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[[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.

自相似一词由本华·曼德勃罗与1964年引入。<ref name=":1" />'''（标注：此处将后文的一句移过来以使行文看起来更连贯。因此参考文献顺序也需要调一下，原来的8提前到3的位置。）'''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：整体和一个或多个部分具有相同的形状）。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不同尺度上表现出相同的统计特性。<ref name="Mandelbrot_Science_1967" /> 自相似是分形的一个典型性质。 '''Scale Invariance 标度不变性'''是自相似的一种精确形式：在任何放大倍数下，物体中总有更小的部分与整体相似。例如，'''Koch Snowflake 科赫雪花'''的一边既对称又具有标度不变性；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。对比一个反例来看，尽管直线的任何部分都可能类似于整体，但是进一步放大之后，却没有更多的细节显露。

自相似一词由'''Benoit Mandelbrot 本华·曼德勃罗'''与1964年引入。<ref name=":1" />'''（标注：此处将后文的一句移过来以使行文看起来更连贯。因此参考文献顺序也需要调一下，原来的8提前到3的位置。）'''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：整体和一个或多个部分具有相同的形状）。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不同尺度上表现出相同的统计特性。<ref name="Mandelbrot_Science_1967" /> 自相似是分形的一个典型性质。 '''Scale Invariance 标度不变性'''是自相似的一种精确形式：在任何放大倍数下，物体中总有更小的部分与整体相似。例如，'''Koch Snowflake 科赫雪花'''的一边既对称又具有标度不变性；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。对比一个反例来看，尽管直线的任何部分都可能类似于整体，但是进一步放大之后，却没有更多的细节显露。

A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity <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 name=":6">{{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 name=":7">{{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 name=":8">{{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.  

A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity <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 name=":6">{{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 name=":7">{{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 name=":8">{{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.  

Peitgen ''et al.'' explain the concept as such:<blockquote>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.</blockquote>

Peitgen ''et al.'' explain the concept as such:<blockquote>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.</blockquote>

如果一个图形的部分是整体的小尺度复制品，就可以认为这一图形是自相似的；如果图形分解产生的部分都是该图形的精确复制，则这个图形是严格自相似的。任何任意的部分都包含整个图形的精确复制。<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. <nowiki>ISBN 0-387-97346-X</nowiki> and <nowiki>ISBN 3-540-97346-X</nowiki>.</ref>

如果一个图形的部分是整体的小尺度复制品，就可以认为这一图形是自相似的；如果图形分解产生的部分都是该图形的精确复制，则这个图形是严格自相似的。任何任意的部分都包含整个图形的精确复制。<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. <nowiki>ISBN 0-387-97346-X</nowiki> and <nowiki>ISBN 3-540-97346-X</nowiki>.</ref>

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.

自相似对于计算机网络的设计有着重要的意义，因为典型的网络流量具有自相似的特性。例如，在电信流量工程中，分组交换数据流量模式似乎在统计上是自相似的<ref name=":1" />。这种性质意味着使用泊松分布的简单模型是不准确的，而没有考虑自相似性的网络很可能以意想不到的方式运行。

自相似对于计算机网络的设计有着重要的意义，因为典型的网络流量具有自相似的特性。例如，在电信流量工程中，分组交换数据流量模式似乎在统计上是自相似的<ref name=":1" />。这种性质意味着使用'''Poisson Distribution 泊松分布'''的简单模型是不准确的，而没有考虑自相似性的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 name=":2">Peitgen, et al (1991), p.2-3.</ref> [[Andrew Lo]]  describes stock market log return self-similarity in  [[econometrics]].<ref name=":3">Campbell, Lo and MacKinlay (1991)  "[[Econometrics]] of Financial Markets ", Princeton University Press! {{ISBN|978-0691043012}}</ref>

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 name=":2">Peitgen, et al (1991), p.2-3.</ref> [[Andrew Lo]]  describes stock market log return self-similarity in  [[econometrics]].<ref name=":3">Campbell, Lo and MacKinlay (1991)  "[[Econometrics]] of Financial Markets ", Princeton University Press! {{ISBN|978-0691043012}}</ref>

类似地，人们在描述股票市场时认为其波动具有自仿射性，也就是说，当根据显示的细节程度，通过适当的仿射变换进行转换时，它们显示出自相似性<ref name=":2" />。 罗安儒描述了计量经济学中股票市场的对数回报自相似性<ref name=":3" />。

类似地，人们在描述股票市场时认为其波动具有自仿射性，也就是说，当根据显示的细节程度，通过适当的仿射变换进行转换时，它们显示出自相似性<ref name=":2" />。 '''Andrew Lo 罗安儒'''描述了计量经济学中股票市场的对数回报自相似性<ref name=":3" />。

[[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.

有限细分规则是一种建立自相似集的强大方法，包括康托集和谢尔宾斯基三角形。[[File:RepeatedBarycentricSubdivision.png|thumb|A triangle subdivided repeatedly using [[barycentric subdivision]]. The complement of the large circles becomes a [[Sierpinski carpet]]

有限细分规则是一种建立自相似集的强大方法，包括 '''Cantor Set 康托集'''和'''Sierpinski Triangle 谢尔宾斯基三角'''。[[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。]]

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。]]

罗马花椰菜的特写镜头。]]

罗马花椰菜的特写镜头。]]

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.