Polynomiography with Non-standard Iterations

Abstract

In the paper visualizations of some modifications based on the Newton's root finding of complex polynomials are presented. Instead of the standard Picard iteration several different iterative processes described in the literature, that we call as non-standard ones, are used. Following Kalantari such visualizations are called polynomiographs. Polynomiographs are interesting from scientific, educational and artistic points of view. By the usage of different kinds of iterations we obtain quite new, comparing to the standard Picard iteration, polynomiographs that look aesthetically pleasing. As examples we present some polynomiographs for complex polynomial equation z^3 - 1 = 0. Polynomiographs graphically present dynamical behaviour of different iterative processes. But we are not interested in it. We are focused on polynomiographs from the artistic point of view. We believe that the new polynomiographs can be interesting as a source of aesthetic patterns created automatically. They also can be used to increase functionality of the existing polynomiography software.