Voronoi Diagrams and Polynomial Root-Finding

Voronoi diagram of points in the Euclidean plane and its computation is foundational to computational geometry. Polynomial root-finding is the origin of fundamental discoveries in all of mathematics and sciences. There is an intrinsic connection between polynomial root-finding in the complex plane and the approximation of Voronoi cells of its roots via a fundamental family of iteration functions, the basic family. For instance, the immediate basin of attraction of a root of a complex polynomial under Newton's method is a rough approximation to its Voronoi cell. We formally introduce these connections through the Basic Family of iteration functions, its properties with respect to Voronoi diagrams, and a corresponding visualization called polynomiography. Polynomiography is a medium for art, math, education and science. By making use of the Basic Family we introduce a layering of the points within each Voronoi cell of a polynomial root and study its properties and potential applications. In particular, we prove some novel results about the basic family in connection with Voronoi diagrams.