Selfsimilarity
In mathematics, a selfsimilar object is exactly or approximately 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 coastlines, are statistically selfsimilar: parts of them show the same statistical properties at many scales.^{2} Selfsimilarity is a typical property of fractals.
Scale invariance is an exact form of selfsimilarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scaleinvariant; it can be continually magnified 3x without changing shape.
The nontrivial 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.
Definition
A compact topological space X is selfsimilar if there exists a finite set S indexing a set of nonsurjective homeomorphisms for which
If , we call X selfsimilar if it is the only nonempty subset of Y such that the equation above holds for . We call
a selfsimilar structure. The homeomorphisms may be 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 padic tree.
The automorphisms of the dyadic monoid is the modular group; the automorphisms can be pictured as hyperbolic rotations of the binary tree.
A more general notion than selfsimilarity is Selfaffinity.
Examples
The Mandelbrot set is also selfsimilar around Misiurewicz points.
Selfsimilarity has important consequences for the design of computer networks, as typical network traffic has selfsimilar properties. For example, in teletraffic engineering, packet switched data traffic patterns seem to be statistically selfsimilar.^{3} This property means that simple models using a Poisson distribution are inaccurate, and networks designed without taking selfsimilarity into account are likely to function in unexpected ways.
Similarly, stock market movements are described as displaying selfaffinity, i.e. they appear selfsimilar when transformed via an appropriate affine transformation for the level of detail being shown.^{4}
Finite subdivision rules are a powerful technique for building selfsimilar sets, including the Cantor set and the Sierpinski triangle.
Andrew Lo describes Stock Market log return selfsimilarity in Econometrics.^{5}
In nature
Selfsimilarity can be found in nature, as well. To the right is a mathematicallygenerated, perfectly selfsimilar image of a fern, which bears a marked resemblance to natural ferns. Other plants, such as Romanesco broccoli, exhibit strong selfsimilarity.
In music
 A Shepard tone is selfsimilar in the frequency or wavelength domains.
 The Danish composer Per Nørgård has made use of a selfsimilar integer sequence named the 'infinity series' in much of his music.
See also
 Droste effect
 Longrange dependency
 Nonwellfounded set theory
 Recursion
 Selfaffinity
 Selfdissimilarity
 Selfreference
 Tweedie distributions
 Zipf's law
References
 ^ Mandelbrot, Benoit B. (1982). The Fractal Geometry of Nature, p.44. ISBN 9780716711865.
 ^ Benoit Mandelbrot (May 1967). "How Long Is the Coast of Britain? Statistical SelfSimilarity and Fractional Dimension". Science Magazine.
 ^ Leland et al. "On the selfsimilar nature of Ethernet traffic", IEEE/ACM Transactions on Networking, Volume 2, Issue 1 (February 1994)
 ^ Benoit Mandelbrot (February 1999). "How Fractals Can Explain What's Wrong with Wall Street". Scientific American.
 ^ Campbell, Lo and MacKinlay (1991) "Econometrics of Financial Markets ", Princeton University Press! iSBN 9780691043012
External links
 "Copperplate Chevrons" — a selfsimilar fractal zoom movie
 "SelfSimilarity" — New articles about the SelfSimilarity. Waltz Algorithm

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