## Physics, Mathematics & Computer Science

#### Event Title

Generalized Mandelbrot Sets

#### Document Type

Oral Presentation

#### Location

Indianapolis, IN

#### Subject Area

Physics, Mathematics & Computer Science

#### Start Date

11-4-2014 9:30 AM

#### End Date

11-4-2014 10:30 AM

#### Sponsor

Bill Johnston (Butler University)

#### Description

A complex point z_{0} is in the famous Mandelbrot Set fractal when an iterative process applied to z_{0} and using the function z^{2} stays bounded. We investigate what happens if we change the function so that z^{2} is now composed with a Mobius transformation, indexed on a parameter a. The Mandelbrot set corresponds to a = 0. What happens when we change a = 0 to other values, repeating the same iterative process and then drawing the sets? Do these Generalized Mandelbrot Sets have similar properties as the original? This presentation describes some surprising results, illustrates the sets in computer-generated movies, and uses transcendental functions to produce further set generalizations.

Generalized Mandelbrot Sets

Indianapolis, IN

A complex point z_{0} is in the famous Mandelbrot Set fractal when an iterative process applied to z_{0} and using the function z^{2} stays bounded. We investigate what happens if we change the function so that z^{2} is now composed with a Mobius transformation, indexed on a parameter a. The Mandelbrot set corresponds to a = 0. What happens when we change a = 0 to other values, repeating the same iterative process and then drawing the sets? Do these Generalized Mandelbrot Sets have similar properties as the original? This presentation describes some surprising results, illustrates the sets in computer-generated movies, and uses transcendental functions to produce further set generalizations.