# Self-similarity

__: Whole of an object being mathematically similar to part of itself__

**Short description**File:KochSnowGif16 800x500 2.gif

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.^{[2]} 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.

A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity
[math]\displaystyle{ f(x,t) }[/math] measured at different times are different but the corresponding dimensionless quantity at given value of [math]\displaystyle{ x/t^z }[/math] remain invariant. It happens if the quantity [math]\displaystyle{ f(x,t) }[/math] exhibits dynamic scaling. The idea is just an extension of the idea of similarity of two triangles.^{[3]}^{[4]}^{[5]} Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide.

Peitgen *et al.* explain the concept as such:

If parts of a figure are small replicas of the whole, then the figure is calledself-similar....A figure isstrictly self-similarif 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.^{[6]}

Since mathematically, a fractal may show self-similarity under indefinite 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.^{[7]}

This vocabulary was introduced by Benoit Mandelbrot in 1964.^{[8]}

## 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 [math]\displaystyle{ \{ f_s : s\in S \} }[/math] for which

- [math]\displaystyle{ X=\bigcup_{s\in S} f_s(X) }[/math]

If [math]\displaystyle{ 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]\displaystyle{ \{ f_s : s\in S \} }[/math]. We call

- [math]\displaystyle{ \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.

## 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.^{[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.

### 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. 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.

### 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
*Denmark*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]}

## See also

## References

- ↑ Mandelbrot, Benoit B. (1982).
*The Fractal Geometry of Nature*, p.44. ISBN:978-0716711865. - ↑ Mandelbrot, Benoit B. (5 May 1967). "How long is the coast of Britain? Statistical self-similarity and fractional dimension".
*Science*. New Series**156**(3775): 636–638. doi:10.1126/science.156.3775.636. PMID 17837158. Bibcode: 1967Sci...156..636M. http://ena.lp.edu.ua:8080/handle/ntb/52473. PDF - ↑ Hassan M. K., Hassan M. Z., Pavel N. I. (2011). "Dynamic scaling, data-collapseand Self-similarity in Barabasi-Albert networks".
*J. Phys. A: Math. Theor.***44**(17): 175101. doi:10.1088/1751-8113/44/17/175101. Bibcode: 2011JPhA...44q5101K. - ↑ Hassan M. K., Hassan M. Z. (2009). "Emergence of fractal behavior in condensation-driven aggregation".
*Phys. Rev. E***79**(2): 021406. doi:10.1103/physreve.79.021406. PMID 19391746. Bibcode: 2009PhRvE..79b1406H. - ↑ Dayeen F. R., Hassan M. K. (2016). "Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice".
*Chaos, Solitons & Fractals***91**: 228. doi:10.1016/j.chaos.2016.06.006. Bibcode: 2016CSF....91..228D. - ↑ 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. - ↑ Peitgen, et al (1991), p.2-3.
- ↑ 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
- ↑ Leland, W.E.; Taqqu, M.S.
*et al*. (January 1995). "On the self-similar nature of Ethernet traffic (extended version)".*IEEE/ACM Transactions on Networking***2**(1): 1–15. doi:10.1109/90.282603. http://ccr.sigcomm.org/archive/1995/jan95/ccr-9501-leland.pdf. - ↑ Benoit Mandelbrot (February 1999). "How Fractals Can Explain What's Wrong with Wall Street".
*Scientific American*. https://www.scientificamerican.com/article/multifractals-explain-wall-street/. - ↑ Campbell, Lo and MacKinlay (1991) "Econometrics of Financial Markets ", Princeton University Press! ISBN:978-0691043012
- ↑ Foote, Jonathan (30 October 1999). "Visualizing music and audio using self-similarity".
*Proceedings of the seventh ACM international conference on Multimedia (Part 1)*. pp. 77–80. doi:10.1145/319463.319472. ISBN 978-1581131512. http://musicweb.ucsd.edu/~sdubnov/CATbox/Reader/p77-foote.pdf. - ↑ Pareyon, Gabriel (April 2011).
*On Musical Self-Similarity: Intersemiosis as Synecdoche and Analogy*. International Semiotics Institute at Imatra; Semiotic Society of Finland. p. 240. ISBN 978-952-5431-32-2. https://tuhat.helsinki.fi/portal/files/15216101/Pareyon_Dissertation.pdf. Retrieved 30 July 2018. (Also see Google Books)

## External links

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

### Self-affinity

- Mandelbrot, Benoit B. (1985). "Self-affinity and fractal dimension".
*Physica Scripta***32**(4): 257–260. doi:10.1088/0031-8949/32/4/001. Bibcode: 1985PhyS...32..257M. http://users.math.yale.edu/mandelbrot/web_pdfs/112selfAffinity.pdf. - Sapozhnikov, Victor; Foufoula-Georgiou, Efi (May 1996). "Self-Affinity in Braided Rivers".
*Water Resources Research***32**(5): 1429–1439. doi:10.1029/96wr00490. Bibcode: 1996WRR....32.1429S. http://efi.eng.uci.edu/papers/efg_023.pdf. 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.

Original source: https://en.wikipedia.org/wiki/Self-similarity.
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