# fractal

fractal
/frak"tl/, n. Math., Physics.
a geometrical or physical structure having an irregular or fragmented shape at all scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a curve or the flow rate in a porous medium, behave as if the dimensions of the structure (fractal dimensions) are greater than the spatial dimensions.
[ < F fractale, equiv. to L fract(us) broken, uneven (see FRACTUS) + -ale -AL2; term introduced by French mathematician Benoit Mandelbrot (born 1924) in 1975]

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in mathematics, any of a class of complex geometric shapes that commonly have “fractional dimension,” a concept first introduced by the mathematician Felix Hausdorff in 1918. Fractals are distinct from the simple figures of classical, or Euclidean, geometry—the square, the circle, the sphere, and so forth. They are capable of describing many irregularly shaped objects or spatially nonuniform phenomena in nature such as coastlines and mountain ranges. The term fractal, derived from the Latin word fractus (“fragmented,” or “broken”), was coined by the Polish-born mathematician Benoit B. Mandelbrot. See the animation of the Mandelbrot fractal set.

Although the key concepts associated with fractals had been studied for years by mathematicians, and many examples, such as the Koch or “snowflake” curve were long known, Mandelbrot was the first to point out that fractals could be an ideal tool in applied mathematics for modeling a variety of phenomena from physical objects to the behavior of the stock market. Since its introduction in 1975, the concept of the fractal has given rise to a new system of geometry that has had a significant impact on such diverse fields as physical chemistry, physiology, and fluid mechanics.

Many fractals possess the property of self-similarity, at least approximately, if not exactly. A self-similar object is one whose component parts resemble the whole. This reiteration of details or patterns occurs at progressively smaller scales and can, in the case of purely abstract entities, continue indefinitely, so that each part of each part, when magnified, will look basically like a fixed part of the whole object. In effect, a self-similar object remains invariant under changes of scale—i.e., it has scaling symmetry. This fractal phenomenon can often be detected in such objects as snowflakes and tree barks. All natural fractals of this kind, as well as some mathematical self-similar ones, are stochastic, or random; they thus scale in a statistical sense.

Another key characteristic of a fractal is a mathematical parameter called its fractal dimension. Unlike Euclidean dimension, fractal dimension is generally expressed by a noninteger—that is to say, by a fraction rather than by a whole number. Fractal dimension can be illustrated by considering a specific example: the snowflake curve defined by Helge von Koch (Koch, Niels Fabian Helge von) in 1904. It is a purely mathematical figure with a six-fold symmetry, like a natural snowflake. It is self-similar in that it consists of three identical parts, each of which in turn is made of four parts that are exact scaled-down versions of the whole. It follows that each of the four parts itself consists of four parts that are-scaled down versions of the whole. There would be nothing surprising if the scaling factor were also four, since that would be true of a line segment or a circular arc. However, for the snowflake curve, the scaling factor at each stage is three. The fractal dimension, D, denotes the power to which 3 must be raised to produce 4—i.e., 3D= 4. The dimension of the snowflake curve is thus D = log 4/log 3, or roughly 1.26. Fractal dimension is a key property and an indicator of the complexity of a given figure.

Fractal geometry with its concepts of self-similarity and noninteger dimensionality has been applied increasingly in statistical mechanics, notably when dealing with physical systems consisting of seemingly random features. For example, fractal simulations have been used to plot the distribution of galaxy clusters throughout the universe and to study problems related to fluid turbulence. Fractal geometry also has contributed to computer graphics. Fractal algorithms have made it possible to generate lifelike images of complicated, highly irregular natural objects, such as the rugged terrains of mountains and the intricate branch systems of trees.

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Universalium. 2010.

### Look at other dictionaries:

• fractal — fractal, ale [ fraktal ] adj. et n. f. • 1975; dér. sav. du lat. fractus « brisé » ♦ Didact. Objet fractal : objet mathématique servant à décrire des objets de la nature dont les formes découpées laissent apparaître à des échelles d observation… …   Encyclopédie Universelle

• fractal — FRACTÁL, Ă, fractáli, e, s.m. şi adj. (Se spune despre o) Curbă sau formă foarte neregulată pentru care orice parte aleasă în mod convenabil devine similară ca formă cu o alta mai mare sau mai mică în momentul în care cea dintâi este mărită,… …   Dicționar Român

• fractal — 1975, from Fr. fractal, from L. fractus interrupted, irregular, lit. broken, pp. of frangere to break (see FRACTION (Cf. fraction)). Coined by French mathematician Benoit Mandelbrot in Les Objets Fractals. Many important spatial patterns of… …   Etymology dictionary

• fractal — adj. 2 g. 1.  [Matemática] Diz se de objetos matemáticos cuja forma é irregular e fragmentada: Objeto fractal; geometria fractal. • s. f. 2.  [Matemática] Conjunto geométrico ou objeto natural cujas partes têm a mesma estrutura (irregular e… …   Dicionário da Língua Portuguesa

• fractal — (Del fr. fractal, voz inventada por el matemático francés B. Mandelbrot en 1975, y este del lat. fractus, quebrado). m. Fís. y Mat. Figura plana o espacial, compuesta de infinitos elementos, que tiene la propiedad de que su aspecto y distribución …   Diccionario de la lengua española

• Fractal —   [engl.], Fraktal …   Universal-Lexikon

• fractal —  Fractal  Фрактал   Бесконечная самоподобная геометрическая фигура, каждый фрагмент которой повторяется при уменьшении масштаба. Фракталами также называют самоподобные множества нецелой размерности. Самоподобное множество множество, которое можно …   Толковый англо-русский словарь по нанотехнологии. - М.

• fractal — ☆ fractal [frak′təl ] n. [< L fractus (see FRACTUS) + AL] Geom. an extremely irregular line or surface formed of an infinite number of similarly irregular sections: fractals have fractional dimension between one and two, or between two and… …   English World dictionary

• Fractal — A fractal is generally a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced size copy of the whole, [cite book last = Mandelbrot first = B.B. title = The Fractal Geometry of… …   Wikipedia

• Fractal — Este artículo o sección sobre matemáticas necesita ser wikificado con un formato acorde a las convenciones de estilo. Por favor, edítalo para que las cumpla. Mientras tanto, no elimines este aviso puesto el 16 de octubre de 2010. También puedes… …   Wikipedia Español