Self-similarity

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thumb|right|250px|A_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 as coastlines, 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 of fractals.

Scale invariance is an exact form of self-similarity 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 scale-invariant; it can be continually magnified 3x without changing shape.

Definition

A compact topological space "X" is self-similar if there exists a finite set "S" indexing a set of non-surjective homeomorphisms $\{\; f\_s\; \}\_\{sin\; S\}$ 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 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 tree.

The automorphisms of the dyadic monoid is the modular 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 around Misiurewicz points.

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 a Poisson 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 displaying self-affinity, i.e. they appear self-similar when transformed via an appropriate affine 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