  By imposing certain combined inversion and rotation symmetries on the
rational maps for SU(2) BPS monopoles we construct geodesics in the monopole
moduli space. In the moduli space approximation these geodesics describe a
novel kind of monopole scattering. During these scattering processes axial
symmetry is instantaneously attained and, in some, monopoles with the
symmetries of the regular solids are formed. The simplest example corresponds
to a charge three monopole invariant under a combined inversion and 90 degree
rotation symmetry. In this example three well-separated collinear unit charge
monopoles coalesce to form first a tetrahedron, then a torus, then the dual
tetrahedron and finally separate again along the same axis of motion. We
explicitly construct the spectral curves in this case and use a numerical ADHMN
construction to compute the energy density at various times during the motion.
We find that the dynamics of the zeros of the Higgs field is extremely rich and
we discover a new phenomenon; there exist charge k SU(2) BPS monopoles with
more than k zeros of the Higgs field.
