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
through a fundamental family of iteration functions, the ``Basic
Family. '' We introduce these connections via the Basic Family
properties with respect to Voronoi diagrams, and a corresponding
visualization called ``Polynomiography.'' Polynomiography is a medium
for art, math, education and science. Then, by making use of properties
of the Basic Family we introduce a layering of points within each
Voronoi cell of a polynomial root, and prove several novel and
nontrivial results. These reveal deeper connections between Voronoi
diagrams and polynomial root-finding, and offer applications with
respect to ordinary Voronoi diagrams and nontrivial generalizations to
be defined. We will pose research problems that may be considered as
problems lying at the intersection of computational geometry,
polynomial root-finding, discrete dynamical systems and more.