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Fractals(fractals)A fractal is a geometric object which is 'broken up' in aradical way. The term fractal was coined in 1975 by Benoît Mandelbrot , from the Latin fractus or 'broken', inorder to call attention to such objects. They are in a number of major aspects different from the more usual 'smooth' objects oftraditional geometry. This is immediately apparent, visually. In many cases a fractal can be generated (for example on a computer screen) by arepeating pattern, typically a recursive or iterative process. This may give it many interesting features, most notably self-similarity and infinite detail regardless of magnification. Fractals can combine structureand irregularity. Fractals of many kinds were originally studied as mathematical objects,and the term "fractal" has been given various precise definitions by mathematicians. Fractal geometry is thebranch of mathematics which studies fractals and the special way they behave. It has often been applied in science , technology , and computer-generated art . Conceptual roots of the theory can be traced to attempts to measure a fractal's perimeter (or area, or volume) in cases wheredefinitions based on calculus fail. Traditional mathematical methods zoomin, in order to simplify the local picture. In constrast, the existence of fractals points up the ways in which thatapproach may fail, if unlimited amounts of ever-finer detail becomes apparent.
HistoryObjects that are now called fractals were discovered and explored long before the word was coined. In 1872 Karl Weierstrass found an example of a function with the non-intuitiveproperty that it is everywhere continuous but nowhere differentiable - the graph of this function would now be called a fractal. In1904 Helge von Koch , dissatisfied with Weierstrass's very abstract andanalytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake . The idea of self-similar curves was taken further by Paul Pierre Lévy who, in his 1938 paper Plane or Space Curves andSurfaces Consisting of Parts Similar to the Whole, described two fractal curves, the Lévy C curve and the Lévy dragon curve . Georg Cantor gave examples of subsets of the real line with unusualproperties - these Cantor sets are also now recognised as fractals. In anattempt to understand objects such as Cantor sets, mathematicians such as Constantin Carathéodory and FelixHausdorff generalised the intuitive concept of dimension to include non-integer values. Iterated functions in the complexplane had been investigated in the late 19th and early 20th centuries by Henri Poincaré , Felix Klein , Pierre Fatou , and GastonJulia . However, without the aid of modern computer graphics, they lacked the means to visualise the beauty of the objectsthat they had discovered. In the 1960s Benoît Mandelbrot started investigatingself-similarity in papers such as How Long Is the Coast of Britain?Statistical Self-Similarity and Fractional Dimension . Taking a highly visual approach, Mandelbrot recognised connectionsbetween these previously unrelated strands of mathematics. In 1975 Mandelbrot coined the word fractal to describeself-similar objects which had no clear dimension. He derived the word fractal from the Latin fractus, meaning broken or irregular, and not from the word fractional, asis commonly believed. Once computer visualization was applied to fractal geometry, it presented a powerful visual argument for fractal geometryconnecting far larger domains of mathematics and science than had previously been considered, particularly in the realm of non-linear dynamics , chaos theory (though a few use the term xaos instead to differentiate between ordered non-linearbehavior and the common meaning of the word), and complexity . One example isplotting Newton's method as a fractal, showing how the boundariesbetween different solutions are fractal, and that the solutions themselves are strange attractors . Fractal geometry was also used for data compression and for modelling complex organic and geological systems, for example the growth of treesor the development of river basins. How Long is the Coast of Britain?Lewis Fry Richardson was a pacifist and a mathematician , studying the cause of war between two countries. He decided to search for a relation between the size of its mutualborder and the probability of two countries going to war. As part of thisresearch, he investigated how the measured length of a border changes as the unit of measurement is changed. Richardson publishedempirical statistics which led to a conjectured relationship. This research was quoted by Mandelbrot in his 1967 paper How Long Is the Coast of Britain? . Suppose the coast of Britain is measured using a 200 km ruler, specifying that both ends of the ruler must touch the coast.Now cut the ruler in half and repeat the measurement, then repeat again:
Notice that the smaller the ruler, the bigger the result. It might be supposed that these values would converge to a finite number representing the "true" length of the coastline, but Richardson proved that themeasurements of the coastline actually tended to infinity . However, to say that thecoastline of Britain is actually infinitely long must be incorrect, since it would require the ability to construct infinitelysmall rulers, something which particle physics says cannot be done,as there is a lower limit to the smallness of a measurement. At the time, Richardson's research was ignored by the scientific community. Today, many see it as one element in the birth ofthe modern study of fractals. Categories of fractals
Fractals can be grouped into three broad categories. These categories are determined from how the fractal is defined orgenerated:
Fractals can also be classified according to their self-similarity. There are three types of self-similarity found infractals:
It should be noted that not all self-similar objects are fractals — e.g., the real line (a straight Euclidean line) is exactly self-similar, but the argument that euclidean objects arefractal is a distinct minority position. Mandelbrot argued that a definition of "fractal" that should include not only "true"fractals, but also traditional Euclidean objects, because irrational numbers on the number line represent complex, non-repeatingproperties. Because a fractal possesses infinite granularity, no natural object can be a fractal. However, natural objects can displayfractal-like properties across a limited range of scales. DefinitionsThe defining characteristics of fractals, while intuitively appealing, are remarkably hard to condense into a mathematicallyprecise definition. Problems with defining fractals include:
The following definitions of fractal have all been proposed, but each one has shortcomings :-
ExamplesTrees and ferns are fractal in nature and can be modelled on a computer using a recursive algorithm . This recursive nature is clear in theseexamples — take a branch from a tree or a frond from a fern and you will see it isa miniature replica of the whole. Not identical, but similar in nature. A relatively simple class of examples is the Cantor sets , in which short andthen shorter (open) intervals are struck out of the unit interval [0, 1],leaving a set that might (or might not) actually be self-similar under enlargement, and might (or might not) have dimensiond that has 0 < d < 1. A simple recipe, such as excluding the digit 7 from decimal expansions ,is self-similar under 10-fold enlargement , and also has dimension log 9/log10 (this value is the same, no matter what base is chosen), showing the connection of the two concepts. Fractals are generally irregular (not smooth) in shape, and thus are not objects definable by traditional geometry . That means that fractals tend to have significant detail, visible at anyarbitrary scale; when there is self-similarity, this can occur because "zooming in" simply shows similar pictures. Such sets areusually defined instead by recursion . For example, a normal Euclidean shape, such as a circle, looks flatter andflatter as it is magnified. At infinite magnification it would be impossible to tell the difference between the circle and astraight line. Fractals are not like this. The conventional idea of curvature ,which represents the reciprocal of the radius of an approximating circle, cannot usefully apply because it scales away. Instead, with a fractal, increasingthe magnification reveals more detail that was previously invisible. Some common examples of fractals include the Mandelbrot set , Lyapunov fractal , Cantor set , Sierpinski carpet and triangle , Peanocurve , and the Koch snowflake . Fractals can be deterministic orstochastic. Chaotic dynamical systems are often (if not always) associatedwith fractals. The Mandelbrot set contains whole discs, so has dimension 2. This is not surprising. What is truly surprising isthat the boundary of the Mandelbrot set also has Hausdorff dimension 2 and topological dimension 1. Approximate fractals are easily found in nature. These objects display complex structure over an extended, but finite, scalerange. These naturally occurring fractals (like clouds, mountains, river networks, and systems of blood vessels) have both lowerand upper cut-offs, but they are separated by several orders ofmagnitude . Despite being ubiquitous, fractals were not much studied until well into the twentieth century , and general definitions came later. Harrison [1] extended Newtonian calculus to fractal domains , including the theorems of Gauss , Green , and Stokes . Fractals are usually calculated by computers with fractal software. See below for a list. Random fractals have the greatest practical use because they can be used to describe many highly irregular real-world objects.Examples include clouds, mountains, turbulence , coastlines, and trees. Fractaltechniques have also been employed in fractal imagecompression , as well as a variety of scientific disciplines. See also
References, further reading
Fractal generators
External links
fratals, self, fractal, similarity, fracatls, mandelbrot, fractas, dimension, farctals, definition, fractasl, non, fractls, example, fracals, long, frcatals, britain, , defined, ractals, word, frctals, detail, rfactals, nature, fractlas, object, factals, include, fratcals, line This article is completely or partly from Wikipedia - The Free Online Encyclopedia. Original Article. The text on this site is made available under the terms of the GNU Free Documentation Licence. We take no responsibility for the content, accuracy and use of this article. Anoca.org Encyclopedia 0.04s |