Fractals

Fractal, in?mathematics, any of a class of complex geometric shapes that commonly have “fractional dimensions.?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.

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.

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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?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