We compute explicit orthonormal bases for functions invariant under the rotational symmetries of a Platonic solid. Each function in the basis is a linear combination of spherical harmonics. For each symmetry (icosahedral, octahedral, tetrahedral) the calculation has three steps: First derive a bilinear equation for the coefficients by comparing the expansion of a symmetrized delta functioii in both spherical harmonics and the symmetric harmonics. The equation is parameterized by the location (Θ0,Ø0) of the delta function and must be ~at~isfiefodr all locations. Second, express t,he dependence on the delta function location in a Fourier (Ø0) and Taylor (Θ0) series and thereby derive a new system of bilinear equations by comparing selected coefficients. Third, derive a recursive solution of the new system and explicitly solve the recursion with the aid of symbolic computation. The results for the icosahedral case are important for structural studies of small spherical viruses.


spherical harmonics; rotational symmetries, finite; Platonic solids, icosahedron, dodecahedron, octahedron, cube, tetrahedron

Date of this Version

October 1994