Pattern formation on a sphere

Pattern formation on the surface of a sphere is described by equations involving interactions of spherical harmonics of degree l. When l is even, the leading-order equations are determined uniquely by the symmetry, regardless of the physical context. Existence and stability results are found for even l up to l = 12. Using either a variational or eigenvalue criterion, the preferred solution has icosahedral symmetry for l = 6, l = 10, and l = 12. Numerical simulations of a model pattern-forming equation are in agreement with these theoretical predictions near onset and show more complex patterns further from onset.