# Evolutionary Exploration and Generalization on Mandelbrot Sets

Submitted to the Journal of Mathematics and Art

### Daniel Ashlock and Brooke Jamieson

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.