The Mandelbrot set is a famous example of a
fractal.
A fractal is an irregular geometric shape that can be divided into parts in such a manner that the shape of each part resembles the shape of the whole. This property is called selfsimilarity. However, not all selfsimilar objects are fractals. For example, a straight Euclidean line (or real line) is formally selfsimilar, but it is regular enough to be described in Euclidean terms.
Images of fractals can be created using fractal generating software. Such software allows scientists to construct computer models of natural objects with irregular shapes that approximate fractals to some degree. These objects include clouds, coastlines, mountain ranges, lightning bolts, and snowflakes.
A closer view of the Mandelbrot set.
Etymology
The term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin word fractus, meaning "broken" or "fractured." In his book The Fractal Geometry of Nature, Mandelbrot describes a fractal as "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reducedsize copy of the whole."^{[1]}
Features
A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.^{[2]}
A fractal often has the following features:^{[3]}
 It has a fine structure at arbitrarily small scales.
 It is too irregular to be easily described in traditional Euclidean geometric language.
 It is selfsimilar (at least approximately or stochastically).
 It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by spacefilling curves such as the Hilbert curve).
 It has a simple and recursive definition.
History
Animated construction of a Sierpiński Triangle, only going nine generations of infinite—click for larger image.
To create a Koch snowflake, one begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral "bump." One then performs the same replacement on every line segment of the resulting shape, ad infinitum. With every iteration, the perimeter of this shape increases by one third of the previous length. The Koch snowflake is the result of an infinite number of these iterations, and has an infinite length, while its area remains
finite. For this reason, the Koch snowflake and similar constructions were sometimes called "monster curves."
The mathematics behind fractals began to take shape in the seventeenth century when mathematician and philosopher Leibniz considered recursive selfsimilarity (although he made the mistake of thinking that only the straight line was selfsimilar in this sense).
It took until 1872 before a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the nonintuitive property of being everywhere continuous but nowhere differentiable. 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. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions. In 1918, Bertrand Russell had recognized a "supreme beauty" within the mathematics of fractals that was then emerging.^{[2]} 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 also 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 were investigated in the late nineteenth and early twentieth 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, which built on earlier work by Lewis Fry Richardson. Finally, 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 Julia set, a fractal related to the Mandelbrot set
A class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, spacefilling curve, and 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 a Hausdorff dimension of two.
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 a Hausdorff dimension equal to its topological dimension of two—but what is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of two (while the topological dimension of one), a result proved by Mitsuhiro Shishikura in 1991. A closely related fractal is the Julia set.
Even simple smooth curves can exhibit the fractal property of selfsimilarity. For example the powerlaw curve (also known as a Pareto distribution) produces similar shapes at various magnifications.
Generating fractals



Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set. 
Four common techniques for generating fractals are:

 Escapetime fractals — (also known as "orbits" fractals) These are defined by a formula or 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, the Nova fractal and the Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escapetime formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.
 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.
 Strange attractors — Generated by iteration of a map or the solution of a system of initialvalue differential equations that exhibit chaos.
Classification
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.
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, crystals, mountain ranges, lightning, river networks, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels. Coastlines may be loosely considered fractal in nature.
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 connection between fractals and leaves are currently being used to determine how much carbon is really contained in trees. This connection is hoped to help determine and solve the environmental issue of carbon emission and control. ^{[4]}
In 1999, certain self similar fractal shapes were shown to have a property of "frequency invariance"—the same electromagnetic properties no matter what the frequency—from Maxwell's equations (see fractal antenna).^{[5]}

A fractal that models the surface of a mountain (animation)

A fractal fern computed using an Iterated function system


Fractal pentagram drawn with a vector iteration program
In creative works
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.^{[6]}
Decalcomania, a technique used by artists such as Max Ernst, can produce fractallike patterns.^{[7]} It involves pressing paint between two surfaces and pulling them apart.
Fractals are also prevalent in African art and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.^{[8]}

A fractal is formed when pulling apart two gluecovered acrylic sheets.

High voltage breakdown within a 4″ block of acrylic creates a fractal Lichtenberg figure.

Fractal branching occurs in a fractured surface such as a microwaveirradiated DVD^{[9]}

Romanesco broccoli showing very fine natural fractals

A DLA cluster grown from a copper(II) sulfate solution in an electrodeposition cell


A magnification of the phoenix set


A fractal flame created with the program Apophysis

Fractal made by the program Sterling
Applications
As described above, random fractals can be used to describe many highly irregular realworld objects. Other applications of fractals include:^{[10]}
 Classification of histopathology slides in medicine
 Fractal landscape or Coastline complexity
 Enzyme/enzymology (MichaelisMenten kinetics)
 Generation of new music
 Generation of various art forms
 Signal and image compression
 Creation of digital photographic enlargements
 Seismology
 Fractal in soil mechanics
 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
 Small angle scattering theory of fractally rough systems
 Tshirts and other fashion
 Generation of patterns for camouflage, such as MARPAT
 Digital sundial
 Technical analysis of price series (see Elliott wave principle)
See also
 Bifurcation theory
 Butterfly effect
 Chaos theory
 Complexity
 Constructal theory
 Diamondsquare algorithm
 Fractal compression
 Fractal cosmology
 Fractal flame
 Fractal landscape
 Fractint
 Fracton
 Graftal
 Greeble
 Lacunarity
 Newton fractal
 Recursionism
 Sacred geometry
 Selfreference
 Strange loop
 Turbulence
Notes
 ↑ Mandelbrot, B.B. 1982. The Fractal Geometry of Nature. San Francisco, CA: W.H. Freeman and Company. ISBN 0716711869.
 ↑ ^{2.0} ^{2.1} John Briggs, 1992, Fractals:The Patterns of Chaos. London, UK: Thames and Hudson. ISBN 0500276935. 148.
 ↑ Kenneth Falconer, 2003, Fractal Geometry: Mathematical Foundations and Applications. Chichester, UK: John Wiley & Sons, Ltd. ISBN 0470848626. xxv.
 ↑ "Hunting the Hidden Dimension." Nova. PBS. WPMBMaryland.
 ↑ R. Hohlfeld, and N. Cohen. 1999. Selfsimilarity and the geometric requirements for frequency independence in antennae. Fractals. 7(1):7984.
 ↑ Taylor, Richard, Adam P. Micolich, and David Jonas. Fractal Expressionism : Can Science Be Used To Further Our Understanding Of Art? phys.unsw.edu.au. Retrieved January 14, 2009.
 ↑ Michael Frame, and Benoît B. Mandelbrot. A Panorama of Fractals and Their Uses. Yale. Retrieved January 14, 2009.
 ↑ Ron Eglash, 1999, African Fractals: Modern Computing and Indigenous Design. New Brunswick, NJ: Rutgers University Press. ISBN 9780813526140. Retrieved January 14, 2009.
 ↑ Peng, Gongwen, Decheng Tian. 1990. The fractal nature of a fracture surface. Journal of Physics A. 23(14):3257–3261. Retrieved January 14, 2009.
 ↑ Applications. ThinkQuest. Retrieved January 14, 2009.
References
 Barnsley, Michael F., and Hawley Rising. 1993. Fractals Everywhere. Boston, MA: Academic Press Professional. ISBN 0120790610.
 Falconer, Kenneth. 1997. Techniques in Fractal Geometry. New York, NY: John Willey and Sons. ISBN 0471922870.
 Gouyet, JeanFrançois. 1996. Physics and Fractal Structures. Paris, FR: Masson; New York, NY: Springer. ISBN 2225851301.
 Jones, Jesse. 1993. Fractals for the Macintosh. Corte Madera, CA: Waite Group Press. ISBN 1878739468.
 Jürgens, Hartmut, HeinsOtto Peitgen, and Dietmar Saupe. 1992. Chaos and Fractals: New Frontiers of Science. New York, NY: SpringerVerlag. ISBN 0387979034.
 Lauwerier, Hans, and Sophia GillHoffstadt trans. 1991. Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton University Press. ISBN 069108551X.  "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.
 LesmoirGordon, Nigel, Ian Stewart, Paul Sinclair, David Gilmour, and Arthur C Clarke. The Colours of Infinity: The Beauty, The Power and the Sense of Fractals. Bath, UK: Clear. ISBN 1904555055. (The book comes with a related DVD of the Arthur C. Clarke documentary introduction to the fractal concept and the Mandelbrot set.
 Mandelbrot, Benoît B. 1982. The Fractal Geometry of Nature. New York, NY: W.H. Freeman and Co. ISBN 0716711869.
 Peitgen, HeinzOtto, and Dietmar Saupe, eds. 1988. The Science of Fractal Images. New York, NY: SpringerVerlag. ISBN 0387966080.
 Pickover, Clifford A. ed. 1998. Chaos and Fractals: A Computer Graphical Journey  A 10 Year Compilation of Advanced Research. Amsterdam, NL; New York, NY: Elsevier. ISBN 0444500022.
 Sprott, Julien Clinton. 2003. Chaos and TimeSeries Analysis. Oxford, UK: Oxford University Press. ISBN 0198508395.
 Wahl, Bernt, Peter Van Roy, Michael Larsen, and Eric Kampman. 1995. Exploring Fractals on the Macintosh. Reading, MA: Addison Wesley. ISBN 0201626306. Retrieved January 14, 2009.
External links
All links retrieved November 12, 2013.
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