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The Mandelbrot Set: Colors of Infinity

By Nathan Reed - www.reedbeta.com
and
Jérôme Guinot - www.oZone3D.Net

Initial draft: January 25, 2006

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 1 - Fractals2 - Infinite Detail3 - The Mandelbrot Set4 - Mandelbrot on the GPU5 - Implementation with Hyperion6 - Further Possibilities7 - Further Resources8 - Downloads9 - About the Authors

1 - Fractals

The word fractal is a term that was coined by the mathematician Benoit Mandelbrot in 1975.
Fractal, from the Latin fractus meaning fractured, refers to a unique type of geometric shape. Fractals have two distinct properties: they tend to be infinitely detailed, and they look like themselves at various scales, a property called self-similiarity. Because of this, fractals are almost never smoothly curving shapes but are characterized by sharp, jagged edges and infinitestimal features that have the same shape as larger-scale features.

Fractals are not just an abstract mathematical topic; many forms and shapes in nature tend to exhibit distinctly fractal-like characteristics.  One good example is a tree: a tree's main truck sprouts branches, and those branches sprout smaller branches, which sprout twigs, which sprout smaller twigs, which sprout leaves.  Any given branch or twig, taken with all of the branches and twigs attached to it, looks a great deal like the entire tree; thus, trees are self-similiar—certain small parts of them resemble the tree as a whole.  Clouds are another fractal-like phenomenon; the bumpy surface of a cloud viewed up close looks very similiar to the large-scale structure of the same cloud.  Fractals appear in computer graphics in the form of fractal Brownian noise (fBn), which is a mathematical function frequently used for procedural texture and terrain generation.

In this article, we will explore the Mandelbrot set, a famous fractal object originally discovered in 1905 by Pierre Fatou.  In the 1970s, Benoit Mandelbrot produced the first computer-generated images of this set, which popularized it among mathematicians, computer graphics researchers, and the general public alike.  First we'll examine the classical algorithm for rendering images of the Mandelbrot set, and then we'll look at ways to adapt this algorithm to run on a GPU.

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