fractal geometry, branch of mathematics mathematics, deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or
..... Click the link for more information. concerned with irregular patterns made of parts that are in some way similar to the whole, e.g., twigs and tree branches, a property called self-similarity or self-symmetry. Unlike conventional geometry geometry [Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.
..... Click the link for more information. , which is concerned with regular shapes and whole-number dimensions, such as lines (one-dimensional) and cones (three-dimensional), fractal geometry deals with shapes found in nature that have non-integer, or fractal, dimensions—linelike rivers with a fractal dimension of about 1.2 and conelike mountains with a fractal dimension between 2 and 3.

Fractal geometry developed from Benoit Mandelbrot's Mandelbrot, Benoit B. (bənwä` măn`dəlbrō', Fr. mäNdĕlbrô`), 1924–, French mathematician, b.
..... Click the link for more information. study of complexity complexity, in science, field of study devoted to the process of self-organization. The basic concept of complexity is that all things tend to organize themselves into patterns, e.g.
..... Click the link for more information. and chaos (see chaos theory chaos theory, in mathematics, physics, and other fields, a set of ideas that attempts to reveal structure in aperiodic, unpredictable dynamic systems such as cloud formation or the fluctuation of biological populations.
..... Click the link for more information. ). Beginning in 1961, he published a series of studies on fluctuations of the stock market, the turbulent motion of fluids, the distribution of galaxies in the universe, and on irregular shorelines on the English coast. By 1975 Mandelbrot had developed a theory of fractals that became a serious subject for mathematical study. Fractal geometry has been applied to such diverse fields as the stock market, chemical industry, meteorology, and computer graphics computer graphics, the transfer of pictorial data into and out of a computer . Using analog-to-digital conversion techniques, a variety of devices—such as curve tracers, digitizers, and light pens—connected to graphic computer terminals , computer-aided
..... Click the link for more information. .

Bibliography

See B. B. Mandelbrot, The Fractal Geometry of Nature (1983); K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications (1990); H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science (1992).

fractal geometry

In mathematics, the study of complex shapes with the property of self-similarity, known as fractals. Rather like holograms that store the entire image in each part of the image, any part of a fractal can be repeatedly magnified, with each magnification resembling all or part of the original fractal. This phenomenon can be seen in objects like snowflakes and tree bark. The term fractal was coined by Benoit B. Mandelbrot in 1975. This new system of geometry has had a significant impact on such diverse fields as physical chemistry, physiology, and fluid mechanics; fractals can describe irregularly shaped objects or spatially nonuniform phenomena that cannot be described by Euclidean geometry. Fractal simulations have been used to plot the distributions of galactic clusters and to generate lifelike images of complicated, irregular natural objects, including rugged terrains and foliage used in films. See also chaos theory.