Self-similarity
Whole of an object being mathematically similar to part of itself
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Peitgen et al. explain the concept as such:
If parts of a figure are small replicas of the whole, then the figure is called self-similar....A figure is strictly self-similar if the figure can be decomposed into parts which are exact replicas of the whole. Any arbitrary part contains an exact replica of the whole figure.
Since mathematically, a fractal may show self-similarity under arbitrary magnification, it is impossible to recreate this physically. Peitgen et al. suggest studying self-similarity using approximations:
In order to give an operational meaning to the property of self-similarity, we are necessarily restricted to dealing with finite approximations of the limit figure. This is done using the method which we will call box self-similarity where measurements are made on finite stages of the figure using grids of various sizes.
This vocabulary was introduced by Benoit Mandelbrot in 1964.
Self-affinity
In mathematics, self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the x and y directions. This means that to appreciate the self-similarity of these fractal objects, they have to be rescaled using an anisotropic affine transformation.
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Definition
A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms for which
If , we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for . We call.
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 group of automorphisms of the dyadic monoid is the modular group; the automorphisms can be pictured as hyperbolic rotations of the binary tree.
A more general notion than self-similarity is self-affinity.
Examples
The Cantor discontinuum is self-similar since any of its closed subsets is a continuous image of the discontinuum.
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. 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. Andrew Lo describes stock market log return self-similarity in econometrics.
Finite subdivision rules are a powerful technique for building self-similar sets, including the Cantor set and the Sierpinski triangle.
Some space filling curves, such as the Peano curve and Moore curve, also feature properties of self-similarity.
In cybernetics
The viable system model of Stafford Beer is an organizational model with an affine self-similar hierarchy, where a given viable system is one element of the System One of a viable system one recursive level higher up, and for whom the elements of its System One are viable systems one recursive level lower down.
In nature
Self-similarity can be found in nature, as well. Plants, such as Romanesco broccoli, exhibit strong self-similarity.
In music
- Strict canons display various types and amounts of self-similarity, as do sections of fugues.
- A Shepard tone is self-similar in the frequency or wavelength domains.
- The Danish composer Per Nørgård made use of a self-similar integer sequence named the infinity series in much of his music.
- In the research field of music information retrieval, self-similarity commonly refers to the fact that music often consists of parts that are repeated in time. In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.