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标准(平凡)自相似性。

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.

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。

 

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。

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>{{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.  

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。

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>This vocabulary was introduced by Benoit Mandelbrot in 1964.

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>

 

{{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|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>}}

{{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==