更改

图示为一个自仿射分形，其豪斯多夫维数为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]].

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]].

比自相似性更一般的概念是自仿射性。

比自相似性更一般的概念是自仿射性。

==Examples 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|Self-similarity in the [[Mandelbrot set]] shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)

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技术，包括康托集和谢尔平斯基三角形。[[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，包括康托集和谢尔宾斯基三角形。[[File:RepeatedBarycentricSubdivision.png|thumb|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。]]

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 ===

 

=== In [[cybernetics]] 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.

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。

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。

=== In nature 自然界中 ===

=== 自然界中 ===

[[File:Flickr - cyclonebill - Romanesco.jpg|thumb|right|200px|Close-up of a [[Romanesco broccoli]].

[[File:Flickr - cyclonebill - Romanesco.jpg|thumb|right|200px|Close-up of a [[Romanesco broccoli]].

罗马花椰菜的特写镜头。]]

罗马花椰菜的特写镜头。]]

{{further|Patterns in nature}}

{{further|Patterns in nature}}

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。

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 ===

 

 

=== In music 音乐世界 ===

* Strict [[canon (music)|canons]] display various types and amounts of self-similarity, as do sections of [[fugue (music)|fugues]].

* Strict [[canon (music)|canons]] display various types and amounts of self-similarity, as do sections of [[fugue (music)|fugues]].

* 丹麦作曲家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。

* 丹麦作曲家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>{{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>{{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>

* 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。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。

* 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" />

 

 

==See also==

==See also==

* Droste effect

 

* Golden ratio

* Long-range dependency

 

* Non-well-founded set theory

* Recursion

 

* Self-dissimilarity

* [[Droste effect]]

* Self-reference

 

* Self-replication

* [[Golden ratio]]

* Self-Similarity of Network Data Analysis

 

* Teragon

* [[Long-range dependency]]

* Tessellation

 

* Tweedie distributions

* [[Non-well-founded set theory]]

* Zipf's law

 

* [[Recursion]]   

 

* [[Self-dissimilarity]]

 

* [[Self-reference]]

 

* [[Self-replication]]

 

* [[Self-Similarity of Network Data Analysis]]

 

* [[Teragon]]

 

* [[Tessellation]]

 

* [[Tweedie distributions]]

 

* [[Zipf's law]]}}

 

 

==References==

==References==