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Dynamics of a soccer ball. (English) Zbl 1310.37021
Summary: 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.

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
Software:
Dynamics
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