No 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.

A recent effort used two rational maps on the Riemann sphere to produce polyhedral structures with properties exemplified by a soccer ball. A key feature of these maps is their respect for the rotational symmetries of the icosahedron. The present article shows how to build such “dynamical polyhedra” for other icosahedral maps. First, algebra associated with the icosahedron determines a special family of maps with 60 periodic critical points. The topological behavior of each map is then worked out and results in a geometric algorithm out of which emerges a system of edges—the dynamical polyhedron—in natural correspondence to a map’s topology. It does so in a procedure that is more robust than the earlier implementation. The descriptions of the maps’ geometric behavior fall into combinatorial classes the presentation of which concludes the paper.

There is a family of seventh-degree polynomials H whose members possess the symmetries of a simple group of order 168. This group has an elegant action on the complex projective plane. Developing some of the action's rich algebraic and geometric properties rewards us with a special map that also realizes the 168-fold symmetry. The map's dynamics provides the main tool in an algorithm that solves "heptic" equations in H.

The symmetric group Sn acts as a reflection group on CP n−2 (for n ≥ 3) . Associated with each of the n 2 transpositions in Sn is an involution on CP n−2 that pointwise fixes a hyperplane-the mirrors of the action. For each such action, there is a unique Sn-symmetric holomorphic map of degree n+1 whose critical set is precisely the collection of hyperplanes. Since the map preserves each reflecting hyperplane, the members of this family are critically-finite in a very strong sense. Considerations of symmetry and critical-finiteness produce global dynamical results: each map's Fatou set consists of a special finite set of superattracting points whose basins are dense.