Self-similarity

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.

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=\bigcup_{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.

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.{{cite magazine | url=https://www.scientificamerican.com/article/multifractals-explain-wall-street/ | title=How Fractals Can Explain What's Wrong with Wall Street | author=Benoit Mandelbrot | magazine=Scientific American| date=February 1999| author-link=Benoit Mandelbrot}} 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.

References

References

1. Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. {{ISBN. 978-0716711865.
2. Mandelbrot, Benoit B.. (5 May 1967). "How long is the coast of Britain? Statistical self-similarity and fractional dimension". [[Science (journal).
3. Peitgen, Heinz-Otto; Jürgens, Hartmut; Saupe, Dietmar; Maletsky, Evan; Perciante, Terry; and Yunker, Lee (1991). ''Fractals for the Classroom: Strategic Activities Volume One'', p.21. Springer-Verlag, New York. {{ISBN. 0-387-97346-X and {{ISBN. 3-540-97346-X.
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6. Arneodo, A.. (2000). "A wavelet-based method for multifractal image analysis. I. Methodology and test applications on isotropic and anisotropic random rough surfaces". The European Physical Journal B.
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8. "Self-similar carpets over finite fields".
9. [[Kazimierz Kuratowski]] (1972) Leo F. Boron, translator, ''Introduction to Set Theory and Topology'', second edition, ch XVI, § 8 The Cantor Discontinuum, page 210 to 15, [[Pergamon Press]]
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11. Campbell, Lo and MacKinlay (1991) "[[Econometrics]] of Financial Markets ", Princeton University Press! {{ISBN. 978-0691043012
12. Salazar, Munera. (1 July 2016). "Self-Similarity of Space Filling Curves".
13. (30 October 1999). "Proceedings of the seventh ACM international conference on Multimedia (Part 1)".
14. (April 2011). "On Musical Self-Similarity: Intersemiosis as Synecdoche and Analogy". International Semiotics Institute at Imatra; Semiotic Society of Finland.