Polynomiography with non-standard iterations

Abstract

In the paper the visualizations of some modifications applied to the Newton’s root finding of complex polynomials are presented. Namely, instead of the standard Picard iteration, several different iterative processes described in the literature, which we call as non-standard ones, were used. Following Kalantari, such visualizations are called polynomiographs. Polynomiographs are interesting from scientific, educational and artistic points of view. By the use of different iterations kinds we obtain quite new polynomiographs, in comparison to the standard Picard iteration, which look aesthetically pleasing. We present some polynomiographs for complex polynomial equation z3 -1 = 0 as examples. Polynomiographs were defined to graphically present dynamical behaviour of different iterative processes. But we are not interested in that. 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 can also be used to increase functionality of the existing polynomiography software.