Generalized Mandelbrot Sets

Author

Aaron Schlenker, Butler University

Date of Award

5-11-2014

Degree Type

Thesis

Degree Name

Honors Thesis

Department

Computer Science

First Advisor

William Johnston

Abstract

A complex point Z₀ is defined to be a member of the famous Mandelbrot set fractal when the iterative process using the function Z² stays bounded when applied to Z₀. We investigate what happens if we change the iterative process so that Z² is now composed with, for example, a Mobius transformation, indexed on a parameter a. The Mandelbrot set corresponds to a = O. What happens when we change a = 0 to other values, repeating the iterative process and then drawing the sets? Do these Generalized Mandelbrot sets have similar properties to the original Mandelbrot set? This thesis describes some surprising results for these new sets, and it also uses transcendental functions to produce similar generalized sets. Further, it describes the algorithms that were developed and used during the study of each of these sets.

Recommended Citation

Schlenker, Aaron, "Generalized Mandelbrot Sets" (2014). Undergraduate Honors Thesis Collection. 229.
https://digitalcommons.butler.edu/ugtheses/229