fractal

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Related to Fractal curve: fractal dimensionality

fractal

[′frakt·əl]
(mathematics)
A geometrical shape whose structure is such that magnification by a given factor reproduces the original object.

fractal

(mathematics, graphics)
A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a smaller copy of the whole. Fractals are generally self-similar (bits look like the whole) and independent of scale (they look similar, no matter how close you zoom in).

Many mathematical structures are fractals; e.g. Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set and Lorenz attractor. Fractals also describe many real-world objects that do not have simple geometric shapes, such as clouds, mountains, turbulence, and coastlines.

Benoit Mandelbrot, the discoverer of the Mandelbrot set, coined the term "fractal" in 1975 from the Latin fractus or "to break". He defines a fractal as a set for which the Hausdorff Besicovich dimension strictly exceeds the topological dimension. However, he is not satisfied with this definition as it excludes sets one would consider fractals.

sci.fractals FAQ.

See also fractal compression, fractal dimension, Iterated Function System.

Usenet newsgroups: news:sci.fractals, news:alt.binaries.pictures.fractals, news:comp.graphics.

["The Fractal Geometry of Nature", Benoit Mandelbrot].

References in periodicals archive ?
Obviously, [(dl).sup.m], as a m-dimensional volume, is a m-dimensional Hausdorff measure; therefore, formula (7) implies that the differences of order m, [d.sup.m] x(l), can be also thought of as a measure for describing the length of a m-dimensional fractal curve. In this case, the order of differences [d.sup.m] x(l) represents the Hausdorff dimension m.
And the Koch fractal curve appeared in 1904 after a publication of the mathematician Niels Fabian Helge von Koch [14,15].
According to dimensional analysis, for the Euclidean length of the fractal curve, C, the perimeter of the aggregate particle, [P.sub.E], can be expressed as
Electrical properties of a wire antenna based on the fractal curve are also associated with these shape parameters.
In the framework of SRT we assume that the movements of complex system entities take place on continuous but nondifferentiable curves (fractal curves) so that all physical phenomena involved in the dynamics depend not only on the space-time coordinates but also on the space-time scales resolution.
In this paper, we investigate terahertz characteristics of SRRs combined with square Sierpinski (SS) fractal curve. Firstly, in Section 2, in order to analyze the influence of different geometrical parameters on the performance of the SS-SRRs with different fractal order, we simulated the structures using the commercial CST Microwave Studio 2011 with the frequency domain solver.
A model fractal curve gives a linear graph, its negative slope can be interpreted as the quantity (1-D) according to M.
This proposed multifractal derives from Sierpinski Gasket and Kochlike fractal curve. The two fractals were combined in superiority-inferiority order with individual iterative.
For fractal curves, the curvature radius [r.sub.c] depends on the scale at which the observation is made.
Fractal curves can be quantified using fractal dimension, a noninteger unit between 1 and 2, with higher integer reflecting the increased of complexity as represented by the density of the space-filling pattern of the retinal vascular tree.
From the above demonstration, we conclude that proper interactive adjustments of the scaling factors give us a wide variety of monotonic fractal curves that can be used in various scientific and engineering problems for aesthetic modifications or where the classical rational quadratic interpolant is insufficient and unsatisfactory.