Editing Self-similarity

{{Short description|Whole of an object being mathematically similar to part of itself}}
{{Use dmy dates|date=April 2017}}
[[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A [[Koch snowflake]] has an infinitely repeating self-similarity when it is magnified.]] 
[[File:Standard self-similarity.png|thumb|300px|Standard (trivial) self-similarity<ref>Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. {{ISBN|978-0716711865}}.</ref>]]

In [[mathematics]], a '''self-similar''' object is exactly or approximately [[similarity (geometry)|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 [[coastline]]s, are statistically self-similar: parts of them show the same statistical properties at many scales.<ref name="Mandelbrot_Science_1967">{{cite journal | title=How long is the coast of Britain? Statistical self-similarity and fractional dimension | journal=[[Science (journal)|Science]] | date=5 May 1967 | author=Mandelbrot, Benoit B. | pages=636–638 | volume=156 | number=3775 | doi=10.1126/science.156.3775.636 | series=New Series | pmid=17837158 | bibcode=1967Sci...156..636M | s2cid=15662830 | url=http://ena.lp.edu.ua:8080/handle/ntb/52473 | access-date=12 November 2020 | archive-date=19 October 2021 | archive-url=https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473 | url-status=dead }} [http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf PDF]</ref> Self-similarity is a typical property of [[fractal]]s. [[Scale invariance]] is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is [[Similarity (geometry)|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.

[[Heinz-Otto Peitgen|Peitgen]] ''et al.'' explain the concept as such:

{{Quote|If parts of a figure are small replicas of the whole, then the figure is called ''self-similar''....A figure is ''strictly self-similar'' if 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.<ref>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}}.</ref>}}Since mathematically, a fractal may show self-similarity under arbitrary magnification, it is impossible to recreate this physically. Peitgen ''et al.'' suggest studying self-similarity using approximations:{{Quote|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.<ref name="Classroom">Peitgen, et al (1991), p.2-3.</ref>}}

This vocabulary was introduced by [[Benoit Mandelbrot]] in 1964.<ref>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</ref>

==Self-affinity==
<!--[[Self-affinity]] redirects directly here.-->
[[Image:Self-affine set.png|thumb|right|A self-affine fractal with [[Hausdorff dimension]] = 1.8272]]

In [[mathematics]], '''self-affinity''' is a feature of a [[fractal]] whose pieces are [[scaling (geometry)|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 space|compact]] [[topological space]] ''X'' is self-similar if there exists a [[finite set]] ''S'' indexing a set of non-[[surjective]] [[homeomorphism]]s <math>\{ f_s : s\in S \}</math> for which

:<math>X=\bigcup_{s\in S} f_s(X)</math>

If <math>X\subset Y</math>, we call ''X'' self-similar if it is the only [[Non-empty set|non-empty]] [[subset]] of ''Y'' such that the equation above holds for <math>\{ f_s : s\in S \} </math>. We call

:<math>\mathfrak{L}=(X,S,\{ f_s : s\in S \} )</math>

a ''self-similar structure''. The homeomorphisms may be [[iterated function|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 number|p-adic]] tree.

The [[automorphism]]s of the dyadic monoid is the [[modular group]]; the automorphisms can be pictured as [[Hyperbolic coordinates|hyperbolic rotation]]s of the binary tree.

A more general notion than self-similarity is [[self-affinity]].

==Examples==
[[Image:Feigenbaumzoom.gif|left|thumb|201px|Self-similarity in the [[Mandelbrot set]] shown by zooming in on the Feigenbaum point at (−1.401155189...,&nbsp;0)]]
[[Image:Fractal fern explained.png|thumb|right|300px|An image of the [[Barnsley fern]] which exhibits [[affine transformation|affine]] self-similarity]]

The [[Mandelbrot set]] is also self-similar around [[Misiurewicz point]]s.

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.<ref>{{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}}</ref>  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.<ref>{{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}}</ref> [[Andrew Lo]]  describes stock market log return self-similarity in  [[econometrics]].<ref>Campbell, Lo and MacKinlay (1991)  "[[Econometrics]] of Financial Markets ", Princeton University Press! {{ISBN|978-0691043012}}</ref>

[[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.<ref>{{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}}</ref>

[[File:RepeatedBarycentricSubdivision.png|thumb|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 ===
[[File:Flickr - cyclonebill - Romanesco.jpg|thumb|right|200px|Close-up of a [[Romanesco broccoli]]]]
{{further|Patterns in nature}}
Self-similarity can be found in nature, as well. Plants, such as [[Romanesco broccoli]], exhibit strong self-similarity.

=== In music ===
* Strict [[canon (music)|canons]] display various types and amounts of self-similarity, as do sections of [[fugue (music)|fugues]].
* A [[Shepard tone]] is self-similar in the frequency or wavelength domains.
* The [[Denmark|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.<ref>{{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 }}</ref> In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.<ref>{{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])</ref>

==See also==
{{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]]}}

==References==
{{Reflist}}

==External links==
*[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

===Self-affinity===
*{{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]]