  Recently, Peter Doyle and Curt McMullen devised an iterative solution to the
fifth degree polynomial. At the method's core is a rational mapping of the
Riemann sphere with the icosahedral symmetry of a general quintic. Moreover,
this map posseses "reliable" dynamics: for almost any initial point, the its
trajectory converges to one of the periodic cycles that comprise an icosahedral
orbit. This symmetry-breaking provides for a reliable or "generally-convergent"
quintic-solving algorithm: with almost any fifth-degree equation, associate a
rational mapping that has reliable dynamics and whose attractor consists of
points from which one computes a root.
  An algorithm that solves the sixth-degree equation requires a dynamical
system with the symmetry of the alternating group on six things. This group
does not act on the Riemmann sphere, but does act on the complex projective
plane--this is the Valentiner group. The present work exploits the resulting
2-dimensional geometry in finding a Valentiner-symmetric rational mapping whose
elegant dynamics experimentally appear to be reliable in the above
sense---transferred to the 2-dimensional setting. This map provides the central
feature of a conjecturally-reliable sextic-solving algorithm analogous to that
employed in the quintic case.
