Robert Spann
I have always been fascinated with the behavior of dynamical systems -- be they mathematical, physical, or financial. Where do different points end up and how do they get there? Do they reach an equilibrium or do they continue moving randomly forever? Computer graphics allows one to see both the numerical and aesthetic properties of these systems. Recently I became interested in the properties of complex polynomials with complex, rather than integer, exponents. The images I have produced are the basins of attraction for the roots of complex polynomials with complex exponents under Newton's method.
This image is obtained by applying Newton's method to the function z^(3+2i) - i =0, where i is the square root of -1. Each point in the complex plane is assigned a color based on which root that point converges to under iteration by Newton's method for finding the roots of an equation.
This image is obtained by applying Newton's method to the function z^(6i) - 1.045z(^4i) + .045z^(2i) -8.23x10^(-5) =0, where i is the square root of -1. Each point in the complex plane is assigned a color based on which root that point converges to under iteration by Newton's method for finding the roots of an equation.
This image is obtained by applying Newton's method to the function z^(3i) - 1.251z(^2i) + .261z^i - .009 =0, where i is the square root of -1. Each point in the complex plane is assigned a color based on which root that point converges to under iteration by Newton's method for finding the roots of an equation.