Polynomiography: The Art and Science of Approximating Roots Using Complex Iterations
Karl McMurtry
Many of us have seen fractals and may have asked, “Is that more than just a pretty picture, and if so, what does it mean?” Fractal images are basically images that depict how long it takes for the functional iteration to converge towards the root. The computer assigns different colors for how long or for how many iterations it takes to get to that root. A certain value may not even converge. This leads to areas that are called basins of attraction. Values within the basin converge quickly to the root. The interesting things happen at the edges of the basin, where a small shift in value creates a large change in the convergence of the root.
The Mandelbrot Set
The basic process of making a fractal image involves guessing a starting point and using an algorithm to find a better point. This process is done recursively until the value is found within a certain buffer size, or after a certain number of iterations. When a starting point is picked, one hopes that the iterations will lead to the correct root. However, this is not always the case and this leads to the basis of fractals.
A polynomiograph of a degree 36 polynomial[i]
Fractals
are related to polynomials through a branch of mathematics called
polynomiography. Polynomiography is
defined as, “the art of science of visualization in approximation of the zeros
of complex polynomials, via fractal and non-fractal images created using the
mathematical convergence properties of iteration functions.”[ii] However, a polynomiograph is not
necessarily a fractal image but is a graphical representation of the
polynomial’s roots. A good deal of work
has been done on the process of finding the roots of small to moderate degree
polynomials, but not much work has been done on finding the roots to large
polynomials because large polynomials do not occur very frequently in mathematical
studies. In addition, the larger the
polynomial, the more difficult it becomes to find its roots. Polynomiography changes all of this by using
an algorithm to estimate the roots of the polynomial graphically. As a result, it has become much easier to
approximate and find the roots of large polynomials. Polynomiography makes use of several well-known iteration
functions in approximating the roots of polynomials, most notably are Newton’s
[
] and Halley’s.
Kalantari has discovered that the easiest
algorithms for approximation are given by what he calls the “Basic
Family.” This family of iteration
functions reveals a pointwise convergence to the family of roots of a
polynomial. The Basic Family is
represented by 
. Where the first
member of the sequence is Newton’s iteration function and the second member is
Halley’s.
If we let 
be a polynomial of
degree 
with complex
coefficients, and 
, and for each 
we define
We can also define 
recursively by:
Therefore, the members of the Basic Family can be defined by:
.
One of the
interesting properties of the Basic Family is the following connection to
Voronoi regions: the basins of attractions of the roots of a complex
polynomial, as computed with respect to 
, 
converge to the
Voronoi regions of the roots of 
.[iii] Now you may be wondering what Voronoi
regions are, so I will briefly explain.
Given a set of points plotted in a plane, the Voronoi regions are the
regions around each point that contain the area closest to that point. Graphically, a Voronoi region looks like
this:
The
fact that 
converges to Voronoi
regions implies that the problem of finding the Voronoi regions of a given set
of points can be approximated by making the set the roots of a polynomial and
recursively solving the problem using the root finding methods discussed
above. The next figure depicts this
process for 
: random points and 
2, 4, 10, 50 (respectively).
The
applications of polynomiography include art, cryptography, industry, and
more. Art is the most obvious
application. A great deal of time,
energy and detail has been put into the development of fractal image
generation. Polynomiography would give
new methods to produce these beautiful images.
Furthermore, polynomiography could significantly change industries like
the textile business. The reason for
this is because, ”…in polynomiography there is a ‘reverse root-finding
problem’: given a known set of points, find an iteration method whose
corresponding polynomiograph would take a desired pattern.”[iv] A textile artist could create a pattern
using a set of points and then would be able to share or repeat this pattern by
a simple mathematical formula.
Cryptography
is one of those interesting areas where polynomiography can advance our
understanding and options. If we are
given a set of numbers 
we can represent the
information graphically like a digital fingerprint by using a corresponding
polynomial 
and finding its
polynomiograph. For example, the
government could encrypt our social security numbers through this process.
The
American mathematician S. Smale said, “There is a sense in which an important
result in mathematics is never finished.”[v] I think that this statement adequately sums
up the study of polynomiography. The
study of the roots of polynomials has been the subject of interest of
mathematicians for centuries, and polynomiography gives us new insights and new
ways to approach this problem.