Self-similarity

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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 $\{ f_s \}_{s\in S}$ for which

$X=\cup_{s\in S} f_s(X) \,$

If $X\subset Y$ , we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for $\{ f_s \}_{s\in S}$ . We call

$\mathfrak{L}=(X,S,\{ f_s \}_{s\in S})$

a self-similar structure.

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.