Evolutionary Exploration and Generalization on Mandelbrot Sets
Submitted to the Journal of Mathematics and Art

Daniel Ashlock and Brooke Jamieson

Abstract PDF eprint

Mandelbrot sets are infinitely complex fractals defined by a simple iterative algorithm operating on the complex numbers. Views of Mandelbrot sets are a common form of fractal art. Presented here is a an evolutionary algorithm that permits the search of Mandelbrot sets for interesting views. The hand of the artist is present in the design of the fitness function used to drive evolution. A collection of fitness functions is presented that permit three-parameter evolutionary search of Mandelbrot sets to locate interesting views. These fitness functions are based on a mathematical envelope that specifies the character of the fractal landscape desired. The fitness function is easily reconfigured with minimal programming skill and without knowledge of complex arithmetic. This paper is the second exploring this system for evolutionary exploration of Mandelbrot sets. It tests for the first time a technique that generalizes from a given view of a Mandelbrot set. Having located one desirable view, a fitness function is constructed from the view that yields similar views in other parts of the Mandelbrot set. This study also presents the first application of the Mandelbrot evolutionary search software to Mandelbrot sets other than the traditional quadratic Mandelbrot set.