Fractal
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225824777
2008-07-15T15:59:45Z
Epbr123
1395162
Reverted edits by [[Special:Contributions/Pink!Teen|Pink!Teen]] to last version by TheRingess (using [[WP:HG|Huggle]])
[[Image:Mandel zoom 00 mandelbrot set.jpg|300px|right|thumb|The [[Mandelbrot set]] is a famous example of a '''fractal'''.]]
[[Image:Mandelpart2_red.png|right|300px|thumb|A closer view of the Mandelbrot set.]]
A '''fractal''' is generally "a rough or fragmented [[Shape|geometric shape]] that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"<ref>{{cite book
| last = Mandelbrot
| first = B.B.
| title = The Fractal Geometry of Nature
| publisher = W.H. Freeman and Company.
| date = 1982
| id = ISBN 0-7167-1186-9}}</ref> a property called [[self-similarity]]. The term was coined by [[Benoît Mandelbrot]] in 1975 and was derived from the [[Latin]] ''[[wikt:fractus|fractus]]'' meaning "broken" or "fractured."
A fractal often has the following features:<ref>{{cite book
| last = Falconer
| first = Kenneth
| title = Fractal Geometry: Mathematical Foundations and Applications
| publisher = John Wiley & Sons, Ltd.
| date = 2003
| pages = xxv
| id = ISBN 0-470-84862-6}}</ref>
* It has a fine structure at arbitrarily small scales.
* It is too irregular to be easily described in traditional [[Euclidean geometry|Euclidean geometric]] language.
* It is [[self-similarity|self-similar]] (at least approximately or [[stochastic]]ally).
* It has a [[Hausdorff dimension]] which is greater than its [[Lebesgue covering dimension|topological dimension]] (although this requirement is not met by [[space-filling curve]]s such as the [[Hilbert curve]]).
* It has a simple and [[recursive definition]].
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals—for example, the [[real line]] (a straight [[Euclidean]] line) is formally self-similar but fails to have other fractal characteristics.
==History==
[[image:animated construction of Sierpinski Triangle.gif|left|thumb|200px|Animated construction of a [[Sierpiński Triangle]], only going nine generations of [[infinity|infinite]]—click for larger image.]]
[[Image:Von Koch curve.gif|right|thumbnail|250px|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 grows by 1/3rd. The Koch snowflake is the result of an infinite number of these iterations, and has an infinite length, while its area remains [[Wiktionary:finite|finite]]. For this reason, the Koch snowflake and similar constructions were sometimes called "monster curves." <!-- This is NOT merely a matter of "approaching infinity but never reaching it". The actual length of the boundary of the union is infinite. [[User:Michael Hardy]] -->]]
The [[mathematics]] behind fractals began to take shape in the 17th century when mathematician and philosopher [[Gottfried Leibniz|Leibniz]] considered [[recursion|recursive]] self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).
It took until 1872 before a function appeared whose [[Graph of a function|graph]] would today be considered fractal, when [[Karl Weierstrass]] gave an [[Weierstrass function|example]] of a function with the non-[[intuition (knowledge)|intuitive]] property of being everywhere [[continuous function|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 [[Sierpinski triangle|triangle]] and, one year later, his [[Sierpinski carpet|carpet]]. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions. The idea of self-similar 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 [[subset]]s of the real line with unusual properties—these [[Cantor set]]s are also now recognized as fractals.
Iterated functions in the [[complex plane]] were 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 self-similarity in papers such as ''[[How Long Is the Coast of Britain? Statistical Self-Similarity 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 [[Hausdorff dimension|Hausdorff-Besicovitch dimension]] is greater than its [[topological dimension]]. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".
==Examples==
[[Image:Julia set (indigo).png|thumb|A [[Julia set]], a fractal related to the Mandelbrot set]]
A relatively simple class of examples is given by the [[Cantor set]]s, [[Sierpinski triangle]] and [[Sierpinski carpet|carpet]], [[Menger sponge]], [[dragon curve]], [[space-filling curve]], and [[Koch snowflake|Koch curve]]. Additional examples of fractals include the [[Lyapunov fractal]] and the limit sets of [[Kleinian group]]s. Fractals can be [[deterministic]] (all the above) or [[stochastic]] (that is, non-deterministic). For example, the trajectories of the [[Brownian motion]] in the plane have a Hausdorff dimension of 2.
[[Chaos theory|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 2—but what is truly surprising is that the [[Boundary (topology)|boundary]] of the Mandelbrot set also has a Hausdorff dimension of 2 (while the topological dimension of 1), 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 self-similarity. For example the [[power-law]] curve (also known as a [[Pareto distribution]]) produces similar shapes at various magnifications.
==Generating fractals ==
<table style="float:right;width:130px;padding-left:20px">
<tr><td>[[Image:Mandelbrot-similar-x1.jpg|The whole Mandelbrot set]]
<tr><td>[[Image:Mandelbrot-similar-x6.jpg|Mandelbrot zoomed 6x]]
<tr><td>[[Image:Mandelbrot-similar-x100.jpg|Mandelbrot Zoomed 100x]]
<tr><td>[[Image:Mandelbrot-similar-x2000.jpg|Mandelbrot Zoomed 2000x]] <small>Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set.</small></table>
Three common techniques for generating fractals are:
:* '''Escape-time fractals''' — (also known as "orbits" 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]], the [[Nova fractal]] and the [[Lyapunov fractal]]. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.
:* '''[[Iterated function system]]s''' — These have a fixed geometric replacement rule. [[Cantor set]], [[Sierpinski carpet]], [[Sierpinski gasket]], [[Peano curve]], [[Koch snowflake]], [[dragon curve|Harter-Heighway dragon curve]], [[T-Square (fractal)|T-Square]], [[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 so-called mass- or dendritic fractals, for example, [[diffusion-limited aggregation]] or [[reaction-limited aggregation]] clusters.
==Classification ==
Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:
:*'''Exact self-similarity''' — This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by [[iterated function]] systems often display exact self-similarity.
:*'''Quasi-self-similarity''' — This is a loose form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by [[recurrence relation]]s are usually quasi-self-similar but not exactly self-similar.
:*'''Statistical self-similarity''' — This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of "fractal" trivially imply some form of statistical self-similarity. ([[Fractal dimension]] itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar.
<br clear="all">
==In nature==
[[Image:Animated fractal mountain.gif|right|thumb|200px|A fractal that models the surface of a mountain (animation)]]
Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, [[snow|snow flakes]], [[crystal]]s, [[mountain|mountain range]]s, [[lightning]], [[river|river networks]], [[cauliflower]] or [[broccoli]], and systems of [[blood vessel]]s and [[pulmonary vessels]]. [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|Coastlines]] may be loosely considered fractal in nature.
[[Image:Bransleys fern.png|left|thumb|120px|A fractal fern computed using an [[Iterated function system]]]]Trees and ferns are fractal in nature and can be modeled on a computer by using a [[recursion|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.
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]]).<ref>Hohlfeld,R., and Cohen, N.,"SELF-SIMILARITY AND THE GEOMETRIC REQUIREMENTS FOR FREQUENCY INDEPENDENCE IN ANTENNAE ", Fractals, Vol. 7, No. 1 (1999) 79-84</ref>
[[Image:PentagramFractal.PNG|right|thumb|120px|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.<ref>[http://www.phys.unsw.edu.au/PHYSICS_!/FRACTAL_EXPRESSIONISM/fractal_taylor.html Richard Taylor, Adam P. Micolich and David Jonas. ''Fractal Expressionism : Can Science Be Used To Further Our Understanding Of Art?'']</ref>
[[Decalcomania]], a technique used by artists such as [[Max Ernst]], can produce fractal-like patterns.<ref>[http://classes.yale.edu/Fractals/Panorama/ A Panorama of Fractals and Their Uses] by Michael Frame and Benoit B. Mandelbrot</ref> 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.<ref>[http://www.rpi.edu/~eglash/eglash.dir/afractal/afractal.htm Ron Eglash. ''African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers University Press 1999.'']</ref>
<gallery caption= widths="200px" heights="200px">
Image:Glue1_800x600.jpg|A fractal is formed when pulling apart two glue-covered [[acryl]]ic sheets.
Image:Square1.jpg|High voltage breakdown within a 4″ block of acrylic creates a fractal [[Lichtenberg figure]].
Image:Microwaved-DVD.jpg|Fractal branching occurs in a fractured surface such as a microwave-irradiated [[DVD]]<ref name='J. Phys. A 21 July 1990'> {{cite journal|title=The fractal nature of a fracture surface|journal=[[Journal of Physics A]]|date=21 July 1990|first=Gongwen|last=Peng|coauthors=Decheng Tian|volume=23|issue=14|pages=3257–3261|doi=10.1088/0305-4470/23/14/022|url=http://www.iop.org/EJ/abstract/0305-4470/23/14/022|format=|accessdate=2007-06-02}}</ref>
Image:Fractal_Broccoli.jpg|[[Romanesco broccoli]] showing very fine natural fractals
Image:DLA_Cluster.JPG|A [[Diffusion-limited aggregation|DLA cluster]] grown from a [[copper(II) sulfate]] solution in an [[electrodeposition]] cell
Image:Woodburn_fractal.jpg|A "woodburn" fractal
Image:Phoenix(Julia).gif|A magnification of the phoenix set
Image:Complex fractle image.PNG|Pascal generated fractal
Image:Lines Apophysis Fractal Flame.jpg | A [[fractal flame]] created with the program [[Apophysis (software)|Apophysis]]
</gallery>
==Applications==
As described above, random fractals can be used to describe many highly irregular real-world objects. Other applications of fractals include:<ref>{{cite web|url=http://library.thinkquest.org/26242/full/ap/ap.html|title=Applications|accessdate=2007-10-21}}</ref>
* [[Categorisation|Classification]] of [[histopathology]] slides in [[medicine]]
* [[Fractal landscape]] or [[Coast]]line complexity
* Enzyme/enzymology ([[Michaelis-Menten kinetics]])
* Generation of new music
* Generation of various [[art]] forms
* [[Signal (information theory)|Signal]] and [[Fractal compression|image compression]]
* [[Seismology]]
* [[Fractal in soil mechanics]]
* [[game design|Computer and video game design]], especially [[computer graphics]] for [[life|organic]] environments and as part of [[procedural generation]]
* Fractography and [[fracture mechanics]]
* [[Fractal antenna]]s — Small size antennas using fractal shapes
* [[SAXS|Small angle scattering theory of fractally rough systems]]
* Neo-[[hippie]]s' [[t-shirt]]s and other [[fashion]]
* Generation of patterns for camouflage, such as [[MARPAT]]
* [[Digital sundial]]
* [[Technical analysis]] of price series (see [[Elliott wave principle]])
==See also==
{{multicol}}
* [[Bifurcation theory]]
* [[Butterfly effect]]
* [[Chaos theory]]
* [[Complexity]]
* [[Constructal theory]]
* [[Contraction mapping theorem| Contraction mapping theorem]]
* [[Diamond-square algorithm]]
* [[Droste effect]]
* [[Feigenbaum function]]
{{multicol-break}}
* [[Fractal art]]
* [[Fractal compression]]
* [[Fractal flame]]
* [[Fractal landscape]]
* [[Fracton]]
* [[Graftal]]
* [[List of fractals by Hausdorff dimension]]
* [[List of publications in mathematics#Fractal geometry|Publications in fractal geometry]]
* [[Newton fractal]]
{{multicol-break}}
* [[Recursion]]
* [[Recursionism]]
* [[Reentrant]]
* [[Sacred geometry]]
* [[Self-reference]]
* [[Strange loop]]
* [[Turbulence]]
{{multicol-end}}
==References==
{{reflist}}
==Further reading==
* Barnsley, Michael F., and Hawley Rising. ''Fractals Everywhere''. Boston: Academic Press Professional, 1993. ISBN 0-12-079061-0
* Falconer, Kenneth. '' Techniques in Fractal Geometry''. John Willey and Sons, 1997. ISBN 0-471-92287-0
* Jürgens, Hartmut, Heins-Otto Peitgen, and Dietmar Saupe. ''Chaos and Fractals: New Frontiers of Science''. New York: Springer-Verlag, 1992. ISBN 0-387-97903-4
*[[Benoît B. Mandelbrot]] ''The Fractal Geometry of Nature''. New York: W. H. Freeman and Co., 1982. ISBN 0-7167-1186-9
* Peitgen, Heinz-Otto, and Dietmar Saupe, eds. ''The Science of Fractal Images''. New York: Springer-Verlag, 1988. ISBN 0-387-96608-0
* [[Clifford A. Pickover]], ed. ''Chaos and Fractals: A Computer Graphical Journey - A 10 Year Compilation of Advanced Research''. Elsevier, 1998. ISBN 0-444-50002-2
* [[Jesse Jones]], ''Fractals for the Macintosh'', Waite Group Press, Corte Madera, CA, 1993. ISBN 1-878739-46-8.
* [[Hans Lauwerier]], ''Fractals: Endlessly Repeated Geometrical Figures'', Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991. ISBN 0-691-08551-X, cloth. ISBN 0-691-02445-6 paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.
* {{cite book | last = Sprott | first = Julien Clinton | title = Chaos and Time-Series Analysis | publisher = Oxford University Press | year = 2003 | id = ISBN 0-19-850839-5 and ISBN 978-0-19-850839-7}}
* Bernt Wahl, Peter Van Roy, Michael Larsen, and Eric Kampman [http://www.fractalexplorer.com ''Exploring Fractals on the Macintosh''], Addison Wesley, 1995. ISBN 0-201-62630-6
*Nigel Lesmoir-Gordon. "The Colours of Infinity: The Beauty, The Power and the Sense of Fractals." ISBN 1-904555-05-5 (The book comes with a related DVD of the [[Arthur C. Clarke]] documentary introduction to the fractal concept and the [[Mandelbrot set]].
* Gouyet, Jean-François. '' Physics and Fractal Structures'' (Foreword by B. Mandelbrot); Masson, 1996. ISBN 2-225-85130-1, and New York: Springer-Verlag, 1996. ISBN 0-387-94153-1. Out-of-print. Available in PDF version at [http://www.jfgouyet.fr/fractal/fractauk.html].
==External links==
{{Spoken Wikipedia|Fractal.ogg|2005-06-16}}
{{Commons|Fractal}}
{{Wiktionarypar|fractal}}
{{Wikibooks|Fractals }}
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