fractal geometry


Also found in: Dictionary, Thesaurus, Medical, Financial, Wikipedia.
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
..... 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 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.
..... 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'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
..... Click the link for more information.
 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.
..... Click the link for more information.
 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.
..... 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 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
..... 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

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 ?
(16) designed a new methodology based on fractal geometry that includes the creation of the concept of degrees of similarity, which allows the comparison of the fractal dimensions of the ventricular contours in systole, diastole and totality from images with a diagnosis of cardiac ventricular normality and severe abnormality.
Li, "Analytical derivation of Brooks-Corey type capillary pressure models using fractal geometry and evaluation of rock heterogeneity," Journal of Petroleum Science and Engineering, vol.
Fractal geometry for the characterisation of urban-related states: Greater Montreal Case, Harmonic and Fractal Image Analysis--HarFA e-journal, 30-34.
Ramakrishnan, "A fractal geometry model for evaluating permeabilities of porous preforms used in liquid composite molding," International Journal of Heat and Mass Transfer, vol.
The extension of the concepts of fractal geometry toward the biomedical sciences has led to significant progress in understanding complex functional properties and structural features [23-29].
From the perspective of the complex systems theory, we can find, in the abovedescribed phenomena, the main characteristics specific to complex systems: nonlinear dynamics, fractal geometry, with a potential latent informational energy, along with a dynamics of a practically infinite diversity, obtained through topological transformations within the complex space of the phase.
They build on the global popularity of the movie and its music to introduce fractal geometry into Primary mathematics.
Sharp claims a number of compositional influences, "from Coltrane to Zappa to Xenakis and beyond." He also infuses his compositions with mathematics from chaos theory and fractal geometry. The MAE make this and the Sanford work above sound "easy," both technically and in their ability to communicate the intensity of the music.
Fractal geometry consists of infinite repeating patterns.
Whereas Euclidean geometry consists of smooth, straight lines, fractal geometry consists of rough or fragmented shapes that can be split into parts, each of which is a reduced-size copy of the whole.