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
\({\displaystyle f(x,t)}\) measured at different times are different but the corresponding dimensionless quantity at given value of \({\displaystyle x/t^{z}}\) remain invariant. It happens if the quantity \({\displaystyle f(x,t)}\) 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 called

self-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]}

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.^{[8]} 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.^{[9]} Andrew Lo describes stock market log return self-similarity in econometrics.^{[10]}

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.
^{[11]}In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.^{[12]}

**^**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. Bibcode:1967Sci...156..636M . doi:10.1126/science.156.3775.636 . PMID 17837158 . 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. arXiv:1101.4730 . Bibcode:2011JPhA...44q5101K . doi:10.1088/1751-8113/44/17/175101 .CS1 maint: multiple names: authors list (link)**^**Hassan M. K., Hassan M. Z. (2009). "Emergence of fractal behavior in condensation-driven aggregation".*Phys. Rev. E*.**79**(2): 021406. arXiv:0901.2761 . Bibcode:2009PhRvE..79b1406H . doi:10.1103/physreve.79.021406 . PMID 19391746 .**^**Dayeen F. R., Hassan M. K. (2016). "Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice".*Chaos, Solitons & Fractals*.**91**: 228. arXiv:1409.7928 . Bibcode:2016CSF....91..228D . doi:10.1016/j.chaos.2016.06.006 .**^**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.**^**Leland, W.E.; Taqqu, M.S.; et al. (January 1995). "On the self-similar nature of Ethernet traffic (extended version)" (PDF).*IEEE/ACM Transactions on Networking*.**2**(1): 1–15. doi:10.1109/90.282603 .**^**Benoit Mandelbrot (February 1999). "How Fractals Can Explain What's Wrong with Wall Street" .*Scientific American*.**^**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*(PDF).*Multimedia '99 Proceedings of the Seventh ACM International Conference on Multimedia (Part 1)*. pp. 77–80. CiteSeerX 10.1.1.223.194 . doi:10.1145/319463.319472 . ISBN 978-1581131512. Archived (PDF) from the original on 9 August 2017.**^**Pareyon, Gabriel (April 2011).*On Musical Self-Similarity: Intersemiosis as Synecdoche and Analogy*(PDF). International Semiotics Institute at Imatra; Semiotic Society of Finland. p. 240. ISBN 978-952-5431-32-2. Archived from the original (PDF) on 8 February 2017. Retrieved 30 July 2018. (Also see Google Books )

- "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 . - Sapozhnikov, Victor; Foufoula-Georgiou, Efi (May 1996). "Self-Affinity in Braided Rivers" (PDF).
*Water Resources Research*.**32**(5): 1429–1439. 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*. ISBN 978-0387989938.

**Categories:** Fractals | Scaling symmetries | Homeomorphisms | Self-reference

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