Self-similarity
From Wikinfo
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.
A curve is said to be self-similar if, for every piece of the curve, there is a smaller piece that is similar to it. For instance, a side of the Koch snowflake is self-similar; it can be divided into two halves, each of which is similar to the whole. Self-similarity is closely related to scale invariance.
A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms <math>\{ f_s \}_{s\in S}</math> for which
- <math>X=\cup_{s\in S} f_s(X) \,</math>
If <math>X\subset Y</math>, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for <math>\{ f_s \}_{s\in S}</math>. We call
- <math>\mathfrak{L}=(X,S,\{ f_s \}_{s\in S})</math>
Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals.
Self-similarity also has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in telecommunications traffic engineering, packet switched data traffic patterns seem to be statistically self-similar. 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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See also
- Fractal
- Scale invariance
- [[Beno�t Mandelbrot]]
- How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
- Self-reference
- Zipfs law
Reference
- Leland et. al. On the self-similar nature of Ethernet traffic IEEE/ACM Transactions on Networking Volume 2, Issue 1 (February 1994)
External links
- "Copperplate Chevrons" - a self-similar fractal zoom movie[[de:Selbst�hnlichkeit]][[sv:Sj�lvliknande]]
References
- Adapted from the Wikipedia article, "Self-similarity" http://en.wikipedia.org/wiki/Self-similarity, used under the GNU Free Documentation License
