Cubature formulas on a sphere invariant under the icosahedral rotation group

Abstract

An algorithm is proposed for constructing cubature formulas on a sphere that are invariant under the icosahedral rotation group. The algorithm is applied to construct new cubature formulas of algebraic order of accuracy n = 19, 20, 21, 23, 24, 25. Parameters of these cubature formulas are given to 16 significant digits. A table, which containsmain characteristics of all optimal cubature formulas for the icosahedral rotation group exact to order 35, is included. A real version of F. Klein’s formula, which establishes a connection between basic invariant forms of the icosahedral rotation group, is presented.