- Fractal art
Fractal art is created by calculating
fractalobjects and representing the calculation results as still images, animations, music, or other media.Fractal art is usually created indirectly with the assistance of a computer, iterating through three phases: setting parameters of appropriate fractal software, executing the possibly lengthy calculation and evaluating the product.
Fractal objects fall into four main categories depending on how an artist can manipulate their construction and rendering to exercise artistic control over the resulting fractal art:
Escape time fractals
Escape time fractals that are manipulated with the choice of the formula to be iterated and its parameters, the choice of what points are iterated (usually a tiny region of the complex plane containing interesting shapes) and how they are mapped to an image, the choice of how to compute a colour from the set of sequences of iterations. All these components have an explicitly mathematical and nonvisual nature and they can often be very complex.Examples of this type of fractal are the Mandelbrot and Julia sets, the
Lyapunov fractal, the Newton fractal, its relative the Nova fractal, and the Burning Ship fractal; some fractal image creation programs for this type of fractal art are Ultra Fractal, Gnofract4D, ChaosProand Fractint.
Lindenmayer systems and other constructions based on replacement rules
Examples of this type include the
Peano curveand the Hilbert curve, the Sierpinski gasketand the Menger sponge, and the Koch snowflake. Stochastic systems where the replaced shapes and/or the choice of rules are random are very popular, especially to recreate trees and other natural objects.Design relies on simple geometry (angles and lengths) and being able to predict the shapes resulting from a rule system, and the possibility of fast or realtime previews of the result greatly facilitates small adjustements of sizes, angles and probabilities.
Iterated function systems
Iterated function systems and variants thereof ( fractal flames in particular); shapes and colours are determined by easily understood transformations of shrunk copies of the whole pattern, and since the transformation matrices and deformations have no particular significance they are usually input in fractal software visually and often with a realtime preview; another trend is manual editing starting from a random fractal (the arbitrary parameters are many and mostly independent). Apophysis is a popular and very sophisticated example of this category.
Stochasticsynthesis of fractal noise(typically fractal landscapes) controlled through few simple high level parameters and by trying different Pseudorandom number generatorseeds.
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.
Fractals are sometimes combined with human-assisted
evolutionary algorithms, either by iteratively choosing good-looking specimens in a set of random variations of a fractal artwork and producing new variations, to avoid dealing with cumbersome or unpredictable parameters, or collectively like in the Electric Sheepproject, where people use fractal flames rendered with distributed computingas their screensaverand "rate" the flame they are viewing, influencing the server which reduces the traits of the undesirables, and increases those of the desirables to produce a computer-generated, community-created piece of art.
*cite book|first=Clifford|last=Pickover|authorlink=Clifford A. Pickover|title=Computers, Pattern, Chaos and Beauty|isbn=0-486-41709-3
*cite book|first=Manfred|last=Schroeder|title=Fractals, Chaos, Power Laws|isbn=0-7167-2357-3
*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 
*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 
* [http://commons.wikimedia.org/wiki/Mandelbrot_set#Some_details_of_the_Mandelbrot_set Art and the Mandelbrot set (in commons.wikimedia)]
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