Fractal
2007 Schools Wikipedia Selection. Related subjects: Mathematics
In colloquial usage, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced/size copy of the whole". The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning broken or fractured.
A fractal as a geometric object generally has the following features:
 fine structure at arbitrarily small scales
 is too irregular to be easily described in traditional Euclidean geometric language.
 is selfsimilar (at least approximatively or stochastically)
 has a Hausdorff dimension that is greater than its topological dimension (although this requirement is not met by spacefilling curves such as the Hilbert curve)
 has a simple and recursive definition.
Due to them appearing similar at all levels of magnification, fractals are often considered to be 'infinitely complex'. Obvious examples include clouds, mountain ranges and lightning bolts. However, not all selfsimilar objects are fractals — for example, the real line (a straight Euclidean line) is formally selfsimilar but fails to have other fractal characteristics.
History
Objects that are now described as fractals were discovered and described centuries ago. Ethnomathematics like Ron Eglash's African Fractals ( ISBN 0813526132) describes pervasive fractal geometry in indigenous African craft work. In 1525, the German Artist Albrecht Dürer published The Painter's Manual, in which one section is on "Tile Patterns formed by Pentagons". The Dürer's Pentagon largely resembled the Sierpinski carpet, but based on pentagons instead of squares..
The idea of "recursive selfsimilarity" was originally developed by the philosopher Leibniz and he even worked out many of the details. In 1872, Karl Weierstrass found an example of a function with the nonintuitive property that it is everywhere continuous but nowhere differentiable — the graph of this function would now be called a fractal. In 1904, Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake. In 1915, Waclaw Sierpinski constructed his triangle and one year later his carpet. Actually, these fractals were described as curves, which is hard to realize with the wellknown modern constructions. The idea of selfsimilar curves was taken further by Paul Pierre Lévy who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, the Lévy C curve.
Georg Cantor gave examples of subsets of the real line with unusual properties — these Cantor sets are also now recognized as fractals. Iterated functions in the complex plane had been investigated in the late 19th and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatou and Gaston Julia. However, without the aid of modern computer graphics they lacked the means to visualize the beauty of many of the objects that they had discovered.
In the 1960s Benoît Mandelbrot started investigating selfsimilarity in papers such as How Long Is the Coast of Britain? Statistical SelfSimilarity and Fractional Dimension. This built on earlier work by Lewis Fry Richardson. In 1975, Mandelbrot coined the word fractal to denote an object whose HausdorffBesicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computerconstructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".
Examples
A relatively simple class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, spacefilling curve, Koch curve. Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals can be deterministic (all the above) or stochastic (that is, nondeterministic). For example the trajectories of the Brownian motion in the plane have Hausdorff dimension 2.
Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals (see attractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so it has the Hausdorff dimension equal to its topological dimension of 2 —but what is truly surprising is that the boundary of the Mandelbrot set also has the Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by M. Shishikura in 1991. A closely related fractal is the Julia set.
Selfsimilarity dimension
The selfsimilarity dimension is a simplification of the Hausdorff dimension which can be applied to exactly selfsimilar objects.
The following analysis of the Koch Snowflake suggests how selfsimilarity can be used to analyze fractal properties.
The total length of a number, N, of small steps, L, is the product NL. Applied to the boundary of the Koch snowflake this gives a boundless length as L approaches zero. But this distinction is not satisfactory, as different Koch snowflakes do have different sizes. A solution is to measure, not in meter, m, nor in square meter, m², but in some other power of a meter, m^{x}. Now 4N(L/3)^{x} = NL^{x}, because a three times shorter steplength requires four times as many steps, as is seen from the figure. Solving that equation gives x = (log 4)/(log 3) ≈ 1.26186. So the unit of measurement of the boundary of the Koch snowflake is approximately m^{1.26186}.
More generally, suppose that a fractal consists of N identical parts that are similar to the entire fractal with the scale factor of L and that the intersection between part is of the Lebesgue measure 0. Then the Hausdorff dimension of the fractal is . For example, the Hausdorff dimension of
 the Cantor set is ,
 the Sierpinski gasket is ,
 the Sierpinski carpet is ,
and so on. Even more generally one may assume that each of N parts is similar to the fractal with a different scale factor L_{i}, i = 1...N. Then the Hausdorff dimension can be calculated by solving the following equation in the variable s:
Generating fractals
Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set. 
Three common techniques for generating fractals are:

 Escapetime fractals — These are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal and the Lyapunov fractal.
 Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, HarterHeighway dragon curve, TSquare, Menger sponge, are some examples of such fractals.
 Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. The latter yields socalled mass or dendritic fractals, for example, diffusionlimited aggregation or reactionlimited aggregation clusters.
Classification of fractals
Fractals can also be classified according to their selfsimilarity. There are three types of selfsimilarity found in fractals:

 Exact selfsimilarity — This is the strongest type of selfsimilarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact selfsimilarity.
 Quasiselfsimilarity — This is a loose form of selfsimilarity; the fractal appears approximately (but not exactly) identical at different scales. Quasiselfsimilar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasiselfsimilar but not exactly selfsimilar.
 Statistical selfsimilarity — This is the weakest type of selfsimilarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of "fractal" trivially imply some form of statistical selfsimilarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically selfsimilar, but neither exactly nor quasiselfsimilar.
Fractals in nature
Approximate fractals are easily found in nature. These objects display selfsimilar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, mountains, river networks, cauliflower or broccoli, and systems of blood vessels.
Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples — a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature.
The surface of a mountain can be modeled on a computer by using a fractal: Start with a triangle in 3D space and connect the central points of each side by line segments, resulting in 4 triangles. The central points are then randomly moved up or down, within a defined range. The procedure is repeated, decreasing at each iteration the range by half. The recursive nature of the algorithm guarantees that the whole is statistically similar to each detail.
Fractal patterns have been found in the paintings of American artist Jackson Pollock. While Pollock's paintings appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in his work.
Fractal branching occurs on a microwaveirradiated DVD 

Applications
As described above, random fractals can be used to describe many highly irregular realworld objects. Other applications of fractals include:
 Classification of histopathology slides in medicine
 Generation of new music
 Generation of various art forms
 Signal and image compression
 Seismology
 Computer and video game design, especially computer graphics for organic environments and as part of procedural generation
 Fractography and fracture mechanics
 Fractal antennas — Small size antennas using fractal shapes
 Neohippies tshirts and other fashion.
 Generation of patterns for camouflage, such as MARPAT.
 Digital sundial