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 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.

Critically-Finite Dynamics on the Icosahedron

“…The behavior of a map's critical orbit-where the critical points go when the map is applied iteratively-bears strongly on the map's global dynamics (See [2], Chapter 9). In the particular case of periodic critical points, their basins of attraction have full measure on the sphere.…”

Completely solving the quintic by iteration

“…Recent work determined all icosahedrally-symmetric maps with internally periodic critical sets of size 60. [1,2] Internal periodicity means that the map acts on its critical set as a permutation. Here, we build a quintic-solving device around the dynamics of one such map g whose critical points have period five.…”

“…Computational results are produced by Mathematica and basins-of-attraction plots are the product of Dynamics 2. [7] 2 Icosahedral algebra: invariants, and equivariants An account of the algebraic objects that emerge from the icosahedral action on CP 1 appears in other places ( [6], [5], and [1]). Results relevant to the task at hand appear without discussion.…”

“…(In [2], I describe how this map was found as well as the other critically-finite degree-31 I-maps.) Its analytic form is approximated by As discussed in [1] and [2], g's geometric behavior gives rise to a polyhedral system of "edges" E g that forms a forward invariant set. This collection of edges fills in the B 62 structure whose faces consist of twelve pentagons, twenty triangles, and thirty quadrilaterals that realize fivefold, three-fold, and two-fold rotational symmetry respectively.…”

“…to be projectively semi-conjugate to g(x). Accordingly, the g Z -orbit of a random initial condition w 0 in CP 1 w is asymptotic to a superattracting five-cycle (ω 1 , . .…”