187 Fractals Electronica ♫ Zoom Widescreen

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• published: 26 May 2010
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These are my favorites. Music by Autechre - Bike Most areas of mathematics are too esoteric for layman to artistically appreciate, but with fractals we get a glimpse of the beauty that permeates a mathematician's creative world. Fractals can be classified according to their self-similarity. There are three types of self-similarity found in fractals: Exact self-similarity This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity. Quasi-self-similarity This is a loose form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar but not exactly self-similar. Statistical self-similarity This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of "fractal" trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar. Fractal objects in fractal art 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 are defined by a recurrence relation at each point in a space (such as the complex plane). These fractals 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, and the choice of how to compute a colour from the numbers produced by the calculations. 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. Since few people can predict the shape of a fractal created by a particular formula (except for the famous Mandelbrot Set and perhaps a few others), this kind of fractal art has a pseudo-random element: the artist selects a formula and is presented with an image, which can then be altered in various ways. The initial image acts as a seed to the process of creating fractal images. The process is analogous to starting with a photograph and then applying various effects to it using a graphics program. The most common method of changing the fractal image is zooming into it, especially into a part that looks interesting or promising. Lindenmayer systems and other constructions based on replacement rules Examples of this type include the Peano curve, the Hilbert curve, the Sierpinski gasket, 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 make simulations of 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. The possibility of fast or realtime previews of the result greatly facilitates small adjustments of sizes, angles and probabilities. Iterated function systems Iterated function systems and variants thereof have a fixed geometric replacement rule. An example is the fractal flame. 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. Stochastic synthesis Stochastic synthesis of fractal noise (typically fractal landscapes) is controlled through a few simple high level parameters and by trying different pseudorandom number generator seeds.

187 Fractals Electronica ♫ Zoom Widescreen

• Report rights infringement
• published: 26 May 2010
• views: 6068

These are my favorites. Music by Autechre - Bike Most areas of mathematics are too esoteric for layman to artistically appreciate, but with fractals we get a glimpse of the beauty that permeates a mathematician's creative world. Fractals can be classified according to their self-similarity. There are three types of self-similarity found in fractals: Exact self-similarity This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity. Quasi-self-similarity This is a loose form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar but not exactly self-similar. Statistical self-similarity This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of "fractal" trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar. Fractal objects in fractal art 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 are defined by a recurrence relation at each point in a space (such as the complex plane). These fractals 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, and the choice of how to compute a colour from the numbers produced by the calculations. 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. Since few people can predict the shape of a fractal created by a particular formula (except for the famous Mandelbrot Set and perhaps a few others), this kind of fractal art has a pseudo-random element: the artist selects a formula and is presented with an image, which can then be altered in various ways. The initial image acts as a seed to the process of creating fractal images. The process is analogous to starting with a photograph and then applying various effects to it using a graphics program. The most common method of changing the fractal image is zooming into it, especially into a part that looks interesting or promising. Lindenmayer systems and other constructions based on replacement rules Examples of this type include the Peano curve, the Hilbert curve, the Sierpinski gasket, 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 make simulations of 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. The possibility of fast or realtime previews of the result greatly facilitates small adjustments of sizes, angles and probabilities. Iterated function systems Iterated function systems and variants thereof have a fixed geometric replacement rule. An example is the fractal flame. 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. Stochastic synthesis Stochastic synthesis of fractal noise (typically fractal landscapes) is controlled through a few simple high level parameters and by trying different pseudorandom number generator seeds.