Fractal

A fractal is a geometric object which is rough or irregular on all scales of length, and so which appears to be 'broken up' in a radical way. Some of the best examples can be divided into parts, each of which is similar to the original object. Fractals are said to possess infinite detail, and they may actually have a self-similar structure that occurs at different levels of magnification. In many cases, a fractal can be generated by a repeating pattern, in a typically recursive or iterative process. The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin fractus or "broken". Before Mandelbrot coined his term, the common name for such structures (the Koch snowflake, for example) was monster curve.

Categories of fractals

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Even 100 times magnification of the Mandelbrot set uncovers fine detail resembling the full set.

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Fractals can be grouped into three broad categories. These categories are determined from how the fractal is defined or generated:

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:* Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals.

Related Topics:
Iterated function system - Cantor set - Sierpinski carpet - Sierpinski gasket - Peano curve - Koch snowflake - Harter-Heighway dragon curve - T-Square - Menger sponge

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:* Fractals defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set and the Lyapunov fractal. These are also called escape-time fractals.

Related Topics:
Recurrence - Mandelbrot set - Lyapunov fractal

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:* Random fractals, generated by stochastic rather than deterministic processes, for example, fractal landscapes and Lévy flights.

Related Topics:
Fractal landscapes - Lévy flight

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Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:

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:*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.

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:*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.

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:*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.

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It should be noted that not all self-similar objects are fractals — e.g., the real line (a straight Euclidean line) is exactly self-similar, but the argument that Euclidean objects are fractals is a distinct minority position. Mandelbrot argued that a definition of "fractal" should include not only "true" fractals, but also traditional Euclidean objects, because irrational numbers on the number line represent complex, non-repeating properties.

Related Topics:
Real line - Irrational number

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Because a fractal possesses infinite granularity, no natural object can be a fractal. However, natural objects can display fractal-like properties across a limited range of scales.

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