Self-similarity

 ( Fractals ) 

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 , are statistically self-similar: parts of them show the same statistical properties at many scales. PDF Self-similarity is a typical property of . 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:

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:

This vocabulary was introduced by Benoit Mandelbrot in 1964.Comment j' /ref>

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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 space topological space X is self-similar if there exists a finite set S indexing a set of non-surjective $\backslash \{\; f\_s\; :\; s\backslash in\; S\; \backslash \}$ for which

$X=\backslash bigcup\_\{s\backslash in\; S\}\; f\_s(X)$

If $X\backslash subset\; Y$, we call X self-similar if it is the only Non-empty set subset of Y such that the equation above holds for $\backslash \{\; f\_s\; :\; s\backslash in\; S\; \backslash \}$. We call

$\backslash mathfrak\{L\}=(X,S,\backslash \{\; f\_s\; :\; s\backslash in\; S\; \backslash \}\; )$

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 number tree.

The 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.

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Examples 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.Campbell, Lo and MacKinlay (1991) "Econometrics of Financial Markets ", Princeton University Press!

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.

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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.

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In nature Self-similarity can be found in nature, as well. Plants, such as Romanesco broccoli, exhibit strong self-similarity.

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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.
  (1999). 9781581131512 . ISBN 9781581131512
  In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.
  (2011). 9789525431322, International Semiotics Institute at Imatra; Semiotic Society of Finland. . ISBN 9789525431322
  (Also see Google Books)

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See also

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External links

• "Copperplate Chevrons" — a self-similar fractal zoom movie
• "Self-Similarity" — New articles about Self-Similarity. Waltz Algorithm

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Self-affinity

• Benoît B. Mandelbrot (2025). 9780387989938, Springer. ISBN 9780387989938

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