Mandelbrot set

Fig. 1.11 Fractal structures (computer generated) [28] The four parts of the figure (from left to right) show a series of Mandelbrot sets [26] after 5, 10, 50, and 150 iterations.

In principle. .. [the Mandelbrot Set] could have been discovered as soon as men learned to count. But even if they never grew tired, and never made a mistake, all the human beings who have ever existed would not have sufficed to do the elementary arithmetic required to produce a Mandelbrot Set of quite modest magnification. 

The young man pointed at the monitor. That beautiful image, ladies and gentlemen, is a Mandelbrot set. Herman noticed that the man had said the last two words with a certain degree of reverence, almost awe. The young man went on to explain... 

Herman said, When I walked up, you had the Mandelbrot set on display. Now you have all these spirals. They re pretty. What are they ... 

That s a close-up of the Mandelbrot set, a magnification of a tiny part of the original. He pointed to little black specks within the spirals. Each of these bumpy-warty bits would look just like the original bushy shape if we magnify them. Currently we re looking at a section magnified 1,000 times. It s called the Valley of the Sea Horses because the spirals resemble sea horses tails. ... 

Herman was pulling up a dented, rusty folding chair at Fractal Tours. A young black man was demonstrating a particularly beautiful section of the Mandelbrot set known as tbe Valley of the Sea Horses. With its strange, knotty spirals it was the most beautiful thing Herman had ever seen. It brought him a sudden sense of wholeness. 

Although many mathematicians were against funding the project, some commented that they would support funding of the project more than they would support a project for studying the Mandelbrot set, a famous fractal object. Others said they did not favor the project but had no objection if the Chudnovsky brothers wanted to carry out such a task. 

FIGURE 13.18 Visual presentation of the Mandelbrot set. (Generated by the ManpWin v2.8. http // www.deleeuv.com.au. With permission.)... 

The applications of chaos theory are infinite. Random systems produce patterns of spooky understandable irregularity. From the Mandelbrot set to turbulence, feedback, and strange... 

Fig. 15.6. Fractals, (a) SierpMski carpet, (b) Mandelbrot set. Note that the incredibly complex (and beautiful) set exhibits some features of self-similarity, e.g., the central turtle is repeated many times in different scales and variations, as does the fantasy creature in the form of an S. On top of this, the system resembles the complexity of the Universe using more and more powerful magnifying glasses, we encounter ever new elements that resemble (but not just copy) those we have already seen. From J. Gleick, Chaos , Viking, New York, 1988, reproduced by permission of the author.

Mandelbrot Set Set of aU complex numbers csuch that iteration of the fimction f x) = + c, starting at x = 0, does not go into infinity. 

With the advent of the computer, numerous simulated unique patterns such as the rich Julia and Mandelbrot sets have been created this is computer graphical generation by pure mathematical models. However, these unique patterns were formed only in the computer by the mathematical means. The twin problems of how to realize these patterns in a real physical system and how to bridge the gap between the real physical system and the pure mathematical model have stimulated the interest of many natural scientists. Since the discovery of buckminsterfullerene its physical properties and interactions with atoms, molecules, polymers, and crystalline surfaces have also been the subject of intensive investigations. Moreover, fullerene-doped polymers show particular promise as new materials with novel electrical, optical, and/or optoelectrical properties. The present study focuses on the striking morphological properties of fullerene-TCNQ multilayered thin films formed under proper growth conditions and explores the relationship between the real physical system and the pure mathematical model. 

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

The intricate structure of the set S of scattering singularities we encountered in connection with the reaction function in Section 1.1 can be characterized using the concept of fractals introduced by Mandelbrot in 1975 (see also Mandelbrot (1977, 1983)). Fractals are discussed in Section 2.3. 

Surfaces of most materials, including natural and synthetic, porous and non-porous, and amorphous and crystalline, are fractal on a molecular scale. Mandelbrot defines that a fractal object has a dimension D which is greater than the geometric or physical dimension (0 for a set of disconnected points, 1 for a curve, 2 for a surface, and 3 for a solid volume), but less than or equal to the embedding dimension in an enclosed space (embedding Euclidean space dimension is usually 3). Various methods, each with its own advantages and disadvantages, are available to obtain... 

Fractals are geometric structures of fractional dimension their theoretical concepts and physical applications were early studied by Mandelbrot [Mandelbrot, 1982]. By definition, any structure possessing a self-similarity or a repeating motif invariant under a transformation of scale is caWcd fractal and may be represented by a fractal dimension. Mathematically, the fractal dimension Df of a set is defined through the relation ... 

Note that fractals (self-similar sets with fractal dimension) were first studied and described by mathematicians long before the publications of Mandelbrot, when such fundamental definitions as function, line, surface, and shape were analyzed. 

Kerker, M. et ah. Determination of particle size by the minima and maxima in the angular dependence of the scattered light. Range of validity of the method, J. Colloid Set, 19, 193-200, 1964. Mandelbrot, B., The Fractal Geometry of Nature, Freeman, San Franciso, CA, 1983. 

From the above analysis follows the necessity of defining sets of fractional dimension. Commonly, such sets are the fractal sets (Greek fractus — broken) introduced to physics by Mandelbrot. We will confine ourselves... 

Fractals are self-similar objects invariant with respect to local dilatations, i.e., objects that reproduce the same shape during observation at various magnifications. The concept of fractals as self-similar ensembles was introduced by Mandelbrot [1], who defined a fractal as a set for which the Hausdorff-Besicovitch dimension always exceeds the topological dimension. The fractal dimension D of an object inserted in a d-dimensional Euclidean space varies from 1 to d. Fractal objects are the natural filling of the gap between known Euclidean objects with integer dimensions 0,1, 2, 3. .. The majority of naturally existing objects proved to be fractal, which is the main reason for the vigorous development of the methods of fractal analysis. 

Multifractal can be considered as an alternate superposition of a large number of monofractals with different dimensions and mainly characterized by the multifractal spectrum f(o) or the generalized dimension Dq. The multifractal spectrum f(o) is visually described through the multifractal curve f(o)-a, which contains a set of parameters to precisely and intuitively describe multifractal (Evertsz Mandelbrot 1992). 

Mandalbrot sat A fractal that produces complex self-similar patterns. In mathematical terms, it is the set of values of cthat make the seriesz +l = z ) + cconverge, where c and z are complex numbers and z begins at the origin (0,0). It was discovered by and named after the Polish-born French mathematician Benoit Mandelbrot (1924- ). 

This movement led to the pubUcation by Mandelbrot in 1975 [1] of an essay in which he highlighted the similarities among the then-known continuous, nowhere-differentiable sets. He coined for these sets the term fractal , to emphasize the fact that their Hausdorff dimensions are often fractional. In the words of Dyson [54], fractal is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played]... an historical role... in the development of pure mathematics . 

Mandelbrot beUeved initially that one would do better without a precise definition of fractals. His original essay [1] contains none. By 1977, however, he saw the need to produce at least a tentative definition. It is the now classical statement that a fractal is a set for which the Hausdorff dimension strictly exceeds the topological dimension [4, 5,10]. For example, the Cantor set is a fractal, according to this viewpoint, since Dh = 0.631 > Z)r=0. 

The Cantor set is simply the dust of points that remain. The number of these points is infinite, but their total length is zero. Mandelbrot recognized the Cantor set as a model for the occurrences of... 

FIGURE 13.19 Visual presentation of the Cantor set. (Mandelbrot, B.B., The Fractal Geometry of Nature, W.H. Freeman and Company, San Francisco, 1977, pp. 80. With permission.)... 

Originally, Greek philosophers thought that the universe was continuous and that the world could be described by lines, areas, and volumes, according to the geometry (literally earth-measurement ), set down by Euclid, for example, around 300 BC. It became evident a few centuries ago that shapes are not continuous but are composed of similar but smaller shapes as they become more magnified. Thus a tree looks more complex the more it is studied on a finer scale, as shown in Fig. 5.7. This concept is the basis of fractal geometry which has been described by Mandelbrot. ... 

The discovery of a fractal symmetry in random matter was great news for the materials scientists. Until a decade or two ago a systematic study of the influence of synthesis conditions on morphology and therefore on the properties of random matter was difficult and therefore rarely done because a quantitative measure of random matter did not exist. The symmetry in random spatial distributions was recognized after the introduction of the concept of fractality by Mandelbrot. A fractal is a set that has a dimension that usually is not an integer. It is not 0 (a point), 1 (a line), 2 (a plane), or 3 (a volume) but a fractional number somewhere between... 

Benoit Mandelbrot, French mathematician, pioneer of fractal geometry, and author of the book The Fractal Geometry of Nature (1982). Mandelbrot, who worked at IBM for over thirty years, used a computer to plot images of fractals called Julia sets in 1979. (Hank Morgan / Photo Researchers, Inc.)... 

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