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A "Koch curve has an infinitely repeating self-similarity when it is magnified.
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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

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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

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Self-similarity in the "Mandelbrot set shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)
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An image of a 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.

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

### In nature

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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 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. "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. "doi:10.1088/1751-8113/44/17/175101.
4. ^ Hassan M. K., Hassan M. Z. (2009). "Emergence of fractal behavior in condensation-driven aggregation". Phys. Rev. E. 79: 021406. "arXiv:. "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. "doi:10.1016/j.chaos.2016.06.006.
6. ^ Leland et al. "On the self-similar nature of Ethernet traffic", IEEE/ACM Transactions on Networking, Volume 2, Issue 1 (February 1994)
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. ^ J. Foote (1999), Visualizing music and audio using self-similarity. '99 Proceedings of the seventh ACM international conference on Multimedia (Part 1), Pages 77-80, New York, NY, USA
10. ^ G. Pareyon (2011), On Musical Self-Similarity
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