Self-similarity

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A Koch curve has an infinitely repeating self-similarity when it is magnified.
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, e.g., 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[1]. 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.

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[edit] 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 \}_{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. 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.

[edit] Examples

An image of a fern which exhibits affine self-similarity.
An image of a fern which exhibits affine self-similarity.

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[2]. 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.

[edit] See also

[edit] Reference

  1. ^ Benoît Mandelbrot, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
  2. ^ Leland et. al. "On the self-similar nature of Ethernet traffic", IEEE/ACM Transactions on Networking, Volume 2, Issue 1 (February 1994)
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