# 自相似 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。

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. 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.

A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity $\displaystyle{ f(x,t) }$ measured at different times are different but the corresponding dimensionless quantity at given value of $\displaystyle{ x/t^z }$ remain invariant. It happens if the quantity $\displaystyle{ f(x,t) }$ 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 $\displaystyle{ f(x,t) }$ measured at different times are different but the corresponding dimensionless quantity at given value of $\displaystyle{ x/t^z }$ remain invariant. It happens if the quantity $\displaystyle{ f(x,t) }$ 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.

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：

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.

## 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.

## 定义

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

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

$\displaystyle{ X=\bigcup_{s\in S} f_s(X) }$

If $\displaystyle{ X\subset Y }$, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for $\displaystyle{ \{ f_s : s\in S \} }$. We call

$\displaystyle{ X\subset Y }$

$\displaystyle{ \{ f_s : s\in S \} }$

$\displaystyle{ \mathfrak{L}=(X,S,\{ f_s : s\in S \} ) }$

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.

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.

## 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. 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.

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. Andrew Lo describes stock market log return self-similarity in econometrics.

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. 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

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.

### 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. In other words, music is self-similar under temporal translation, rather than (or in addition to) under scaling.
• 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。 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。

## 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

## 参考文献

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