Critically-Finite Dynamics on the Icosahedron

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 &ldquo
dynamical polyhedra&rdquo
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&mdash
the dynamical polyhedron&mdash
in natural correspondence to a map&rsquo
s topology. It does so in a procedure that is more robust than the earlier implementation. The descriptions of the maps&rsquo
geometric behavior fall into combinatorial classes the presentation of which concludes the paper.