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第1行： + + + + − {{short description|The whole of an object being mathematically similar to part of itself}} − __NOTOC__+ − __NOTOC__+ − {{Use dmy dates|date=April 2017}} − + − 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" />]]+ − 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'''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. 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. − 对于一个依赖时间发展的现象，如果其相关观测量<math>f(x,t)</math>在不同时间所测得的数值不同，但是对应的无量纲量在给定的<math>x/t^z</math>下保持不变，则可以说该现象具有自相似性。通常如果<math>f(x,t)</math>显示出'''Dynamic Scaling 动态缩放'''就会出现这种情况。这也是相似三角形概念的拓展和延伸。<ref name=":6" /><ref name=":7" /><ref name=":8" /> 值得注意的是，即使两个三角形的边长不同，但他们的内角相等，则他们也是相似的。+ + − + − '''Peitgen 佩特根'''等曾这样解释这一概念：+ − 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> − Since mathematically, a fractal may show self-similarity under indefinite magnification, it is impossible to recreate this physically. Peitgen ''et al.'' suggest studying self-similarity using approximations:<blockquote>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.</blockquote> − 即使从数学上来说，分形可以在无限放大的条件下显示出自相似性，但是这在物理上是不可能实现的。佩特根等建议使用近似方法来研究自相似性： − − 为了使自相似的性质具有操作意义，我们必须处理有限图形的有限近似。这可以采取盒子自相似性方法来解决，即使用不同尺寸的格子对图形的有限阶段进行测量。<ref>Peitgen, et al (1991), p.2-3.</ref> + − <!--[[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. − − 图示为一个自仿射分形，其豪斯多夫维数为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]]. − − 在数学中，'''Self-affinity 自仿射性''' 是分形的特征之一，分形的各部分在 x 方向和 y 方向上按不同的比例缩放。这意味着要理解这些分形对象的自相似性，必须使用'''Anisotropic Affine Transformation 各向异性仿射变换'''进行缩放。 − + − 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 − − − 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>X\subset Y</math>， − 当且仅当X是Y的唯一非空子集，使得上式对+ − <math>\{ f_s : s\in S \} </math> − 成立，则X是自相似的。而且我们称+ − <math>\mathfrak{L}=(X,S,\{ f_s : s\in S \} )</math> − 是自相似结构。+ − − 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. − − 同胚可以迭代，产生'''Iterated Function System 迭代函数系统'''。函数的组合产生了'''Monoid 幺半群'''的代数结构。当集合S只有两个元素时，幺半群此时称为D'''yadic Monoid二元幺半群'''。二元幺半群可以表示为无限'''Binary Tree 二叉树'''；更一般地说，如果集合S有p个元素，则一元类可以表示为'''P-adic Tree p进树'''。 − − 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'''Modular Group 模群'''，自同构可以描述为二叉树的'''Hyperbolic Rotations 双曲旋转'''。 − − A more general notion than self-similarity is [[Self-affinity]]. − − 比自相似性更一般的概念是自仿射性。 − + − − − − [[Image:Fractal fern explained.png|thumb|right|300px|An image of the [[Barnsley fern]] which exhibits [[affine transformation|affine]] self-similarity − − 具有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. − − '''Mandelbrot Set 曼德勃罗集'''在'''Misiurewicz Point 米约维奇点'''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 name=":1">{{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>+ − 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. − + − 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。]] + − 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. − − '''Stafford Beer 斯塔福德 · 比尔'''的'''Viable System Model 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。]] − − 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. − 自然中也存在自相似性。右边是一个数学生成的，完全自相似的蕨类图像，与自然蕨类有明显的相似之处。其他植物，如罗马花椰菜，表现出强烈的自相似性。 − − * 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. − + − * 丹麦作曲家'''Per Nørgård 诺加德'''在他的很多音乐中都使用了一种名为“无限系列”的自相似整数序列。+ + − * 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 name=":4">{{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 name=":5">{{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> − * 在音乐信息检索的研究领域中，自相似性通常指的是音乐往往由在时间上重复的部分组成。<ref name=":4" /> 换句话说，音乐在时间转换下是自相似的，而不是(或附加)在缩放下。<ref name=":5" /> + + + + + − * 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 − * Fractal − − == 编者推荐 == − 论文解读：几何重整化揭示 多尺度人脑网络的自相似性 (swarma.org) —— 郑木华 https://campus.swarma.org/course/1906 − − 分形的世界 (swarma.org) —— 狄增如 https://campus.swarma.org/course/760 − − 路径 - 分形的学习路径 (swarma.org) —— 海芽 https://pattern.swarma.org/path?id=80 − − + − − 第194行：
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第118行： + + − {{Fractals}}+ + − {{DEFAULTSORT:Self-Similarity}}+ − + − + − + − Category:Fractals − − 分类: 分形 − − − − Category:Scaling symmetries − − 类别: 缩放对称性 − − − − Category:Homeomorphisms − − 范畴: 同胚 − − − − 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]]
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[[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|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|标准(平凡)自相似性。<ref name=":0">Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. {{ISBN|978-0716711865}}.</ref>]
[[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.
自相似一词由[[伯努瓦·曼德布洛特 Benoit Mandelbrot]]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">{{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> 自相似是[[分形]]的一个典型性质。[[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，[[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。
对于一个依赖时间发展的现象，如果其相关观测量<math>f(x,t)</math>在不同时间所测得的数值不同，但是对应的无量纲量在给定的<math>x/t^z</math>下保持不变，则可以说该现象具有自相似性。通常如果<math>f(x,t)</math>显示出'''Dynamic Scaling 动态缩放'''就会出现这种情况。这也是相似三角形概念的拓展和延伸。<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>值得注意的是，即使两个三角形的边长不同，但他们的内角相等，则他们也是相似的。
佩特根 Peitgen等人曾这样解释这一概念：
<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。
</blockquote><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>
即使从数学上来说，分形可以在无限放大的条件下显示出自相似性，但是这在物理上是不可能实现的。佩特根等建议使用近似方法来研究自相似性：
<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>
为了使自相似的性质具有操作意义，我们必须处理有限图形的有限近似。这可以采取盒子自相似性方法来解决，即使用不同尺寸的格子对图形的有限阶段进行测量。
</blockquote><ref>Peitgen, et al (1991), p.2-3.</ref>
==自仿射性==
==自仿射性==
[[Image:Self-affine set.png|thumb|right| 图示为一个自仿射分形，其[[豪斯多夫维数]]为1.8272.]]
在数学中，'''自仿射性 Self-affinity ''' 是分形的特征之一，分形的各部分在 x 方向和 y 方向上按不同的比例缩放。这意味着要理解这些分形对象的自相似性，必须使用'''各向异性仿射变换 Anisotropic Affine Transformation '''进行缩放。
==定义==
==定义==
如果存在一个有限集 ''S'' 对<math>\{ f_s : s\in S \}</math>中的一组'''非满射同胚集 Non-surjective Homeomorphisms '''进行索引，则'''紧致拓扑空间 Compact Topological Space ''' ''X'' 是自相似的，有：
如果存在一个有限集 ''S'' 对<math>\{ f_s : s\in S \}</math>中的一组'''Non-surjective Homeomorphisms 非满射同胚集'''进行索引，则'''Compact Topological Space 紧致拓扑空间''' ''X'' 是自相似的，有：
<math>X=\bigcup_{s\in S} f_s(X)</math>
<math>X=\bigcup_{s\in S} f_s(X)</math>
假设有
假设有<math>X\subset Y</math>，当且仅当''X''是''Y'的唯一非空子集，使得上式对<math>\{ f_s : s\in S \} </math>成立，则''X''是自相似的。而且我们称<math>\mathfrak{L}=(X,S,\{ f_s : s\in S \} )</math>是自相似结构。
同胚可以迭代，产生'''迭代函数系统 Iterated Function System '''。函数的组合产生了'''幺半群 Monoid '''的代数结构。当集合S只有两个元素时，幺半群此时称为'''二元幺半群 Dyadic Monoid'''。二元幺半群可以表示为无限二叉树；更一般地说，如果集合S有p个元素，则一元类可以表示为'''p进树 P-adic Tree '''。
二元幺半群的自同构是'''模群 Modular Group '''，自同构可以描述为二叉树的'''双曲旋转 Hyperbolic Rotations '''。
比自相似性更一般的概念是自仿射性。
==实例==
==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..., 0)
[[Image:Feigenbaumzoom.gif|left|thumb|201px|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。]]
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。]]
[[Image:Fractal fern explained.png|thumb|right|300px|具有仿射自相似性的巴恩斯利蕨类植物的图像]]
[[曼德布洛特集]]在Misiurewicz 点附近也具有自相似性。
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。<ref name=":1">{{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>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。
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">Peitgen, et al (1991), p.2-3.</ref>。Andrew Lo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">Campbell, Lo and MacKinlay (1991) "[[Econometrics]] of Financial Markets ", Princeton University Press! {{ISBN|978-0691043012}}</ref>。
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]]
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[[谢尔宾斯基三角]]。[[File:RepeatedBarycentricSubdivision.png|thumb|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。]]
=== 控制论领域 ===
=== 控制论领域 ===
[[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。
=== 自然界中 ===
=== 自然界中 ===
[[File:Flickr - cyclonebill - Romanesco.jpg|thumb|right|200px|罗马花椰菜的特写镜头。]]
自然中也存在自相似性。右边是一个数学生成的，完全自相似的蕨类图像，与自然蕨类有明显的相似之处。其他植物，如罗马花椰菜，表现出强烈的自相似性。
=== 音乐世界 ===
=== 音乐世界 ===
* 严格的经典表现出各种类型和数量的自相似性，赋格曲部分也是如此。
* 严格的经典表现出各种类型和数量的自相似性，赋格曲部分也是如此。
* 谢帕德音调在频率域或波长域是自相似的。
* 谢帕德音调在频率域或波长域是自相似的。
* The [[Denmark|Danish]] [[composer]] [[Per Nørgård]] has made use of a self-similar [[integer sequence]] named the 'infinity series' in much of his music.
* 丹麦作曲家Per Nørgård'''在他的很多音乐中都使用了一种名为“无限系列”的自相似整数序列。
* 在音乐信息检索的研究领域中，自相似性通常指的是音乐往往由在时间上重复的部分组成。<ref name=":4">{{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>换句话说，音乐在时间转换下是自相似的，而不是(或附加)在缩放下。<ref name=":5">{{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>
==另见==
==另见==
[[递归定理]]
[[自我复制]]
[[Tweedie分布]]
[[Zipf定律]]
[[分形]]
==参考文献==
==参考文献==
{{Reflist}}
{{Reflist}}
==其他链接==
==External links==
*[http://www.ericbigas.com/fractals/cc "Copperplate Chevrons"] — a self-similar fractal zoom movie
*[http://www.ericbigas.com/fractals/cc "Copperplate Chevrons"] — a self-similar fractal zoom movie
===自仿射===
===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|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.}}
== 编者推荐 ==
论文解读：几何重整化揭示 多尺度人脑网络的自相似性 (swarma.org) —— 郑木华 https://campus.swarma.org/course/1906
分形的世界 (swarma.org) —— 狄增如 https://campus.swarma.org/course/760
路径 - 分形的学习路径 (swarma.org) —— 海芽 https://pattern.swarma.org/path?id=80
[[Category:分形]]
[[Category:缩放对称性]]
[[Category:Fractals]]
[[Category:同胚]]
[[Category:自我参照]]
[[Category:Scaling symmetries]]
[[Category:Homeomorphisms]]
[[Category:Self-reference]]