Dynamics of a soccer ball

Abstract

Exploiting the symmetry of the regular icosahedron, Peter Doyle and Curt McMullen constructed a solution to the quintic equation. Their algorithm relied on the dynamics of a certain icosahedral equivariant map for which the icosahedron's twenty face-centers--one of its special orbits--are superattracting periodic points. The current study considers the question of whether there are icosahedrally symmetric maps with superattracting periodic points at a 60-point orbit. The investigation leads to the discovery of two maps whose superattracting sets are configurations of points that are respectively related to a soccer ball and a companion structure. It concludes with a discussion of how a generic 60-point attractor provides for the extraction of all five of the quintic's roots.