自相似

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此词条由栗子CUGB整理和审校。



A Koch curve has an infinitely repeating self-similarity when it is magnified. 当无限放大科赫曲线时,它会展示出无限重复的自相似性。
Standard (trivial) self-similarity.[1]标准(平凡)自相似性。[1]

This vocabulary was introduced by Benoit Mandelbrot in 1964. 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.[2] 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.

自相似一词由Benoit Mandelbrot 本华·曼德勃罗与1964年引入。[3](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。[2] 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]\displaystyle{ f(x,t) }[/math] measured at different times are different but the corresponding dimensionless quantity at given value of [math]\displaystyle{ x/t^z }[/math] remain invariant. It happens if the quantity [math]\displaystyle{ f(x,t) }[/math] exhibits dynamic scaling. The idea is just an extension of the idea of similarity of two triangles.[4][5][6] 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]\displaystyle{ f(x,t) }[/math] measured at different times are different but the corresponding dimensionless quantity at given value of [math]\displaystyle{ x/t^z }[/math] remain invariant. It happens if the quantity [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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。[4][5][6] 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:

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.

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。[7]

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:

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.

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。[8]

自仿射性

! -- 自我关联直接重定向到这里 --

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 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。

定义

A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms [math]\displaystyle{ \{ 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]\displaystyle{ \{ f_s : s\in S \} }[/math] for which

如果存在一个有限集 S[math]\displaystyle{ \{ 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]\displaystyle{ X=\bigcup_{s\in S} f_s(X) }[/math]

If [math]\displaystyle{ 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]\displaystyle{ \{ f_s : s\in S \} }[/math]. We call

假设有

[math]\displaystyle{ X\subset Y }[/math]

当且仅当X是Y的唯一非空子集,使得上式对

[math]\displaystyle{ \{ f_s : s\in S \} }[/math]

成立,则X是自相似的。而且我们称

[math]\displaystyle{ \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, 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.

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Dyadic 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 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.

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。

实例

Self-similarity in the Mandelbrot set shown by zooming in on the Feigenbaum point at (−1.401155189..., 0) 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。
An image of the Barnsley fern which exhibits 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 points.

The Mandelbrot set is also self-similar around Misiurewicz points.

Mandelbrot Set 曼德勃罗集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.[3] 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[3]。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.[9] Andrew Lo describes stock market log return self-similarity in econometrics.[10]

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[9]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[10]

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 Cantor Set 康托集Sierpinski Triangle 谢尔宾斯基三角

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。

控制论领域

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。

自然界中

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。

音乐世界

  • Strict canons display various types and amounts of self-similarity, as do sections of fugues.
  • 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。
  • A Shepard tone is self-similar in the frequency or wavelength domains.
  • 谢帕德音调在频率域或波长域是自相似的。
  • The 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 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.[11] In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.[12]
  • 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。[11] 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。[12]

另见

  • 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

参考文献

  1. 1.0 1.1 Mandelbrot, Benoit B. (1982). The Fractal Geometry of Nature, p.44. .
  2. 2.0 2.1 Mandelbrot, Benoit B. (5 May 1967). "How long is the coast of Britain? Statistical self-similarity and fractional dimension". Science. New Series. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158. PDF
  3. 3.0 3.1 3.2 Leland, W.E.; Taqqu, M.S.; et al. (January 1995). "On the self-similar nature of Ethernet traffic (extended version)" (PDF). IEEE/ACM Transactions on Networking. 2 (1): 1–15. doi:10.1109/90.282603.
  4. 4.0 4.1 Hassan M. K., Hassan M. Z., Pavel N. I. (2011). "Dynamic scaling, data-collapseand Self-similarity in Barabasi-Albert networks". J. Phys. A: Math. Theor. 44 (17): 175101. arXiv:1101.4730. Bibcode:2011JPhA...44q5101K. doi:10.1088/1751-8113/44/17/175101.CS1 maint: multiple names: authors list (link)
  5. 5.0 5.1 Hassan M. K., Hassan M. Z. (2009). "Emergence of fractal behavior in condensation-driven aggregation". Phys. Rev. E. 79 (2): 021406. arXiv:0901.2761. Bibcode:2009PhRvE..79b1406H. doi:10.1103/physreve.79.021406. PMID 19391746.
  6. 6.0 6.1 Dayeen F. R., Hassan M. K. (2016). "Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice". Chaos, Solitons & Fractals. 91: 228. arXiv:1409.7928. Bibcode:2016CSF....91..228D. doi:10.1016/j.chaos.2016.06.006.
  7. 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.
  8. Peitgen, et al (1991), p.2-3.
  9. 9.0 9.1 Peitgen, et al (1991), p.2-3.
  10. 10.0 10.1 Campbell, Lo and MacKinlay (1991) "Econometrics of Financial Markets ", Princeton University Press!
  11. 11.0 11.1 Foote, Jonathan (30 October 1999). Visualizing music and audio using self-similarity. pp. 77–80. doi:10.1145/319463.319472. ISBN 978-1581131512. http://musicweb.ucsd.edu/~sdubnov/CATbox/Reader/p77-foote.pdf. 
  12. 12.0 12.1 Pareyon, Gabriel (April 2011). On Musical Self-Similarity: Intersemiosis as Synecdoche and Analogy. International Semiotics Institute at Imatra; Semiotic Society of Finland. p. 240. ISBN 978-952-5431-32-2. Archived from the original on 8 February 2017. https://web.archive.org/web/20170208034152/https://tuhat.helsinki.fi/portal/files/15216101/Pareyon_Dissertation.pdf. Retrieved 30 July 2018.  (Also see Google Books)


External links


Self-affinity

  • Benoît B. Mandelbrot (2002). Gaussian Self-Affinity and Fractals: Globality, the Earth, 1/F Noise, and R/S. ISBN 978-0387989938. 


模板:Fractals

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


This page was moved from wikipedia:en:Self-similarity. Its edit history can be viewed at 自相似/edithistory