An important piece of information when dealing with a polynomial in the complex plane is its roots, the value or values of x for a given function f such that f(x)=0. Iterative root finding methods, such as Newton’s method, are utilized to discover an approximate value when these values cannot be explicitly solved. This process can be graphically represented for complex-valued functions and has been achieved with relative ease on a 2-Dimensional plane. However, this process can also be embodied on a sphere through the method of stereographic projection, which has not been attempted. In this research, I worked with iterative root finding methods, such as Newton’s and other higher order methods, on the complex plane. Upon mapping out their iterations on the 2-D plane, I wrote a program to stereographically project them upon a sphere in order to be able to analyze their complete dynamics.
Nicklawsky, Andrew, "Visualizing Chaos" (2012). Mathematics Student Work. 1.