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.

History

Contributions from classical analysis

Objects that are now called fractals were discovered and explored long before the word was coined. In 1872 Karl Weierstrass found an example of a function with the non-intuitive property that it is everywhere continuous but nowhere differentiable - the graph of this function would now be called a fractal. In 1904 Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake. The idea of self-similar curves was taken further by Paul Pierre Lévy who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, the Lévy C curve.

Related Topics:
Karl Weierstrass - Continuous - Differentiable - Helge von Koch - Koch snowflake - Paul Pierre Lévy - Lévy C curve

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Georg Cantor gave examples of subsets of the real line with unusual properties - these Cantor sets are also now recognised as fractals. Iterated functions in the complex plane had been investigated in the late 19th and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatou, and Gaston Julia. However, without the aid of modern computer graphics, they lacked the means to visualize the beauty of the objects that they had discovered.

Related Topics:
Georg Cantor - Subset - Cantor set - Complex plane - Henri Poincaré - Felix Klein - Pierre Fatou - Gaston Julia

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Aspects of set description

In an attempt to understand objects such as Cantor sets, mathematicians such as Constantin Carathéodory and Felix Hausdorff generalised the intuitive concept of dimension to include non-integer values. This was part of the general movement in the first part of the twentieth century to create a descriptive set theory; that is, a continuation of the direction of Cantor's research that was able in some way to classify sets of points in Euclidean space. The definition of Hausdorff dimension is geometric in nature, although it is based technically on tools from mathematical analysis. This direction was taken up by Besicovitch, amongst others; it is different in character from the logical investigations that made up much of the descriptive set theory of the 1920s and 1930s. Both of these fields were pursued for some time afterwards, but mainly by specialists.

Related Topics:
Mathematician - Constantin Carathéodory - Felix Hausdorff - Integer - Descriptive set theory - Euclidean space - Hausdorff dimension - Mathematical analysis - Besicovitch

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Mandelbrot's contributions

In the 1960s Benoît Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. This built on earlier work by Lewis Fry Richardson. Taking a highly visual approach, Mandelbrot recognised connections between these previously unrelated strands of mathematics. In 1975 Mandelbrot coined the word fractal to describe self-similar objects which had no clear dimension. He derived the word fractal from the Latin fractus, meaning broken or irregular, and not from the word fractional, as is commonly believed. However, fractional itself is derived ultimately from fractus as well.

Related Topics:
Benoît Mandelbrot - How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension - Lewis Fry Richardson - Latin

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Once computer visualization was applied to fractal geometry, it presented a powerful visual argument for fractal geometry connecting far larger domains of mathematics and science than had previously been considered, particularly in the realm of non-linear dynamics, chaos theory (though a few use the term xaos instead to differentiate between ordered non-linear behaviour and the common meaning of the word), and complexity. One example is plotting Newton's method as a fractal, showing how the boundaries between different solutions are fractal, and that the solutions themselves are strange attractors. Fractal geometry was also used for data compression and for modelling complex organic and geological systems, for example the growth of trees or the development of river basins.

Related Topics:
Non-linear dynamics - Chaos theory - Complexity - Newton's method - Strange attractor - Data compression

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Harrison http://math.berkeley.edu/~harrison/research/publications/ extended Newtonian calculus to fractal domains, including the theorems of Gauss, Green, and Stokes.

Related Topics:
Calculus - Fractal domain - Gauss - Green - Stokes

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