspond to the non-isotropic cogwheel regime of the two-dimensional problem.
We discuss simplified models that reproduce the symmetry properties of the
original system in N dimensions. For the full grasshopper model in two
dimensions we obtain quantitative predictions for optimal perturbations of the
disk. Our analytical results are confirmed by numerical simulations.
