Self-similarity

Image:Self-affine set.png = 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.

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

:$X=\backslash bigcup\_\{s\backslash in\; S\}\; f\_s(X)$

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

:$\backslash mathfrak\{L\}=(X,S,\backslash \{\; f\_s\; :\; s\backslash in\; S\; \backslash \}\; )$

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.

Image:Feigenbaumzoom.gif shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)]]

Image:Fractal fern explained.png 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.{{cite journal|last1=Leland|first1=W.E.|last2=Taqqu|first2=M.S.|last3=Willinger|first3=W.|last4=Wilson|first4=D.V.|display-authors=2|title=On the self-similar nature of Ethernet traffic (extended version)|journal=IEEE/ACM Transactions on Networking|date=January 1995|volume=2|issue=1|pages=1–15|doi=10.1109/90.282603|s2cid=6011907|url=http://ccr.sigcomm.org/archive/1995/jan95/ccr-9501-leland.pdf}} 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.{{cite magazine | url=https://www.scientificamerican.com/article/multifractals-explain-wall-street/ | title=How Fractals Can Explain What's Wrong with Wall Street | author=Benoit Mandelbrot | magazine=Scientific American|

date=February 1999| author-link=Benoit Mandelbrot}} Andrew Lo describes stock market log return self-similarity in econometrics.Campbell, Lo and MacKinlay (1991) "Econometrics of Financial Markets ", Princeton University Press! {{ISBN|978-0691043012}}

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

Some space filling curves, such as the Peano curve and Moore curve, also feature properties of self-similarity.{{Cite web |last=Salazar |first=Munera |last2=Eduardo |first2=Luis |date=July 1, 2016 |title=Self-Similarity of Space Filling Curves |url=https://repository.icesi.edu.co/items/5f9b8cea-4787-7785-e053-2cc003c84dc5 |url-status=live |archive-url=https://web.archive.org/web/20250313193207/https://repository.icesi.edu.co/items/5f9b8cea-4787-7785-e053-2cc003c84dc5 |archive-date=March 13, 2025 |access-date=March 13, 2025 |website=Universidad ICESI}}

File:RepeatedBarycentricSubdivision.png. The complement of the large circles becomes a Sierpinski carpet]]

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.

File:Flickr - cyclonebill - Romanesco.jpg]]

{{further|Patterns in nature}}

Self-similarity can be found in nature, as well. 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.{{cite book |last1=Foote |first1=Jonathan |title=Proceedings of the seventh ACM international conference on Multimedia (Part 1) |chapter=Visualizing music and audio using self-similarity |date=30 October 1999 |pages=77–80 |doi=10.1145/319463.319472 |url=http://musicweb.ucsd.edu/~sdubnov/CATbox/Reader/p77-foote.pdf |url-status=live |archive-url=https://web.archive.org/web/20170809032554/http://musicweb.ucsd.edu/~sdubnov/CATbox/Reader/p77-foote.pdf |archive-date=9 August 2017|isbn=978-1581131512 |citeseerx=10.1.1.223.194 |s2cid=3329298 }} In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.{{cite book |last1=Pareyon |first1=Gabriel |title=On Musical Self-Similarity: Intersemiosis as Synecdoche and Analogy |date=April 2011 |publisher=International Semiotics Institute at Imatra; Semiotic Society of Finland |isbn=978-952-5431-32-2 |page=240 |url=https://tuhat.helsinki.fi/portal/files/15216101/Pareyon_Dissertation.pdf |access-date=30 July 2018 |archive-url=https://web.archive.org/web/20170208034152/https://tuhat.helsinki.fi/portal/files/15216101/Pareyon_Dissertation.pdf |archive-date=8 February 2017}} (Also see [https://books.google.com/books?id=xQIynayPqMQC&pg=PA240&lpg=PA240&focus=viewport&vq=%221/f+noise+substantially+as+a+temporal+phenomenon%22 Google Books])

{{columns-list|colwidth=30em|

• Droste effect
• Golden ratio
• Logarithmic spiral
• Long-range dependency
• Non-well-founded set theory
• Recursion
• Self-dissimilarity
• Self-reference
• Self-replication
• Self-similarity of network data analysis
• Self-similar process
• Teragon
• Tessellation
• Tweedie distributions
• Zipf's law
• Fractal}}

{{Reflist}}

• [http://www.ericbigas.com/fractals/cc "Copperplate Chevrons"] — a self-similar fractal zoom movie
• [http://pi.314159.ru/longlist.htm "Self-Similarity"] — New articles about Self-Similarity. Waltz Algorithm

• {{cite journal|journal=Physica Scripta|volume=32|issue=4|year=1985|pages=257–260|title=Self-affinity and fractal dimension|url=http://users.math.yale.edu/mandelbrot/web_pdfs/112selfAffinity.pdf|doi=10.1088/0031-8949/32/4/001|bibcode=1985PhyS...32..257M|last1=Mandelbrot|first1=Benoit B.|s2cid=250815596 }}
• {{cite journal |last1=Sapozhnikov |first1=Victor |last2=Foufoula-Georgiou |first2=Efi |title=Self-Affinity in Braided Rivers |journal=Water Resources Research |date=May 1996 |volume=32 |issue=5 |pages=1429–1439 |doi=10.1029/96wr00490 |bibcode=1996WRR....32.1429S |url=http://efi.eng.uci.edu/papers/efg_023.pdf |access-date=30 July 2018 |url-status=live |archive-url=https://web.archive.org/web/20180730230931/http://efi.eng.uci.edu/papers/efg_023.pdf |archive-date=30 July 2018}}
• {{cite book|title=Gaussian Self-Affinity and Fractals: Globality, the Earth, 1/F Noise, and R/S|author= Benoît B. Mandelbrot|isbn=978-0387989938|year= 2002|publisher= Springer}}

{{Fractals}}

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Category:Fractals

Category:Scaling symmetries

Category:Homeomorphisms

Category:Self-reference