Octahedral and dodecahedral monopoles

Abstract

It is shown that there exists a charge five monopole with octahedral symmetry\nand a charge seven monopole with icosahedral symmetry. A numerical implementation of the\nADHMN construction is used to calculate the energy density of these monopoles and surfaces\nof constant energy density are displayed. The charge five and charge seven monopoles look\nlike an octahedron and a dodecahedron, respectively. A scattering geodesic for each of these\nmonopoles is presented and discussed using rational maps. This is done with the aid of a new\nformula for the cluster decomposition of monopoles when the poles of the rational map are close\ntogether.