# Self-similarity

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A Koch curve has an infinitely repeating self-similarity when it is magnified.
Standard (trivial) self-similarity.[1]

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 artificial 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.

## Self-affinity

A self-affine fractal with Hausdorff dimension=1.8272.

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 ${\displaystyle \{f_{s}:s\in S\}}$ 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.

## Examples

Self-similarity in the Mandelbrot set shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)
An image of the Barnsley fern which exhibits affine self-similarity

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.[6] 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.[7] Andrew Lo describes stock market log return self-similarity in econometrics.[8]

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

A triangle subdivided repeatedly using barycentric subdivision. The complement of the large circles becomes a Sierpinski carpet

### 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

Close-up of a Romanesco broccoli.

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 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.[9] In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.[10]

## 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. New Series. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158. Retrieved 11 January 2016. PDF
3. ^ 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: 175101. arXiv:1101.4730. Bibcode:2011JPhA...44q5101K. doi:10.1088/1751-8113/44/17/175101.CS1 maint: Multiple names: authors list (link)
4. ^ Hassan M. K., Hassan M. Z. (2009). "Emergence of fractal behavior in condensation-driven aggregation". Phys. Rev. E. 79: 021406. arXiv:0901.2761. Bibcode:2009PhRvE..79b1406H. doi:10.1103/physreve.79.021406.
5. ^ 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.
6. ^ 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.
7. ^ Benoit Mandelbrot (February 1999). "How Fractals Can Explain What's Wrong with Wall Street". Scientific American.
8. ^ Campbell, Lo and MacKinlay (1991) "Econometrics of Financial Markets ", Princeton University Press! ISBN 978-0691043012
9. ^ 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): 77-80. doi:10.1145/319463.319472. Archived (PDF) from the original on 9 August 2017.
10. ^ 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)