更改

{{short description|The whole of an object being mathematically similar to part of itself}}

{{short description|The whole of an object being mathematically similar to part of itself}}

__NOTOC__

__NOTOC__

__NOTOC__

__NOTOC__

[[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.

[[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.

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。'''（标注：此处将后文的一句移过来以使行文看起来更连贯。）'''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。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。

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。

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>{{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.  

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.  

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。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。

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 ''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>

佩特根等曾这样解释这一概念：

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。

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>

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：

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。

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>

 

 

{{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 自仿射性==

==自仿射性==

<!--[[Self-affinity]] redirects directly here.-->

<!--[[Self-affinity]] redirects directly here.-->

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 [[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，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。

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。

==Definition 定义==

==定义==

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 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

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

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进行索引，则紧致拓扑空间 ''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>

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 function|iterated]], resulting in an [[iterated function system]]. The composition of functions creates the algebraic structure of a [[monoid]]. When the set ''S'' has only two elements, the monoid is known as the [[dyadic monoid]]. The dyadic monoid can be visualized as an infinite [[binary tree]]; more generally, if the set ''S'' has ''p'' elements, then the monoid may be represented as a [[p-adic number|p-adic]] tree.

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

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.

The automorphisms of the dyadic monoid is the modular group; the automorphisms can be pictured as hyperbolic rotations of the binary tree.

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

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

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 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.<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.