Abstract
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.
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