Fractal art

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$Computer-generated fractal image.$

Computer-generated fractal image.

Fractal art is created by calculating fractal mathematical functions and transforming the calculation results into still images, animations, music, or other art media. Fractal images are graphs of the calculation results, and fractal animations are sequences of these graphs. Fractal music maps the calculation results to music pitches or other sounds. Fractal art is usually created with the assistance of a computer, in order to speed up the process of calculating the fractal function.

Fractals fall into four broad categories relevant to fractal art:

Those for which membership of a point in a fractal set may be determined by iterative application of a simple function. Examples of this type are the Mandelbrot set, the Lyapunov fractal, and the Burning Ship fractal.
Those for which a geometric replacement rule exists. Examples include Cantor dust, the Sierpinski gasket, the Menger sponge and the Koch snowflake.
Those created with iterated function systems, in particular fractal flames.
Those which are generated by stochastic rather than deterministic processes (examples include fractal landscapes).

Fractals of all four kinds have been used as the basis for digital art and animation. Starting with 2-dimensional details of fractals such as the Mandelbrot Set, fractals have found artistic application in fields as varied as texture generation, plant growth simulation and landscape generation.

$A fractal image generated by Sterling Fractal by Stephen Ferguson$

A fractal image generated by Sterling Fractal by Stephen Ferguson

Fractals are also being used in context with evolutionary algorithms in the Electric Sheep project, as people use fractals rendered with distributed computing as their screensaver, and "rate" the flame they are viewing. Then the server reduces the traits of the undesirables, and increases those of the desirables to produce a computer-generated, community-created piece of art.

Fractal art galleries can be found on the Internet, for example at [this Tierazon and Sterling fractal gallery page]. A good starting point would be the fractal pages of Stephen C. Ferguson who has made several fractal generators like Sterling Fractal, an example image from which is shown to the left. His more classic fractal generators include [Iterations et Flarium] (et means "and").

The two most popular fractal image creation programs are probably Ultra Fractal and Apophysis. The latter is a fractal flame editor and the former a more general purpose fractal program. During the 1990s Fractint for DOS was the most popular fractal rendering software for the PC.

Fractal music is able to produce more realistic natural sounds and subtle tunes than conventional approaches.

Painter Jackson Pollock, famous in the 1950s for abstract expressionist painting, has had fractals applied to his paintings as a way of authenticating originals from forged paintings.

Art and the Mandelbrot set

Some people search the Mandelbrot set for interesting pictures and use them in fractal art. Regions of the Mandelbrot set have popular names evoking their appearance. For example, the seahorse valley evokes seahorse tails and the elephant valley is a region where spirals evoke elephant trunks.

"Seahorse valley"

"Seahorse valley"

"Elephant valley"

Further examples below are regions, to the right is a min-cardioid, connected by filaments (too thin to be seen) to the main set (and other cardioids, some of which are actually in this image but too small to be seen).:

Mini-set with filaments leading elsewhere

There are free fractal-generating programs available, such as those by Stephen Ferguson (Sterling Fractal and the Tierazon series). Some people write fractal programs themselves for the highest level of control over the results.

Since the set (unlike nature) has unlimited detail, it appears less noisy when calculated on a fine grid and down-sampled with anti-aliasing. Here are two color codings of the same calculation and two images made by combining two calculations for each image.

"Spiral Cleft" done with MandelZot and down-sampled with Photoshop

"Spiral Cleft" Simpler colors.

"Islands of Consciousness" distance estimator calculated separately.

"Cypress Underbrush" hairy filaments calculated separately as in previous image.

Following the left of one valley and the right of another cancels out most of the spiraling found on the sides of valleys.

In the following three images, the distance estimator has been used to help in anti-aliasing by eliminating unrepresentative samples. In the last three, these pixels were replaced by bleeding in their neighbors. "Deep Re Twisted" has been processed to enhance the edges, similarly to some in the previous row.

$Wayne's "Pachifractal" with white edges$
Wayne's "Pachifractal" with white edges

"Deep Re Twisted"

"Budding Turbines"

"Persian Rug"

The main artistic challenge is that simple mathematical forms are uninteresting. A greater variety of shapes can be generated by taking a varied path while zooming in, near a point, down a cleft, right and left sides of valleys, etc., and by picking dense areas where most of the pixels are blends of the colors of different dwell values. The difficulty is that deeper dwells and fine detail increase the computer time, so what can be done without parallel arithmetic is limited.

"Persian Rug" was found by selecting a star near the body of the set and then a puff on a horizontal part of a ray of the star (an arm of the candelabra), twice. It is similar with reflection and almost identical with 180 degree rotation. The dwell values are very high, so only the top half was calculated, and that was rotated to approximate the bottom.

References

[1] John Briggs, Fractals ISBN 0671742175
[2] Clifford A. Pickover, Computers, Pattern, Chaos and Beauty ISBN 0486417093
[3] Clifford A. Pickover, Fractal Horizons, ISBN 0312125992
[4] Clifford A. Pickover, Chaos and Fractals ISBN 0444500022
[5] Clifford A. Pickover, Keys to Infinity ISBN 0471193348
[6] Manfred Schroeder, Fractals, Chaos, Power Laws ISBN 0716723573
[7] Michael Michelitsch and Otto E. Rössler, The "Burning Ship" and Its Quasi-Julia Sets, Computers & Graphics Vol. 16, No. 4, pp. 435-438, 1992, reprinted in [9]
[8] Michael Michelitsch and Otto E. Rössler, "A New Feature in Hénon's Map." Comput. & Graphics Vol. 13, No. 2, pp. 263-265, 1989, reprinted in [9]
[9] Clifford A. Pickover Ed., Chaos and Fractals: A Computer Graphical Journey - A 10 Year Compilation of Advanced Research. Amsterdam, Netherlands: Elsevier 1998. ISBN 0-444-50002-2

External links

[Apophysis] – Freeware fractal flame editor for Windows
[The Fractal Art Manifesto] – essay by Kerry Mitchell
[Free Fractal Screensaver] – Free fractal screensaver (50+ fractal images and 100+ various effects).
[Ultra Fractal] – software for creating fractal images and animations
[The Infinite Fractal Loop] – a web ring of Fractal Art
[C82] – images by Nicholas Rougeux
[XaoS] – open source software for creating fractal images and animations
[Fractovia] – images and music by Juan Luis, and listings of fractal art software and tutorials
[Fractal Flames] – fractal flame images by Scott Draves
[Fractal Recursions] – images and animations by Jock Cooper
The [Electric Sheep] Project – a distributed computing project to render animated fractal flames.
[The Structures of Infinity] – images by Jaroslaw Wierny (text in Polish)
[Fractal zoom movies] – animations by Eric Bigas
[Fract-O-Rama] – open source software for creating fractal images

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