Editing Self-similarity (section)

{{Short description|Whole of an object being mathematically similar to part of itself}}
{{Use dmy dates|date=April 2017}}
[[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A [[Koch snowflake]] has an infinitely repeating self-similarity when it is magnified.]] 
[[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>]]

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 | s2cid=15662830 | url=http://ena.lp.edu.ua:8080/handle/ntb/52473 | access-date=12 November 2020 | archive-date=19 October 2021 | archive-url=https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473 | url-status=dead }} [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.

[[Heinz-Otto Peitgen|Peitgen]] ''et al.'' explain the concept as such:

{{Quote|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.<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. {{ISBN|0-387-97346-X}} and {{ISBN|3-540-97346-X}}.</ref>}}Since mathematically, a fractal may show self-similarity under arbitrary magnification, it is impossible to recreate this physically. Peitgen ''et al.'' suggest studying self-similarity using approximations:{{Quote|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.<ref name="Classroom">Peitgen, et al (1991), p.2-3.</ref>}}

This vocabulary was introduced by [[Benoit Mandelbrot]] in 1964.<ref>Comment j'ai découvert les fractales, Interview de [[Benoit Mandelbrot]], La Recherche https://www.larecherche.fr/math%C3%A9matiques-histoire-des-sciences/%C2%AB-comment-jai-d%C3%A9couvert-les-fractales-%C2%BB</ref>