Editing Mandelbrot set (section)

{{Short description|Fractal named after mathematician Benoit Mandelbrot}}
{{More citations needed|date=June 2024}}
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[[File:Mandel zoom 00 mandelbrot set.jpg|upright=1.35|thumb|The Mandelbrot set within a continuously colored environment|alt=]]<!-- The sequence \, is inserted in MATH items to ensure consistency of representation.
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The '''Mandelbrot set''' ({{IPAc-en|ˈ|m|æ|n|d|əl|b|r|oʊ|t|,_|-|b|r|ɒ|t}})<ref>{{Cite encyclopedia |url=http://www.lexico.com/definition/Mandelbrot+set |archive-url=https://web.archive.org/web/20220131051320/https://www.lexico.com/definition/mandelbrot_set?s=t |url-status=dead |archive-date=2022-01-31 |title=Mandelbrot set |dictionary=[[Lexico]] UK English Dictionary |publisher=[[Oxford University Press]]}}</ref><ref>{{cite Merriam-Webster|Mandelbrot set|access-date=2022-01-30}}</ref> is a two-dimensional [[set (mathematics)|set]] that is defined in the [[complex plane]] as the [[complex number]]s <math>c</math> for which the function <math>f_c(z)=z^2+c</math> does not [[Stability theory|diverge]] to infinity when [[Iteration|iterated]] starting at <math>z=0</math>, i.e., for which the sequence <math>f_c(0)</math>, <math>f_c(f_c(0))</math>, etc., remains bounded in [[absolute value]].<ref>{{Cite book |last1=Cooper |first1=S. B. |url=https://books.google.com/books?id=3yqpmHn9zAEC |title=New Computational Paradigms: Changing Conceptions of What is Computable |last2=Löwe |first2=Benedikt |last3=Sorbi |first3=Andrea |date=2007-11-28 |publisher=Springer Science & Business Media |isbn=978-0-387-68546-5 |pages=450 |language=en}}</ref>

This set was first defined and drawn by [[Robert W. Brooks]] and Peter Matelski in 1978, as part of a study of [[Kleinian group]]s.<ref name=":0" /> Afterwards, in 1980, [[Benoit Mandelbrot]] obtained high-quality visualizations of the set while working at [[IBM]]'s [[Thomas J. Watson Research Center]] in [[Yorktown Heights, New York]].<ref>{{Cite book |last=Nakos |first=George |url=https://books.google.com/books?id=OAoFEQAAQBAJ |title=Elementary Linear Algebra with Applications: MATLAB®, Mathematica® and MaplesoftTM |date=2024-05-20 |publisher=Walter de Gruyter GmbH & Co KG |isbn=978-3-11-133185-0 |pages=322 |language=en}}</ref>
[[File:Mandelbrot sequence new.gif|thumb|Zooming into the Mandelbrot set's so-called ‘Seahorse Valley’, with high iteration.]]

Images of the Mandelbrot set exhibit an infinitely complicated [[Boundary (topology)|boundary]] that reveals progressively ever-finer [[Recursion|recursive]] detail at increasing magnifications;<ref>{{Cite book |last=Addison |first=Paul S. |url=https://books.google.com/books?id=l2E4ciBQ9qEC |title=Fractals and Chaos: An illustrated course |date=1997-01-01 |publisher=CRC Press |isbn=978-0-8493-8443-1 |pages=110 |language=en}}</ref><ref>{{Cite book |last=Briggs |first=John |url=https://books.google.com/books?id=i5fLgAtUVucC |title=Fractals: The Patterns of Chaos : a New Aesthetic of Art, Science, and Nature |date=1992 |publisher=Simon and Schuster |isbn=978-0-671-74217-1 |pages=77 |language=en}}</ref> mathematically, the boundary of the Mandelbrot set is a ''[[fractal curve]]''.<ref>{{Cite book |last=Hewson |first=Stephen Fletcher |url=https://books.google.com/books?id=iqrEX8t-Nh8C |title=A Mathematical Bridge: An Intuitive Journey in Higher Mathematics |date=2009 |publisher=World Scientific |isbn=978-981-283-407-2 |pages=155 |language=en}}</ref> The "style" of this recursive detail depends on the region of the set boundary being examined.<ref>{{Cite book |last1=Peitgen |first1=Heinz-Otto |url=https://books.google.com/books?id=aIzsCAAAQBAJ |title=The Beauty of Fractals: Images of Complex Dynamical Systems |last2=Richter |first2=Peter H. |date=2013-12-01 |publisher=Springer Science & Business Media |isbn=978-3-642-61717-1 |pages=166 |language=en |quote="the Mandelbrot set is very diverse in its different regions"}}</ref> Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point <math>c</math>, whether the sequence <math>f_c(0), f_c(f_c(0)),\dotsc</math> [[Sequence#Bounded|goes to infinity]].<ref name=":4">{{Cite book |last=Hunt |first=John |url=https://books.google.com/books?id=O3baEAAAQBAJ |title=Advanced Guide to Python 3 Programming |date=2023-10-01 |publisher=Springer Nature |isbn=978-3-031-40336-1 |pages=117 |language=en}}</ref>{{close paraphrasing inline|date=March 2025}} Treating the [[Real numbers|real]] and [[Imaginary number|imaginary part]]s of <math>c</math> as [[image coordinate]]s on the [[complex plane]], pixels may then be colored according to how soon the sequence <math>|f_c(0)|, |f_c(f_c(0))|,\dotsc</math> crosses an arbitrarily chosen threshold (the threshold must be at least 2, as −2 is the complex number with the largest magnitude within the set, but otherwise the threshold is arbitrary).<ref name=":4" />{{close paraphrasing inline|date=March 2025}} If <math>c</math> is held constant and the initial value of <math>z</math> is varied instead, the corresponding [[Julia set]] for the point <math>c</math> is obtained.<ref>{{Cite web |last=Campuzano |first=Juan Carlos Ponce |date=20 November 2020 |title=Complex Analysis |url=https://complex-analysis.com/content/mandelbrot_set.html |url-status=live |archive-url=https://web.archive.org/web/20241016071311/https://complex-analysis.com/content/mandelbrot_set.html |archive-date=16 October 2024 |access-date=5 March 2025 |website=Complex Analysis — The Mandelbrot Set}}</ref>

The Mandelbrot set is well-known,<ref>{{Cite book |last1=Oberguggenberger |first1=Michael |url=https://books.google.com/books?id=js10DwAAQBAJ |title=Analysis for Computer Scientists: Foundations, Methods, and Algorithms |last2=Ostermann |first2=Alexander |date=2018-10-24 |publisher=Springer |isbn=978-3-319-91155-7 |pages=131 |language=en}}</ref> even outside mathematics,<ref>{{Cite web |title=Mandelbrot Set |url=https://cometcloud.sci.utah.edu/index.php/apps/mandelbrot-set |access-date=2025-03-22 |website=cometcloud.sci.utah.edu}}</ref> for how it exhibits complex fractal structures when visualized and magnified, despite having a relatively simple definition.<ref>{{Cite book |last1=Peitgen |first1=Heinz-Otto |url=https://books.google.com/books?id=GvnxBwAAQBAJ |title=Fractals for the Classroom: Part Two: Complex Systems and Mandelbrot Set |last2=Jürgens |first2=Hartmut |last3=Saupe |first3=Dietmar |date=2012-12-06 |publisher=Springer Science & Business Media |isbn=978-1-4612-4406-6 |pages=415 |language=en}}</ref><ref>{{Cite book |last1=Gulick |first1=Denny |url=https://books.google.com/books?id=k90BEQAAQBAJ |title=Encounters with Chaos and Fractals |last2=Ford |first2=Jeff |date=2024-05-10 |publisher=CRC Press |isbn=978-1-003-83578-3 |pages=§7.2 |language=en}}</ref><ref>{{Cite book |last1=Bialynicki-Birula |first1=Iwo |url=https://books.google.com/books?id=sc0TDAAAQBAJ |title=Modeling Reality: How Computers Mirror Life |last2=Bialynicka-Birula |first2=Iwona |date=2004-10-21 |publisher=OUP Oxford |isbn=978-0-19-853100-5 |pages=80 |language=en}}</ref>