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In mathematics, 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). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales.<ref name="Mandelbrot_Science_1967">Template:Cite journal PDF</ref> 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. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.
Peitgen et al. explain the concept as such:
Template:QuoteSince 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:Template:Quote
This vocabulary was introduced by Benoit Mandelbrot in 1964.<ref>Comment j'ai découvert les fractales, Interview de Benoit Mandelbrot, La Recherche https://www.larecherche.fr/math%C3%A9matiques-histoire-des-sciences/%C2%AB-comment-jai-d%C3%A9couvert-les-fractales-%C2%BB</ref>
Self-affinityEdit
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.
DefinitionEdit
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=\bigcup_{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>
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.
A more general notion than self-similarity is self-affinity.
ExamplesEdit
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.<ref>Template:Cite journal</ref> 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.<ref>Template:Cite magazine</ref> Andrew Lo describes stock market log return self-similarity in econometrics.<ref>Campbell, Lo and MacKinlay (1991) "Econometrics of Financial Markets ", Princeton University Press! Template:ISBN</ref>
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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
In cyberneticsEdit
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 natureEdit
Template:Further Self-similarity can be found in nature, as well. Plants, such as Romanesco broccoli, exhibit strong self-similarity.
In musicEdit
- 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 has 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.<ref>Template:Cite book</ref> In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.<ref>Template:Cite book (Also see Google Books)</ref>
See alsoEdit
ReferencesEdit
External linksEdit
- "Copperplate Chevrons" — a self-similar fractal zoom movie
- "Self-Similarity" — New articles about Self-Similarity. Waltz Algorithm