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COLORING ALGORITHMS FOR DYNAMICAL SYSTEMS IN THE COMPLEX PLANE

 

Javier Barrallo & Damien M. Jones

The University of the Basque Country

Plaza de Oñati, 2.

20009 San Sebastián. SPAIN

mapbacaj@sa.ehu.es

 

 

With the increasing complexity of fractal generation software, the focus of exploration in new fractal techniques is shifting away from fractal formulas and towards coloring algorithms. This paper provides an overview of the coloring algorithms in popular use and a general classification system for these techniques.

 

 

Introduction

Dynamical systems are a well-known branch of mathematics, but until the arrival of computers the sheer number of calculations involved made them impractical for real use. The computerís ability to perform rapid calculations allows us to condense billions of calculations into results we can digest. Benoit Mandelbrot was the first one to use computers to produce graphical representations of dynamical systems in the complex plane, based on the quadratic formulas described by the French mathematician Gaston Julia at the beginning of the 20th century.

During the 1980s, fractal enthusiasts began exploring fractals for their artistic merit, not for their mathematical significance. While mathematics was the tool, the focus was art. As the fractal equation itself was the most obvious mathematical element, fractal artists experimented with new equations, introducing hundreds of different fractal types. By carefully choosing parameters to refine form, color, and location, these explorers introduced the concept of fractal art.

After 1995, few new major fractal types have been introduced. This is because the newest innovations in fractal art do not come from changing the fractal equation, but from new ways of coloring the results of those equations. As these coloring algorithms move from simple to complex, fractal artists are often returning to the simpler, classical fractal equations. With the increased flexibility these sophisticated algorithms provide, there is even more room for personal artistic expression.

 


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