When we were all hard at work on Project 1, some of us might have wondered how root finding might be used to create even more visually stunning displays of art. Professor Bahman Kalantari is one person who has done just that [1]. Using root finding techniques, Kalantari’s process – called polynomiography which is a combination between “polynomial” and “graphy” – takes polynomial equations and a set of rules for coloring and produces impressive visualizations.

Using some of the same techniques we used in class (Newton’s method mentioned specifically), the individual colors are really areas around a root where the equation would end up converging to the root. For example, if graph homework 3’s 3.3.5 where x^3-5x+3=0 using Kalantari’s software and an interval of [-3, +3) we can visually see the area of convergence around the largest root which we are asked to find, as well as the areas of convergence around the other 2 roots [2].

Modifying the software so it maps color to steps, we can see the number of steps it would take to converge to the root depending on the starting iteration point.

Were Kalantari’s software free, it could be used in helping students learn, visualize and analyze root finding techniques. Even if not free, the research underscores the value of root finding to society.

[1] http://www.sciencenews.org/articles/20030419/mathtrek.asp
[2] http://www.cs.rutgers.edu/~kalantar/poly/polywin.html