Polynomiography: Visualization of Polynomial Equations and New Challenges

Bahman Kalantari
Rutgers University

Despite the very old history of the polynomial root-finding problem or the compuer visualizations that have emerged in the past few decades, its applications and mysteries are  far from being exhausted. This is contrary to the common belief, even among some experts in various fields. Polynomiography is the art and science of visualization in the approximation of zeros of complex polynomials via iteration functions. Polynomiography has numerous applications in art, science, and education. Like photography, iteration functions have the ability
to capture numerous “polynomiographs” of the underlying polynomial, images that may or may not exhibit fractal behavior. Like NKS, the iteration functions within polynomiography could be generated via simple programs, yet exhibit unpredictable and complex behavior. The Mathematica software and extensive NKS experiences strengthen the view that we should carry out polynomiography on larger and larger degree polynomials. This visualization not only could bring deeper insight into the nature of various classes of polynomials, but perhaps analogous to the problem of computing the digits of pi, it could bring about new scientific and mathematical challenges. However, unlike the problem of approximating digits of pi, polynomiography of large degree polynomials should bring to life new artwork, never seen before, perhaps worthy of appreciation by the millions.

Created by Mathematica  (April 20, 2004)