Self-similarity

 

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.

A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms ${\displaystyle \{f_{s}:s\in S\}}$  for which

${\displaystyle X=\bigcup _{s\in S}f_{s}(X)}$ 

If ${\displaystyle X\subset Y}$ , we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for ${\displaystyle \{f_{s}:s\in S\}}$ . We call

${\displaystyle {\mathfrak {L}}=(X,S,\{f_{s}:s\in S\})}$ 

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.

 

 

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.^[9] 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.^[10] Andrew Lo describes stock market log return self-similarity in econometrics.^[11]

Finite subdivision rules are a powerful technique for building self-similar sets, including the Cantor set and the Sierpinski triangle.

 

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.

 

Self-similarity can be found in nature, as well. To the right is a mathematically generated, perfectly self-similar image of a fern, which bears a marked resemblance to natural ferns. Other plants, such as Romanesco broccoli, exhibit strong self-similarity.

• 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.^[12] In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.^[13]

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

• Mandelbrot, Benoit B. (1985). "Self-affinity and fractal dimension" (PDF). Physica Scripta. 32 (4): 257–260. Bibcode:1985PhyS...32..257M. doi:10.1088/0031-8949/32/4/001. S2CID 250815596.
• Sapozhnikov, Victor; Foufoula-Georgiou, Efi (May 1996). "Self-Affinity in Braided Rivers" (PDF). Water Resources Research. 32 (5): 1429–1439. Bibcode:1996WRR....32.1429S. doi:10.1029/96wr00490. Archived (PDF) from the original on 30 July 2018. Retrieved 30 July 2018.
• Benoît B. Mandelbrot (2002). Gaussian Self-Affinity and Fractals: Globality, the Earth, 1/F Noise, and R/S. Springer. ISBN 978-0387989938.