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 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 the 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.

2000 AMS Subject Classification::

• 37F10

Keywords::

• icosahedron
• equivariant map
• quintic equation

Notes

¹Files that establish the facts are available at www.csulb.edu/~scrass/math.html.