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当无限放大科赫曲线时，它会展示出无限重复的自相似性。]]  

当无限放大科赫曲线时，它会展示出无限重复的自相似性。]]  

[[File:Standard self-similarity.png|thumb|300px|Standard (trivial) self-similarity.<ref name=":0">Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. {{ISBN|978-0716711865}}.</ref>标准(平凡)自相似性。<ref name=":0" />]]

[[File:Standard self-similarity.png|thumb|300px|Standard (trivial) self-similarity.<ref name=":0">Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. {{ISBN|978-0716711865}}.</ref>标准(平凡)自相似性。<ref name=":0" />]]

This vocabulary was introduced by Benoit Mandelbrot in 1964. 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 }} [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.

This vocabulary was introduced by Benoit Mandelbrot in 1964. 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 }} [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.

比自相似性更一般的概念是自仿射性。

比自相似性更一般的概念是自仿射性。

==Examples==

==Examples 实例==

[[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:Feigenbaumzoom.gif|left|thumb|201px|Self-similarity in the [[Mandelbrot set]] shown by zooming in on the Feigenbaum point at (−1.401155189...,&nbsp;0)

曼德勃罗集在费根鲍姆点（- 1.401155189... ，0）处不断放大显示出其中的自相似性。]]

 

在[[ Mandelbrot 集合放大 Feigenbaum 点在(- 1.401155189... ，0)]中的自相似性

 

 

An image of the [[Barnsley fern which exhibits affine self-similarity]]

 

仿射图[展现仿射自相似性的巴恩斯利蕨类植物]的图像

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

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

The Mandelbrot set is also self-similar around Misiurewicz points.

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.<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|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.

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|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.

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.  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.

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.  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|

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| authorlink=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>

date=February 1999| authorlink=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]].

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

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

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

 

 

[[File:RepeatedBarycentricSubdivision.png|thumb|A triangle subdivided repeatedly using [[barycentric subdivision]]. The complement of the large circles becomes a [[Sierpinski carpet]]]]

使用重心细分重复细分的三角形。大圆圈的补充使其成为谢尔宾斯基地毯。]]

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

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

使用[[重心细分]重复细分的三角形。大圆圈的补充变成了谢尔宾斯基地毯

使用[[重心细分]重复细分的三角形。大圆圈的补充变成了谢尔宾斯基地毯

 

=== In [[cybernetics]] 控制论领域 ===

 

 

=== 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.

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.

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.

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.

斯塔福德 · 比尔的可行系统模型是一个具有仿射自相似层次结构的组织模型，其中给定的可行系统是一个可行系统的一个元素，一个递归层次更高的可行系统的一个元素，对于这个系统的元素是一个递归层次更低的可行系统。

斯塔福德 · 比尔的可行系统模型是一个具有仿射自相似层次结构的组织模型，其中一个给定的可行系统是一个递归更高一级的可行系统之一的一个元素，对于这个系统的元素是一个递归层次更低的可行系统。

 

[[File:Flickr - cyclonebill - Romanesco.jpg|thumb|right|200px|Close-up of a [[Romanesco broccoli]].]]

罗马花椰菜的特写镜头。]]

Close-up of a [[Romanesco broccoli.]]

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.

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 ===

=== In music 音乐世界 ===

* Strict [[canon (music)|canons]] display various types and amounts of self-similarity, as do sections of [[fugue (music)|fugues]].

* 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.

* 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.

* 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=Visualizing music and audio using self-similarity |journal=Multimedia '99 Proceedings of the Seventh ACM International Conference on Multimedia (Part 1) |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 }}</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 |accessdate=30 July 2018 |archiveurl=https://web.archive.org/web/20170208034152/https://tuhat.helsinki.fi/portal/files/15216101/Pareyon_Dissertation.pdf |archivedate=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>

* 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=Visualizing music and audio using self-similarity |journal=Multimedia '99 Proceedings of the Seventh ACM International Conference on Multimedia (Part 1) |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 }}</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 |accessdate=30 July 2018 |archiveurl=https://web.archive.org/web/20170208034152/https://tuhat.helsinki.fi/portal/files/15216101/Pareyon_Dissertation.pdf |archivedate=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>