Constructing fully symmetric cubature formulae for the sphere

We construct symmetric cubature formulae of degrees in the 1339 range for the surface measure on the unit sphere. We exploit a recently published correspondence between cubature formulae on the sphere and on the triangle. Specifically, a fully symmetric cubature formula for the surface measure on the unit sphere corresponds to a symmetric cubature formula for the triangle with weight function (u1u2u3)−1/2, where u1, u2, and u3 are homogeneous coordinates.