fractal geometry

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Related to fractal geometry: Mandelbrot set, Fractals

fractal geometry,

branch of mathematicsmathematics,
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 logical
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 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 geometrygeometry
[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.
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, 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'sMandelbrot, Benoît B.
, 1924–2010, French-American mathematician, b. Warsaw, Poland, Ph.D. Univ. of Paris, 1952. Largely self-taught and considered a maverick in the field of mathematics, he was uncomfortable with the rigorously pure logical analysis prescribed by
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 study of complexitycomplexity,
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.
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 and chaos (see chaos theorychaos 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.
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). 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 graphicscomputer 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 design
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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

the geometry of complex topologies/structures. There have suggestions, especially recently, that social reality is inherently ‘fractal’, its topologies ‘bizarre’ (see Law, 1999). With modern computers, the presentation of these structures visually has become more accessible.
References in periodicals archive ?
Applying fractal geometry to mechanical ventilation may become the new way to ventilate our ARDS/ALI patients.
In his influential book "The Fractal Geometry of Nature" [17], published in 1982, he wrote: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line".
AMS Special Session on Fractal Geometry in Pure and Applied Mathematics: In Memory of Benoit Mandelbrot (2012: Boston, MA.
AIM: The aim of this study was to use fractal geometry to compare the morphometric complexity in the normal andsquamous cell carcinoma of the buccal mucosa, which was fulfilled by estimating the fractal dimensions of ECTI profiles, isolated from histological sections of the tissue specimens.
As the iteration of fractal geometry increases, its resonance frequency decreases, which may lead to an effective antenna miniaturization.
Heretofore, we would naturally conceive the idea of creating a multiband helix with fractal geometry.
Continue reading "Benoit Mandelbrot, the Father of Fractal Geometry, Pens a Disturbing New Memoir" at.
Mathematical processes to create fractal geometry are iterations of order "n".
To varying degrees these include fractal geometry, complexity theory, network theories, chaos theory, power laws (allometric scaling rules), and other "general models" and optimality statements (minimum, maximum, optimum).
Fractal Geometry, closely related to Chaos theory, arose after criticisms of classical Euclidian Geometry.
The people who are making the backgrounds to films like Up are actually using fractal geometry, a very simple bit of code that creates the complexity of the natural world.