Chaos is typically visualized on an infinite 2D plane. By using stereographic projection, my colleague Preston Hardy and I utilized a third dimension to plot basin maps of iterative root finding methods on a subset of the complex plane onto a sphere. These spheres are then shaded in accordance to the speed in which the particular initial point converges, creating images that can be used to visualize all basins of attraction on the complex plane on a finite 3D surface. The resulting images are used to explore efficiency of root finding methods as well as evaluating the choice of addition or subtraction n the denominator of the Hansen-Patrick root finding method. There are many theories suggesting the sign choice for positive alpha values; however, in the case of a negative alpha value, these theories do not hold. Using programs based off of those developed by Andrew Nicklawsky and Dr. Robert Hesse, we developed rules to dictate this choice between addition and subtraction in order to maximize the speed of convergence for negative and imaginary alpha values.
Peterson, Margaret, "Visualizing Chaos" (2013). Honors Theses. 20.