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第1行： − 此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。+ − − {{Use dmy dates|date=April 2017}} 第9行：
第7行： − Notoc+ − − [[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.]] − − A [[Koch curve has an infinitely repeating self-similarity when it is magnified.]] − − [[[科赫曲线被放大后有无限重复的自相似性]] − − − − Standard (trivial) self-similarity. − − − −
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此词条由彩云小译翻译，栗子CUGB人工整理和审校。
{{short description|The whole of an object being mathematically similar to part of itself}}
{{short description|The whole of an object being mathematically similar to part of itself}}
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[[File:Standard self-similarity.png|thumb|300px|Standard (trivial) self-similarity.<ref name=":0">Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. {{ISBN|978-0716711865}}.</ref>标准 (平凡) 自相似性。<ref name=":0" />]]
[[File:Standard self-similarity.png|thumb|300px|Standard (trivial) self-similarity.<ref>Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. {{ISBN|978-0716711865}}.</ref>]]
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.
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.