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自相似 (查看源代码)

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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>

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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：

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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.

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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。

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'''Stafford Beer 斯塔福德 · 比尔'''的'''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 ===

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* The [[Denmark|Danish]] [[composer]] [[Per Nørgård]] has made use of a self-similar [[integer sequence]] named the 'infinity series' in much of his music.

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* ~~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。~~

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* 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.<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>

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