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.

Definitions

The most characteristic property of fractals is that they are generally irregular (not smooth) in shape, and thus are not objects definable by traditional geometry. That means that fractals tend to have significant detail, visible at any arbitrary scale; when there is self-similarity, this can occur because "zooming in" simply shows similar pictures. Such sets are usually defined instead by recursion.

Related Topics:
Smooth - Geometry - Recursion

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For example, a normal Euclidean shape, such as a circle, looks flatter and flatter as it is magnified. At infinite magnification it would be impossible to tell the difference between the circle and a straight line. Fractals are not like this. The conventional idea of curvature, which represents the reciprocal of the radius of an approximating circle, cannot usefully apply because it scales away. Instead, with a fractal, increasing the magnification reveals more detail that was previously invisible.

Related Topics:
Euclidean - Circle - Curvature - Reciprocal - Radius

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The defining characteristics of fractals, while intuitively appealing, are remarkably hard to condense into a mathematically precise definition. Mandelbrot defined fractal as "a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension". For an entirely self-similar fractal, the Hausdorff dimension is equal to the Minkowski-Bouligand dimension.

Related Topics:
Hausdorff-Besicovitch dimension - Topological dimension - Self-similar - Minkowski-Bouligand dimension

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Problems with defining fractals include:

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:*There is no precise meaning of "too irregular".

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:*There is no single definition of "dimension".

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:*There are many ways that an object can be self-similar.

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:*Not every fractal is defined recursively.

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