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 ?
This article was focused on the possible use of fractal geometry for secure authentication of users.
This cursory glance gives us six levels to look for self-similarity, wholeness emerging from fragmentation, order and chaos, and other traits of fractal geometry.
2b, the profile exhibits statistical self similarity, and may be described by fractal geometry.
A fascinating article that explores the possibility that fractal geometry is a design principle in biological systems.
Hayles also challenges our assumptions about the cultural ascendancy of science, as she exposes ways in which the flow of influence has actually gone from the humanities into the sciences in fluid dynamics, chaos theory, and fractal geometry.
Applications of fractal geometry have been used to characterize psychological phenomena such as learning curves (Bendler & Shlesinger, 1991), as well as biological phenomena such as pulmonary blood flow (Glenny & Robertson, 1991), the surface properties of cells (Kenough, Hyam, Pink, & Quinn, 1991), and characteristics of the nervous system (King, 1991).
It may be that fractal geometry in relation to the science of chaos is revealing that which underscores events in the real world.
Application of Percolation Theory and Fractal Geometry to Tablet Compaction
PISRS International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics (2011: Sicily, Italy) AMS Special Session on Fractal Geometry in Pure and Applied Mathematics: In Memory of Benoit Mandelbrot (2012: Boston, MA) AMS Special Sessin on Geometry and Analysis on Fractal Spaces (2012: Honolulu, Hawaii) Ed.
Lectures on fractal geometry and dynamical systems.
Fractal geometry, on the other hand, pertains to the fourth dimension.
Mandelbrot's (1983) formulation of fractal geometry meant that scientists could measure not just triangles, rectangles and circles (Euclidean geometry) but unusual shapes they had previously called: grainy, hydralike, in between, pimply, pocky, ramified, seaweedy, strange, tangled, tortuous, wiggly, wispy, wrinkled and the like.