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Moved page from wikipedia:en:Self-similarity (history)
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{{short description|The whole of an object being mathematically similar to part of itself}}
{{Use dmy dates|date=April 2017}}
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[[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.]]
A [[Koch curve has an infinitely repeating self-similarity when it is magnified.]]
[[[科赫曲线被放大后有无限重复的自相似性]]
[[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>]]
Standard (trivial) self-similarity.
标准(平凡)自相似性。
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.
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. Self-similarity is a typical property of 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.
在数学中，一个自相似的物体与它自身的某一部分完全或近似地相似(例如:。整体具有相同的形状作为一个或多个部分)。现实世界中的许多物体，例如海岸线，在统计学上是自相似的: 它们的某些部分在许多尺度上表现出相同的统计特性。自相似是分形的一个典型性质。尺度不变性是自相似的一种精确形式，在任何放大倍数下，物体中有一小块与整体相似的部分。例如，科赫雪花的一侧既对称又具有尺度不变性; 它可以连续放大3倍而不改变形状。分形中明显的非平凡的相似性是通过它们的精细结构或任意小尺度上的细节来区分的。作为一个反例，尽管直线的任何部分都可能类似于整体，但是进一步的细节没有透露。
A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity
A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity
一个时间发展的现象被称为自相似性，如果某个可观测量的数值
<math>f(x,t)</math> measured at different times are different but the corresponding dimensionless quantity at given value of <math>x/t^z</math> remain invariant. It happens if the quantity <math>f(x,t)</math> exhibits [[dynamic scaling]]. The idea is just an extension of the idea of similarity of two triangles.<ref>{{cite journal | author = Hassan M. K., Hassan M. Z., Pavel N. I. | year = 2011 | title = Dynamic scaling, data-collapseand Self-similarity in Barabasi-Albert networks | url = | journal = J. Phys. A: Math. Theor. | volume = 44 | issue = 17| page = 175101 | doi=10.1088/1751-8113/44/17/175101| arxiv = 1101.4730| bibcode = 2011JPhA...44q5101K}}</ref><ref>{{cite journal | author = Hassan M. K., Hassan M. Z. | year = 2009 | title = Emergence of fractal behavior in condensation-driven aggregation | url = | journal = Phys. Rev. E | volume = 79 | issue = 2| page = 021406 | doi=10.1103/physreve.79.021406| pmid = 19391746 | arxiv = 0901.2761| bibcode = 2009PhRvE..79b1406H}}</ref><ref>{{cite journal | author = Dayeen F. R., Hassan M. K. | year = 2016 | title = Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice | url = | journal = Chaos, Solitons & Fractals | volume = 91 | issue = | page = 228 | doi=10.1016/j.chaos.2016.06.006| arxiv = 1409.7928| bibcode = 2016CSF....91..228D}}</ref> 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.
<math>f(x,t)</math> measured at different times are different but the corresponding dimensionless quantity at given value of <math>x/t^z</math> remain invariant. It happens if the quantity <math>f(x,t)</math> exhibits dynamic scaling. The idea is just an extension of the idea of similarity of two triangles. 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.
在不同时间测量的 math f (x，t) / math 是不同的，但是在给定的 math x / t ^ z / math 值下相应的无量纲量是不变的。如果数学量 f (x，t) / math 表现出动态缩放，就会发生这种情况。这个概念只是两个三角形相似性概念的延伸。请注意，如果两个三角形的边的数值不同，那么它们是相似的，但是对应的无量纲量，例如它们的角重合。
{{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>}}
{{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>}}
==Self-affinity==
<!--[[Self-affinity]] redirects directly here.-->
<!--Self-affinity redirects directly here.-->
! -- 自我关联直接重定向到这里 --
[[Image:Self-affine set.png|thumb|right| A self-affine fractal with [[Hausdorff dimension]]=1.8272.]]
A self-affine fractal with [[Hausdorff dimension=1.8272.]]
一个自仿射分形[豪斯多夫维数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]].
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.
在数学中，自亲和力是分形的一个特征，它的各部分在 x 方向和 y 方向上按不同的比例缩放。这意味着，为了了解这些分形物体的自相似性，它们必须使用各向异性仿射变换重新标度。
==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
A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms <math>\{ f_s : s\in S \}</math> for which
紧致拓扑空间 x 是自相似的，如果存在一个有限集 s 对 s / math 中的一组非满射同胚数学进行索引
:<math>X=\bigcup_{s\in S} f_s(X)</math>
<math>X=\bigcup_{s\in S} f_s(X)</math>
S } f (x) / math 中的数学 x 大杯
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
If <math>X\subset Y</math>, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for <math>\{ f_s : s\in S \} </math>. We call
如果数学 x 子集 y / math，我们称 x 自相似，如果它是 y 的唯一非空子集，使得上面的方程适用于数学 s: s / math。我们打电话
:<math>\mathfrak{L}=(X,S,\{ f_s : s\in S \} )</math>
<math>\mathfrak{L}=(X,S,\{ f_s : s\in S \} )</math>
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.
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.
一个自相似的结构。同胚可以迭代，产生迭代函数系统。函数的组合创建了 monoid 的代数结构。当集合 s 只有两个元素时，这个幺半群称为二元幺半群。二元幺半群可以被视为一棵无限的二叉树，更一般地说，如果集合 s 有 p 个元素，那么幺半群可以被表示为一棵 p-adic 树。
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.
The automorphisms of the dyadic monoid is the modular group; the automorphisms can be pictured as hyperbolic rotations of the binary tree.
二元幺半群的自同构是模群，自同构可以描述为二叉树的双曲旋转。
A more general notion than self-similarity is [[Self-affinity]].
A more general notion than self-similarity is Self-affinity.
比自相似性更一般的概念是自相似性。
==Examples==
[[Image:Feigenbaumzoom.gif|left|thumb|201px|Self-similarity in the [[Mandelbrot set]] shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)]]
Self-similarity in the [[Mandelbrot set shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)]]
在[[ Mandelbrot 集合放大 Feigenbaum 点在(- 1.401155189... ，0)]中的自相似性
[[Image:Fractal fern explained.png|thumb|right|300px|An image of the [[Barnsley fern]] which exhibits [[affine transformation|affine]] self-similarity]]
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 points.
曼德尔布罗特集合在 Misiurewicz 点附近也具有自相似性。
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.
自相似性对于计算机网络的设计有着重要的影响，因为典型的网络流量具有自相似的特性。例如，在远程通信工程中，分组交换数据通信模式在统计学上似乎是自相似的。这个特性意味着使用泊松分佈的简单模型是不准确的，而且在设计网络时没有考虑自相似性，网络很可能以意想不到的方式运行。
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|
类似地，股票市场的运动被描述为表现出自我亲和力，即。当通过适当的仿射变换显示细节的水平时，它们看起来自我相似。 不同的 https://www.scientificamerican.com/article/multifractals-Explain-Wall-Street/ 可以解释什么是华尔街的错误 | 作者本华·曼德博 | 科学美国人 |
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.
日期1999年2月 | authorlink 本华·曼德博} / ref Andrew Lo 在计量经济学中描述了股票市场日志返回自相似性。
[[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.
有限细分规则是建立自相似集的一种强大技术，包括 Cantor 集和谢尔宾斯基三角形集。
[[File:RepeatedBarycentricSubdivision.png|thumb|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
=== 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.
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。
=== In nature ===
[[File:Flickr - cyclonebill - Romanesco.jpg|thumb|right|200px|Close-up of a [[Romanesco broccoli]].]]
Close-up of a [[Romanesco broccoli.]]
宝塔花菜的特写镜头
{{further|Patterns in nature}}
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.
自相似性也存在于自然界中。右边是一个数学生成的完全自相似的蕨类植物图像，它与天然蕨类植物有明显的相似之处。其他植物，如宝塔花菜，表现出很强的自相似性。
=== 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=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>
==See also==
{{columns-list|colwidth=30em|
{{columns-list|colwidth=30em|
{ columns-list | colwidth 30em |
* [[Droste effect]]
* [[Golden ratio]]
* [[Long-range dependency]]
* [[Non-well-founded set theory]]
* [[Recursion]]
* [[Self-dissimilarity]]
* [[Self-reference]]
* [[Self-replication]]
* [[Self-Similarity of Network Data Analysis]]
* [[Teragon]]
* [[Tessellation]]
* [[Tweedie distributions]]
* [[Zipf's law]]}}
==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.}}
*{{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 |url=http://efi.eng.uci.edu/papers/efg_023.pdf |accessdate=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}}
{{Fractals}}
{{DEFAULTSORT:Self-Similarity}}
[[Category:Fractals]]
Category:Fractals
分类: 分形
[[Category:Scaling symmetries]]
Category:Scaling symmetries
类别: 缩放对称性
[[Category:Homeomorphisms]]
Category:Homeomorphisms
范畴: 同胚
[[Category:Self-reference]]
Category:Self-reference
类别: 自我参照
<noinclude>
<small>This page was moved from [[wikipedia:en:Self-similarity]]. Its edit history can be viewed at [[自相似/edithistory]]</small></noinclude>
[[Category:待整理页面]]
{{short description|The whole of an object being mathematically similar to part of itself}}
{{Use dmy dates|date=April 2017}}
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[[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.]]
A [[Koch curve has an infinitely repeating self-similarity when it is magnified.]]
[[[科赫曲线被放大后有无限重复的自相似性]]
[[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>]]
Standard (trivial) self-similarity.
标准(平凡)自相似性。
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.
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. Self-similarity is a typical property of 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.
在数学中，一个自相似的物体与它自身的某一部分完全或近似地相似(例如:。整体具有相同的形状作为一个或多个部分)。现实世界中的许多物体，例如海岸线，在统计学上是自相似的: 它们的某些部分在许多尺度上表现出相同的统计特性。自相似是分形的一个典型性质。尺度不变性是自相似的一种精确形式，在任何放大倍数下，物体中有一小块与整体相似的部分。例如，科赫雪花的一侧既对称又具有尺度不变性; 它可以连续放大3倍而不改变形状。分形中明显的非平凡的相似性是通过它们的精细结构或任意小尺度上的细节来区分的。作为一个反例，尽管直线的任何部分都可能类似于整体，但是进一步的细节没有透露。
A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity
A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity
一个时间发展的现象被称为自相似性，如果某个可观测量的数值
<math>f(x,t)</math> measured at different times are different but the corresponding dimensionless quantity at given value of <math>x/t^z</math> remain invariant. It happens if the quantity <math>f(x,t)</math> exhibits [[dynamic scaling]]. The idea is just an extension of the idea of similarity of two triangles.<ref>{{cite journal | author = Hassan M. K., Hassan M. Z., Pavel N. I. | year = 2011 | title = Dynamic scaling, data-collapseand Self-similarity in Barabasi-Albert networks | url = | journal = J. Phys. A: Math. Theor. | volume = 44 | issue = 17| page = 175101 | doi=10.1088/1751-8113/44/17/175101| arxiv = 1101.4730| bibcode = 2011JPhA...44q5101K}}</ref><ref>{{cite journal | author = Hassan M. K., Hassan M. Z. | year = 2009 | title = Emergence of fractal behavior in condensation-driven aggregation | url = | journal = Phys. Rev. E | volume = 79 | issue = 2| page = 021406 | doi=10.1103/physreve.79.021406| pmid = 19391746 | arxiv = 0901.2761| bibcode = 2009PhRvE..79b1406H}}</ref><ref>{{cite journal | author = Dayeen F. R., Hassan M. K. | year = 2016 | title = Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice | url = | journal = Chaos, Solitons & Fractals | volume = 91 | issue = | page = 228 | doi=10.1016/j.chaos.2016.06.006| arxiv = 1409.7928| bibcode = 2016CSF....91..228D}}</ref> 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.
<math>f(x,t)</math> measured at different times are different but the corresponding dimensionless quantity at given value of <math>x/t^z</math> remain invariant. It happens if the quantity <math>f(x,t)</math> exhibits dynamic scaling. The idea is just an extension of the idea of similarity of two triangles. 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.
在不同时间测量的 math f (x，t) / math 是不同的，但是在给定的 math x / t ^ z / math 值下相应的无量纲量是不变的。如果数学量 f (x，t) / math 表现出动态缩放，就会发生这种情况。这个概念只是两个三角形相似性概念的延伸。请注意，如果两个三角形的边的数值不同，那么它们是相似的，但是对应的无量纲量，例如它们的角重合。
{{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>}}
{{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>}}
==Self-affinity==
<!--[[Self-affinity]] redirects directly here.-->
<!--Self-affinity redirects directly here.-->
! -- 自我关联直接重定向到这里 --
[[Image:Self-affine set.png|thumb|right| A self-affine fractal with [[Hausdorff dimension]]=1.8272.]]
A self-affine fractal with [[Hausdorff dimension=1.8272.]]
一个自仿射分形[豪斯多夫维数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]].
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.
在数学中，自亲和力是分形的一个特征，它的各部分在 x 方向和 y 方向上按不同的比例缩放。这意味着，为了了解这些分形物体的自相似性，它们必须使用各向异性仿射变换重新标度。
==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
A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms <math>\{ f_s : s\in S \}</math> for which
紧致拓扑空间 x 是自相似的，如果存在一个有限集 s 对 s / math 中的一组非满射同胚数学进行索引
:<math>X=\bigcup_{s\in S} f_s(X)</math>
<math>X=\bigcup_{s\in S} f_s(X)</math>
S } f (x) / math 中的数学 x 大杯
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
If <math>X\subset Y</math>, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for <math>\{ f_s : s\in S \} </math>. We call
如果数学 x 子集 y / math，我们称 x 自相似，如果它是 y 的唯一非空子集，使得上面的方程适用于数学 s: s / math。我们打电话
:<math>\mathfrak{L}=(X,S,\{ f_s : s\in S \} )</math>
<math>\mathfrak{L}=(X,S,\{ f_s : s\in S \} )</math>
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.
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.
一个自相似的结构。同胚可以迭代，产生迭代函数系统。函数的组合创建了 monoid 的代数结构。当集合 s 只有两个元素时，这个幺半群称为二元幺半群。二元幺半群可以被视为一棵无限的二叉树，更一般地说，如果集合 s 有 p 个元素，那么幺半群可以被表示为一棵 p-adic 树。
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.
The automorphisms of the dyadic monoid is the modular group; the automorphisms can be pictured as hyperbolic rotations of the binary tree.
二元幺半群的自同构是模群，自同构可以描述为二叉树的双曲旋转。
A more general notion than self-similarity is [[Self-affinity]].
A more general notion than self-similarity is Self-affinity.
比自相似性更一般的概念是自相似性。
==Examples==
[[Image:Feigenbaumzoom.gif|left|thumb|201px|Self-similarity in the [[Mandelbrot set]] shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)]]
Self-similarity in the [[Mandelbrot set shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)]]
在[[ Mandelbrot 集合放大 Feigenbaum 点在(- 1.401155189... ，0)]中的自相似性
[[Image:Fractal fern explained.png|thumb|right|300px|An image of the [[Barnsley fern]] which exhibits [[affine transformation|affine]] self-similarity]]
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 points.
曼德尔布罗特集合在 Misiurewicz 点附近也具有自相似性。
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.
自相似性对于计算机网络的设计有着重要的影响，因为典型的网络流量具有自相似的特性。例如，在远程通信工程中，分组交换数据通信模式在统计学上似乎是自相似的。这个特性意味着使用泊松分佈的简单模型是不准确的，而且在设计网络时没有考虑自相似性，网络很可能以意想不到的方式运行。
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|
类似地，股票市场的运动被描述为表现出自我亲和力，即。当通过适当的仿射变换显示细节的水平时，它们看起来自我相似。 不同的 https://www.scientificamerican.com/article/multifractals-Explain-Wall-Street/ 可以解释什么是华尔街的错误 | 作者本华·曼德博 | 科学美国人 |
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.
日期1999年2月 | authorlink 本华·曼德博} / ref Andrew Lo 在计量经济学中描述了股票市场日志返回自相似性。
[[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.
有限细分规则是建立自相似集的一种强大技术，包括 Cantor 集和谢尔宾斯基三角形集。
[[File:RepeatedBarycentricSubdivision.png|thumb|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
=== 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.
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。
=== In nature ===
[[File:Flickr - cyclonebill - Romanesco.jpg|thumb|right|200px|Close-up of a [[Romanesco broccoli]].]]
Close-up of a [[Romanesco broccoli.]]
宝塔花菜的特写镜头
{{further|Patterns in nature}}
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.
自相似性也存在于自然界中。右边是一个数学生成的完全自相似的蕨类植物图像，它与天然蕨类植物有明显的相似之处。其他植物，如宝塔花菜，表现出很强的自相似性。
=== 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=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>
==See also==
{{columns-list|colwidth=30em|
{{columns-list|colwidth=30em|
{ columns-list | colwidth 30em |
* [[Droste effect]]
* [[Golden ratio]]
* [[Long-range dependency]]
* [[Non-well-founded set theory]]
* [[Recursion]]
* [[Self-dissimilarity]]
* [[Self-reference]]
* [[Self-replication]]
* [[Self-Similarity of Network Data Analysis]]
* [[Teragon]]
* [[Tessellation]]
* [[Tweedie distributions]]
* [[Zipf's law]]}}
==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.}}
*{{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 |url=http://efi.eng.uci.edu/papers/efg_023.pdf |accessdate=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}}
{{Fractals}}
{{DEFAULTSORT:Self-Similarity}}
[[Category:Fractals]]
Category:Fractals
分类: 分形
[[Category:Scaling symmetries]]
Category:Scaling symmetries
类别: 缩放对称性
[[Category:Homeomorphisms]]
Category:Homeomorphisms
范畴: 同胚
[[Category:Self-reference]]
Category:Self-reference
类别: 自我参照
<noinclude>
<small>This page was moved from [[wikipedia:en:Self-similarity]]. Its edit history can be viewed at [[自相似/edithistory]]</small></noinclude>
[[Category:待整理页面]]