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| [[Image:Kochsim.gif|thumb|right|250px|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.]]
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| [[File:Standard self-similarity.png|thumb|300px|Standard (trivial) self-similarity.<ref>Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. ISBN 978-0716711865.</ref>]]
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| In [[mathematics]], a '''self-similar''' object is exactly or approximately [[similarity (geometry)|similar]] to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as [[coastline]]s, are statistically self-similar: parts of them show the same statistical properties at many scales.<ref>{{cite web|
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| url=http://www.sciencemag.org/content/156/3775/636|
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| title=How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|
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| author=Benoit Mandelbrot|
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| publisher=Science Magazine|
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| date=May 1967|
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| authorlink=Benoit Mandelbrot}}</ref> Self-similarity is a typical property of [[fractal]]s.
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| [[Scale invariance]] is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is [[Similarity (geometry)|similar]] to the whole. For instance, a side of the [[Koch snowflake]] is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape.
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| The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a [[counterexample]], whereas any portion of a [[straight line]] may resemble the whole, further detail is not revealed.
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| ==Definition==
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| A [[Compact space|compact]] [[topological space]] ''X'' is self-similar if there exists a [[finite set]] ''S'' indexing a set of non-[[surjective]] [[homeomorphism]]s <math>\{ f_s \}_{s\in S}</math> for which
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| :<math>X=\cup_{s\in S} f_s(X)</math>
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| If <math>X\subset Y</math>, we call ''X'' self-similar if it is the only [[Non-empty set|non-empty]] [[subset]] of ''Y'' such that the equation above holds for <math>\{ f_s \}_{s\in S}</math>. We call
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| :<math>\mathfrak{L}=(X,S,\{ f_s \}_{s\in S})</math>
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| a ''self-similar structure''. The homeomorphisms may be [[iterated function|iterated]], resulting in an [[iterated function system]]. The composition of functions creates the algebraic structure of a [[monoid]]. When the set ''S'' has only two elements, the monoid is known as the [[dyadic monoid]]. The dyadic monoid can be visualized as an infinite [[binary tree]]; more generally, if the set ''S'' has ''p'' elements, then the monoid may be represented as a [[p-adic number|p-adic]] tree. | |
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| The [[automorphism]]s of the dyadic monoid is the [[modular group]]; the automorphisms can be pictured as [[Hyperbolic coordinates|hyperbolic rotation]]s of the binary tree.
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| ==Examples==
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| [[Image:Feigenbaumzoom.gif|left|thumb|201px|Self-similarity in the [[Mandelbrot set]] shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)]]
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| [[Image:Fractal fern explained.png|thumb|right|200px|An image of a fern which exhibits [[affine transformation|affine]] self-similarity]]
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| The [[Mandelbrot set]] is also self-similar around [[Misiurewicz point]]s.
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| Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in [[teletraffic engineering]], [[packet switched]] data traffic patterns seem to be statistically self-similar.<ref>Leland ''et al.'' "On the self-similar nature of Ethernet traffic", ''IEEE/ACM Transactions on Networking'', Volume '''2''', Issue 1 (February 1994)</ref> This property means that simple models using a [[Poisson distribution]] are inaccurate, and networks designed without taking self-similarity into account are likely to function in unexpected ways.
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| Similarly, [[stock market]] movements are described as displaying [[self-affinity]], i.e. they appear self-similar when transformed via an appropriate [[affine transformation]] for the level of detail being shown.<ref>{{cite web|
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| url=http://www.sciam.com/article.cfm?id=multifractals-explain-wall-street|
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| title=How Fractals Can Explain What's Wrong with Wall Street|
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| author=Benoit Mandelbrot|
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| publisher=Scientific American|
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| date=February 1999|
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| authorlink=Benoit Mandelbrot}}</ref>
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| [[Finite subdivision rules]] are a powerful technique for building self-similar sets, including the [[Cantor set]] and the [[Sierpinski triangle]]. | |
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| [[File:RepeatedBarycentricSubdivision.png|thumb|A triangle subdivided repeatedly using [[barycentric subdivision]]. The [[complement]] of the large circles is becoming a [[Sierpinski carpet]]]]
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| {{clear|left}} [[Andrew Lo]] describes Stock Market log return self-similarity in [[Econometrics]].<ref>Campbell, Lo and MacKinlay (1991) "[[Econometrics]] of Financial Markets ", Princeton University Press! iSBN 978-0691043012</ref>
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| === In nature ===
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| [[File:Flickr - cyclonebill - Romanesco.jpg|thumb|right|200px|Close-up of a [[Romanesco broccoli]].]]
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| Self-similarity can be found in nature, as well. To the right is a mathematically-generated, perfectly self-similar image of a fern, which bears a marked resemblance to natural ferns. Other plants, such as [[Romanesco broccoli]], exhibit strong self-similarity.
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| === In music ===
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| *A [[Shepard tone]] is self-similar in the frequency or wavelength domains.
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| * The [[Denmark|Danish]] [[composer]] [[Per Nørgård]] has made use of a self-similar [[integer sequence]] named the 'infinity series' in much of his music.
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| ==See also==
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| * [[Droste effect]]
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| * [[Long-range dependency]]
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| * [[Non-well-founded set theory]]
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| * [[Recursion]]
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| * [[Self-dissimilarity]]
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| * [[Self-reference]]
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| * [[Tweedie distributions]]
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| * [[Zipf's law]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://www.ericbigas.com/fractals/cc "Copperplate Chevrons"] — a self-similar fractal zoom movie
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| *[http://pi.314159.ru/longlist.htm "Self-Similarity"] — New articles about the Self-Similarity. Waltz Algorithm
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| {{Fractals}}
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| {{DEFAULTSORT:Self-Similarity}}
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| [[Category:Fractals]]
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| [[Category:Scaling symmetries]]
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| [[Category:Homeomorphisms]]
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