- Self-similarity
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Koch curve has an infinitely repeating self-similarity when it is magnified.]In
mathematics , a self-similar 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 ascoastline s, are statistically self-similar: parts of them show the same statistical properties at many scales. [Benoît Mandelbrot , "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension "] Self-similarity is a typical property offractal s.Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that issimilar to the whole. For instance, a side of theKoch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape.Definition
A compact
topological space "X" is self-similar if there exists afinite set "S" indexing a set of non-surjective homeomorphism s f_s }_{sin S} for which:X=cup_{sin S} f_s(X)
If Xsubset Y, we call "X" self-similar if it is the only non-empty
subset of "Y" such that the equation above holds for f_s }_{sin S}. We call:mathfrak{L}=(X,S,{ f_s }_{sin S})
a "self-similar structure". The homeomorphisms may be iterated, resulting in an
iterated function system . The composition of functions creates the algebraic structure of amonoid . When the set "S" has only two elements, the monoid is known as thedyadic monoid . The dyadic monoid can be visualized as an infinitebinary tree ; more generally, if the set "S" has "p" elements, then the monoid may be represented as a p-adic tree.The
automorphism s of the dyadic monoid is themodular group ; the automorphisms can be pictured as hyperbolic rotations of the binary tree.Examples
[
Mandelbrot set shown by zooming on a round feature while panning in the negative-X direction. The display center pans from (-1,0) to (-1.31,0) while the view magnifies from .5 x .5 to .12 x .12.]The
Mandelbrot set is also self-similar aroundMisiurewicz point s.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 [Leland "et al." "On the self-similar nature of Ethernet traffic", "IEEE/ACM Transactions on Networking", Volume 2, Issue 1 (February 1994)] . This property means that simple models using aPoisson distribution are inaccurate, and networks designed without taking self-similarity into account are likely to function in unexpected ways.Similarly,
stock market movements are described as displayingself-affinity , i.e. they appear self-similar when transformed via an appropriateaffine transformation for the level of detail being shown. [cite web
url=http://www.sciam.com/article.cfm?id=multifractals-explain-wall-street
title=How Fractals Can Explain What's Wrong with Wall Street
author=Benoit Mandelbrot
publisher=Scientific American
date=February 1999]ee also
*
Droste effect
*Self-reference
*Zipf's law
*Self-affinity References
External links
* [http://www.ericbigas.com/fractals/cc "Copperplate Chevrons"] — a self-similar fractal zoom movie
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