In mathematics, a selfsimilar 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 selfsimilar: parts of them show the same statistical properties at many scales.^{[1]} Selfsimilarity is a typical property of fractals.
Scale invariance is an exact form of selfsimilarity 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 scaleinvariant; it can be continually magnified 3x without changing shape.
Definition
A compact topological space X is selfsimilar if there exists a finite set S indexing a set of nonsurjective homeomorphisms for which
If , we call X selfsimilar if it is the only nonempty subset of Y such that the equation above holds for . We call
a selfsimilar 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 padic tree.
The automorphisms of the dyadic monoid is the modular group; the automorphisms can be pictured as hyperbolic rotations of the binary tree.
Examples
The Mandelbrot set is also selfsimilar around Misiurewicz points.
Selfsimilarity has important consequences for the design of computer networks, as typical network traffic has selfsimilar properties. For example, in teletraffic engineering, packet switched data traffic patterns seem to be statistically selfsimilar^{[2]}. This property means that simple models using a Poisson distribution are inaccurate, and networks designed without taking selfsimilarity into account are likely to function in unexpected ways.
Similarly, stock market movements are described as displaying selfaffinity, i.e. they appear selfsimilar when transformed via an appropriate affine transformation for the level of detail being shown.^{[3]}
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