half of an oscillation period, the interface alternates between a pattern and its dual. For the Platonic solids, the dual of an octahedron is a cube, that of an icosahedron is a dodecahedron and the dual of a tetrahedron is another tetrahedron.

The patterns are highly dependent on the value of $\ell$ considered. Axisymmetric (sometimes called zonal) solutions are never stable if $\ell\geq 2$ [@Chossat1991]. For odd $\ell$, solutions exist which are stable at onset. In contrast, for even $\ell$, all solutions are unstable near the bifurcation point. In this case, the preferred solution can be considered to be that with the smallest number of unstable eigenvalues [@Matth2003] or which extremizes a functional [@Busse1975]. Unstable solution branches produced at a transcritical bifurcation can be stabilized, for example at a saddle-node bifurcation some distance from the threshold. The theory of pattern selection with $O(3)$ symmetry differs from the framework of our simulations in some important ways. First, the theory applies to steady bifurcations rather than oscillatory solutions resulting from time-periodic forcing. The archetypical application is Rayleigh-Bénard convection in a sphere [@Busse1975; @BR1982; @Riahi1984] and the steady symmetry-breaking bifurcations it undergoes. However, many of the conclusions can be generalized to the time-periodic 