To solve a polynomial you need a means of breaking its symmetry. In the case of a generic equation of degree n, this is the symmetric group Sn. If we extract the square root of the polynomial's discriminant, the group reduces to the alternating group An. An iterative algorithm for solving the equation has two ingredients: Geometric: a complex projective space S upon which the polynomial's symmetry group acts faithfully Dynamical: a mapping of S that respects the group action-sends group-orbits to group-orbits. This paper discusses these two aspects in the cases of the fifth and sixth degree equations. Motivating the project is a desire to develop an algorithm with especially elegant qualities.