Editing Mandelbrot set

{{Short description|Fractal named after mathematician Benoit Mandelbrot}}
{{More citations needed|date=June 2024}}
{{Use dmy dates|date=September 2021}}
[[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>

==History==
[[File:Mandel.png|thumb|upright=1.35|The first published picture of the Mandelbrot set, by [[Robert W. Brooks]] and Peter Matelski in 1978]]
The Mandelbrot set has its origin in [[complex dynamics]], a field first investigated by the [[French mathematicians]] [[Pierre Fatou]] and [[Gaston Julia]] at the beginning of the 20th century. The fractal was first defined and drawn in 1978 by [[Robert W. Brooks]] and Peter Matelski as part of a study of [[Kleinian group]]s.<ref name=":0">Robert Brooks and Peter Matelski, ''The dynamics of 2-generator subgroups of PSL(2,C)'', in {{cite book|url=https://abel.math.harvard.edu/archive/118r_spring_05/docs/brooksmatelski.pdf|title=Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference|author=Irwin Kra|date=1981|publisher=Princeton University Press|others=[[Bernard Maskit]]|isbn=0-691-08267-7|editor=Irwin Kra|access-date=1 July 2019|archive-url=https://web.archive.org/web/20190728201429/http://www.math.harvard.edu/archive/118r_spring_05/docs/brooksmatelski.pdf|archive-date=28 July 2019|url-status=dead}}</ref> On 1 March 1980, at [[IBM]]'s [[Thomas J. Watson Research Center]] in [[Yorktown Heights, New York|Yorktown Heights]], [[New York (state)|New York]], [[Benoit Mandelbrot]] first visualized the set.<ref name="bf">{{cite journal |url=http://sprott.physics.wisc.edu/pubs/paper311.pdf |title=Biophilic Fractals and the Visual Journey of Organic Screen-savers |author=R.P. Taylor & J.C. Sprott |access-date=1 January 2009 |year=2008 |journal=Nonlinear Dynamics, Psychology, and Life Sciences |volume=12 |issue=1 |pages=117–129 |publisher=Society for Chaos Theory in Psychology & Life Sciences |pmid=18157930 }}</ref>

Mandelbrot studied the [[parameter space]] of [[quadratic polynomial]]s in an article that appeared in 1980.<ref>{{cite journal |first=Benoit |last=Mandelbrot |title=Fractal aspects of the iteration of <math>z\mapsto\lambda z(1-z)</math> for complex <math>\lambda, z</math> |journal=Annals of the New York Academy of Sciences |volume=357 |issue=1 |pages=249–259 |year=1980 |doi=10.1111/j.1749-6632.1980.tb29690.x |s2cid=85237669 }}</ref> The mathematical study of the Mandelbrot set really began with work by the mathematicians [[Adrien Douady]] and [[John H. Hubbard]] (1985),<ref name="John H. Hubbard 1985">Adrien Douady and John H. Hubbard, ''Etude dynamique des polynômes complexes'', Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)</ref> who established many of its fundamental properties and named the set in honor of Mandelbrot for his influential work in [[fractal geometry]].

The mathematicians [[Heinz-Otto Peitgen]] and [[Peter Richter]] became well known for promoting the set with photographs, books (1986),<ref>{{cite book |title=The Beauty of Fractals |last=Peitgen |first=Heinz-Otto |author2=Richter Peter |year=1986 |publisher=Springer-Verlag |location=Heidelberg |isbn=0-387-15851-0 |title-link=The Beauty of Fractals }}</ref> and an internationally touring exhibit of the German [[Goethe-Institut]] (1985).<ref>[[Frontiers of Chaos]], Exhibition of the Goethe-Institut by H.O. Peitgen, P. Richter, H. Jürgens, M. Prüfer, D.Saupe. Since 1985 shown in over 40 countries.</ref><ref>{{cite book |title=Chaos: Making a New Science |last=Gleick |first=James |year=1987 |publisher=Cardinal |location=London |pages=229 |title-link=Chaos: Making a New Science }}</ref>

The cover article of the August 1985 ''[[Scientific American]]'' introduced the [[algorithm]] for computing the Mandelbrot set. The cover was created by Peitgen, Richter and [[Dietmar Saupe|Saupe]] at the [[University of Bremen]].<ref>{{Cite journal|date=August 1985|title=Exploring The Mandelbrot Set|url=https://www.jstor.org/stable/24967754|journal=Scientific American|volume=253|issue=2|pages=4|jstor=24967754}}</ref> The Mandelbrot set became prominent in the mid-1980s as a [[Demo (computer programming)|computer-graphics demo]], when [[personal computer]]s became powerful enough to plot and display the set in high resolution.<ref>{{cite magazine |last=Pountain |first=Dick |date=September 1986 |title= Turbocharging Mandelbrot |url=https://archive.org/stream/byte-magazine-1986-09/1986_09_BYTE_11-09_The_68000_Family#page/n370/mode/1up |magazine= [[Byte (magazine)|Byte]] |access-date=11 November 2015 }}</ref>

The work of Douady and Hubbard occurred during an increase in interest in [[complex dynamics]] and [[abstract mathematics]],<ref name=rees>{{cite journal
 | last1 = Rees
 | first1 = Mary
    | author-link = Mary Rees
 | date = January 2016
    | title = One hundred years of complex dynamics
 | journal = [[Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences]]
 | volume =  472
 | issue = 2185
 | pages =
 | doi = 10.1098/rspa.2015.0453
 | pmid = 26997888
 | pmc = 4786033
 | bibcode = 2016RSPSA.47250453R
 }}</ref> and the topological and geometric study of the Mandelbrot set remains a key topic in the field of complex dynamics.<ref>{{Cite book |last=Schleicher |first=Dierk |url=https://books.google.com/books?id=Ek3rBgAAQBAJ |title=Complex Dynamics: Families and Friends |date=2009-11-03 |publisher=CRC Press |isbn=978-1-4398-6542-2 |pages=xii |language=en}}</ref>

==Formal definition==
The Mandelbrot set is the [[uncountable set]] of values of ''c'' in the [[complex plane]] for which the [[Orbit (dynamics)|orbit]] of the [[Complex quadratic polynomial#Critical point|critical point]] <math display="inline">z = 0</math> under [[Iterated function|iteration]] of the [[quadratic map]]

:<math>z \mapsto z^2 + c</math> <ref>{{Cite web |last=Weisstein |first=Eric W. |title=Mandelbrot Set |url=https://mathworld.wolfram.com/ |access-date=2024-01-24 |website=mathworld.wolfram.com |language=en}}</ref>

remains [[Bounded sequence|bounded]].<ref>{{cite web|url=http://math.bu.edu/DYSYS/explorer/def.html|title=Mandelbrot Set Explorer: Mathematical Glossary|access-date=7 October 2007}}</ref> Thus, a [[complex number]] ''c'' is a member of the Mandelbrot set if, when starting with <math>z_0 = 0</math> and applying the iteration repeatedly, the [[absolute value]] of <math>z_n</math> remains bounded for all <math>n > 0</math>.

For example, for ''c'' = 1, the [[sequence]] is 0, 1, 2, 5, 26, ..., which tends to [[infinity]], so 1 is not an element of the Mandelbrot set. On the other hand, for <math>c=-1</math>, the sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set.

The Mandelbrot set can also be defined as the [[connectedness locus]] of the family of [[Quadratic equation|quadratic]] [[polynomial]]s <math>f(z) = z^2 + c</math>, the subset of the space of parameters <math>c</math> for which the [[Julia set]] of the corresponding polynomial forms a [[connected set]].<ref>{{Citation |last=Tiozzo |first=Giulio |title=Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set |date=2013-05-15 |arxiv=1305.3542 }}</ref> In the same way, the [[boundary (topology)|boundary]] of the Mandelbrot set can be defined as the [[bifurcation locus]] of this quadratic family, the subset of parameters near which the dynamic behavior of the polynomial (when it is [[Iterated function|iterated]] repeatedly) changes drastically.

==Basic properties==
The Mandelbrot set is a [[compact set]], since it is [[closed set|closed]] and contained in the [[closed disk]] of radius 2 centred on [[Origin (mathematics)|zero]]. A point <math>c</math> belongs to the Mandelbrot set if and only if <math>|z_n|\leq 2</math> for all <math>n\geq 0</math>. In other words, the [[absolute value]] of <math>z_n</math> must remain at or below 2 for <math>c</math> to be in the Mandelbrot set, <math>M</math>, and if that absolute value exceeds 2, the sequence will escape to infinity. Since <math>c=z_1</math>, it follows that <math>|c|\leq 2</math>, establishing that <math>c</math> will always be in the closed disk of radius 2 around the origin.<ref>{{cite web|url=https://mrob.com/pub/muency/escaperadius.html|title=Escape Radius|access-date=17 January 2024}}</ref>

[[File:Verhulst-Mandelbrot-Bifurcation.jpg|thumb|Correspondence between the Mandelbrot set and the [[bifurcation diagram]] of the [[Complex quadratic polynomial|quadratic map]]]]
[[File:Logistic Map Bifurcations Underneath Mandelbrot Set.gif|thumb|With <math>z_{n}</math> iterates plotted on the vertical axis, the Mandelbrot set can be seen to bifurcate at the period-2<sup>k</sup> components.]]
The [[intersection (set theory)|intersection]] of <math>M</math> with the real axis is the interval <math>\left[-2,\frac{1}{4}\right]</math>. The parameters along this interval can be put in [[one-to-one correspondence]] with those of the real [[logistic map|logistic family]],
:<math>x_{n+1} = r x_n(1-x_n),\quad r\in[1,4].</math>
The correspondence is given by

:<math>r = 1+\sqrt{1- 4 c},
\quad
c = \frac{r}{2}\left(1-\frac{r}{2}\right),
\quad
z_n = r\left(\frac{1}{2} - x_n\right).</math>

This gives a correspondence between the entire [[parameter space]] of the logistic family and that of the Mandelbrot set.<ref>{{Cite web |last=thatsmaths |date=2023-12-07 |title=The Logistic Map is hiding in the Mandelbrot Set |url=https://thatsmaths.com/2023/12/07/the-logistic-map-is-hiding-in-the-mandelbrot-set/ |access-date=2024-02-18 |website=ThatsMaths |language=en}}</ref>

Douady and Hubbard showed that the Mandelbrot set is [[connected space|connected]]. They constructed an explicit [[holomorphic function|conformal isomorphism]] between the complement of the Mandelbrot set and the complement of the [[closed unit disk]]. Mandelbrot had originally conjectured that the Mandelbrot set is [[Disconnected (topology)|disconnected]]. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of <math>M</math>. Upon further experiments, he revised his conjecture, deciding that <math>M</math> should be connected. A [[Topology|topological]] proof of the connectedness was discovered in 2001 by [[Jeremy Kahn]].<ref>{{cite web|url=http://www.math.brown.edu/~kahn/mconn.pdf|title=The Mandelbrot Set is Connected: a Topological Proof|last=Kahn|first=Jeremy|date=8 August 2001}}</ref>
[[File:Wakes near the period 1 continent in the Mandelbrot set.png|thumbnail|right|External rays of wakes near the period 1 continent in the Mandelbrot set]]
The dynamical formula for the [[uniformization theorem|uniformisation]] of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of <math>M</math>, gives rise to [[external ray]]s of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the [[Jean-Christophe Yoccoz#Mathematical work|Yoccoz parapuzzle]].<ref>''The Mandelbrot set, theme and variations''. Tan, Lei. Cambridge University Press, 2000. {{isbn|978-0-521-77476-5}}. Section 2.1, "Yoccoz para-puzzles", [https://books.google.com/books?id=-a_DsYXquVkC&pg=PA121 p.&nbsp;121]</ref>

The [[boundary (topology)|boundary]] of the Mandelbrot set is the [[bifurcation locus]] of the family of quadratic polynomials. In other words, the boundary of the Mandelbrot set is the set of all parameters <math>c</math> for which the dynamics of the quadratic map <math>z_n=z_{n-1}^2+c</math> exhibits sensitive dependence on <math>c,</math> i.e. changes abruptly under arbitrarily small changes of <math>c.</math> It can be constructed as the limit set of a sequence of [[algebraic curves|plane algebraic curves]], the ''Mandelbrot curves'', of the general type known as [[polynomial lemniscate]]s. The Mandelbrot curves are defined by setting <math>p_0=z,\ p_{n+1}=p_n^2+z</math>, and then interpreting the set of points <math>|p_n(z)| = 2</math> in the complex plane as a curve in the real [[Cartesian coordinate system|Cartesian plane]] of degree <math>2^{n+1}</math>in ''x'' and ''y''.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Mandelbrot Set Lemniscate |url=https://mathworld.wolfram.com/MandelbrotSetLemniscate.html |access-date=2023-07-17 |website=Wolfram Mathworld |language=en}}</ref> Each curve <math>n > 0</math> is the mapping of an initial circle of radius 2 under <math>p_n</math>. These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.

==Other properties==

===Main cardioid and period bulbs===
<!--[[Douady rabbit]] links directly here.-->[[File:Mandelbrot Set – Periodicities coloured.png|right|thumb|Periods of hyperbolic components]]

The ''main [[cardioid]]'' is the period 1 continent.<ref>{{Cite book |last1=Brucks |first1=Karen M. |url=https://books.google.com/books?id=p-amwZp0R-0C |title=Topics from One-Dimensional Dynamics |last2=Bruin |first2=Henk |date=2004-06-28 |publisher=Cambridge University Press |isbn=978-0-521-54766-6 |pages=264 |language=en}}</ref> It is the region of parameters <math>c</math> for which the map <math>f_c(z) = z^2 + c</math> has an [[Periodic points of complex quadratic mappings|attracting fixed point]].<ref>{{Cite book |last=Devaney |first=Robert |url=https://books.google.com/books?id=YEIPEAAAQBAJ |title=An Introduction To Chaotic Dynamical Systems |date=2018-03-09 |publisher=CRC Press |isbn=978-0-429-97085-6 |pages=147 |language=en}}</ref> It consists of all parameters of the form <math> c(\mu) := \frac\mu2\left(1-\frac\mu2\right)</math> for some <math>\mu</math> in the [[open unit disk]].<ref name=":5">{{Cite book |last1=Ivancevic |first1=Vladimir G. |url=https://books.google.com/books?id=mbtCAAAAQBAJ |title=High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction |last2=Ivancevic |first2=Tijana T. |date=2007-02-06 |publisher=Springer Science & Business Media |isbn=978-1-4020-5456-3 |pages=492–493 |language=en}}</ref>{{close paraphrasing inline|date=March 2025}}

To the left of the main cardioid, attached to it at the point <math>c=-3/4</math>, a circular bulb, the ''period-2 bulb'' is visible.<ref name=":5" />{{close paraphrasing inline|date=March 2025}} The bulb consists of <math>c</math> for which <math>f_c</math> has an [[Periodic points of complex quadratic mappings|attracting cycle of period 2]]. It is the filled circle of radius 1/4 centered around −1.<ref name=":5" />{{close paraphrasing inline|date=March 2025}}

[[File:Animated cycle.gif|left|thumb|Attracting cycle in 2/5-bulb plotted over [[Julia set]] (animation)]]
More generally, for every positive integer <math>q>2</math>, there are <math>\phi(q)</math> circular bulbs tangent to the main cardioid called ''period-q bulbs'' (where <math>\phi</math> denotes the [[Euler's totient function|Euler phi function]]), which consist of parameters <math>c</math> for which <math>f_c</math> has an attracting cycle of period <math>q</math>.{{Citation needed|date=March 2025}} More specifically, for each primitive <math>q</math>th root of unity <math>r=e^{2\pi i\frac{p}{q}}</math> (where <math>0<\frac{p}{q}<1</math>), there is one period-q bulb called the <math>\frac{p}{q}</math> bulb, which is tangent to the main cardioid at the parameter <math> c_{\frac{p}{q}} := c(r) = \frac{r}2\left(1-\frac{r}2\right),</math> and which contains parameters with <math>q</math>-cycles having combinatorial rotation number <math>\frac{p}{q}</math>.<ref>{{Cite book |last1=Devaney |first1=Robert L. |url=https://books.google.com/books?id=4XrHCQAAQBAJ |title=Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets: The Mathematics Behind the Mandelbrot and Julia Sets |last2=Branner |first2=Bodil |date=1994 |publisher=American Mathematical Soc. |isbn=978-0-8218-0290-8 |pages=18–19 |language=en}}</ref> More precisely, the <math>q</math> periodic [[Classification of Fatou components|Fatou components]] containing the attracting cycle all touch at a common point (commonly called the ''<math>\alpha</math>-fixed point''). If we label these components <math>U_0,\dots,U_{q-1}</math> in counterclockwise orientation, then <math>f_c</math> maps the component <math>U_j</math> to the component <math>U_{j+p\,(\operatorname{mod} q)}</math>.<ref name=":5" />{{close paraphrasing inline|date=March 2025}}

[[File:Juliacycles1.png|right|thumb|Attracting cycles and [[Julia set]]s for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs]]

The change of behavior occurring at <math>c_{\frac{p}{q}}</math> is known as a [[bifurcation theory|bifurcation]]: the attracting fixed point "collides" with a repelling period-''q'' cycle. As we pass through the bifurcation parameter into the <math>\tfrac{p}{q}</math>-bulb, the attracting fixed point turns into a repelling fixed point (the <math>\alpha</math>-fixed point), and the period-''q'' cycle becomes attracting.<ref name=":5" />{{close paraphrasing inline|date=March 2025}}

===Hyperbolic components===
Bulbs that are interior components of the Mandelbrot set in which the maps <math>f_c</math> have an attracting periodic cycle are called ''hyperbolic components''.<ref>{{cite thesis |last=Redona |first=Jeffrey Francis |title=The Mandelbrot set |year=1996 |type=Masters of Arts in Mathematics
|publisher=Theses Digitization Project |url=https://scholarworks.lib.csusb.edu/etd-project/1166}}</ref>

It is conjectured that these are the ''only'' [[Interior (topology)|interior regions]] of <math>M</math> and that they are [[dense set|dense]] in <math>M</math>. This problem, known as ''density of hyperbolicity'', is one of the most important open problems in [[complex dynamics]].<ref>{{cite arXiv|eprint=1709.09869 |author1=Anna Miriam Benini |title=A survey on MLC, Rigidity and related topics |year=2017 |class=math.DS }}</ref> Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components.<ref>{{cite book |title=Exploring the Mandelbrot set. The Orsay Notes |first1=Adrien |last1=Douady |first2=John H. |last2=Hubbard |page=12 }}</ref><ref>{{cite thesis |first=Wolf |last=Jung |year=2002 |title=Homeomorphisms on Edges of the Mandelbrot Set |type=Doctoral thesis |publisher=[[RWTH Aachen University]] |id={{URN|nbn|de:hbz:82-opus-3719}} }}</ref> For real quadratic polynomials, this question was proved in the 1990s independently by Lyubich and by Graczyk and Świątek. (Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in the [[Bifurcation diagram|Feigenbaum diagram]]. So this result states that such windows exist near every parameter in the diagram.)

Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. Such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).
[[File:Centers8.png|thumb|Centers of 983 hyperbolic components of the Mandelbrot set.]]
Each of the hyperbolic components has a ''center'', which is a point ''c'' such that the inner Fatou domain for <math>f_c(z)</math> has a super-attracting cycle—that is, that the attraction is infinite. This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. Therefore, <math>f_c^n(0) = 0</math> for some ''n''. If we call this polynomial <math>Q^{n}(c)</math> (letting it depend on ''c'' instead of ''z''), we have that <math>Q^{n+1}(c) = Q^{n}(c)^{2} + c</math> and that the degree of <math>Q^{n}(c)</math> is <math>2^{n-1}</math>. Therefore, constructing the centers of the hyperbolic components is possible by successively solving the equations <math>Q^{n}(c) = 0, n = 1, 2, 3, ...</math>.{{Citation needed|date=July 2023}} The number of new centers produced in each step is given by Sloane's {{oeis|A000740}}.

===Local connectivity===
It is conjectured that the Mandelbrot set is [[locally connected]]. This conjecture is known as ''MLC'' (for ''Mandelbrot locally connected''). By the work of [[Adrien Douady]] and [[John H. Hubbard]], this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important ''hyperbolicity conjecture'' mentioned above.{{Citation needed|date=July 2023}}

The work of [[Jean-Christophe Yoccoz]] established local connectivity of the Mandelbrot set at all finitely [[Renormalization|renormalizable]] parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies.<ref name="yoccoz">{{cite book
 | last = Hubbard | first = J. H.
 | contribution = Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz
 | contribution-url = https://pi.math.cornell.edu/~hubbard/Yoccoz.pdf
 | location = Houston, TX
 | mr = 1215974
 | pages = 467–511
 | publisher = Publish or Perish
 | title = Topological methods in modern mathematics (Stony Brook, NY, 1991)
 | year = 1993}}. Hubbard cites as his source a 1989 unpublished manuscript of Yoccoz.</ref> Since then, local connectivity has been proved at many other points of <math>M</math>, but the full conjecture is still open.

===Self-similarity===
[[File:Self-Similarity-Zoom.gif|right|thumb|[[Self-similarity]] in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-''x'' direction. The display center pans left from the fifth to the seventh round feature (−1.4002,&nbsp;0) to (−1.4011,&nbsp;0) while the view magnifies by a factor of 21.78 to approximate the square of the [[Feigenbaum constants|Feigenbaum ratio]].]]
The Mandelbrot set is [[self-similar]] under magnification in the neighborhoods of the [[Misiurewicz point]]s. It is also conjectured to be self-similar around generalized [[Feigenbaum point]]s (e.g., −1.401155 or −0.1528&nbsp;+&nbsp;1.0397''i''), in the sense of converging to a limit set.<ref>{{cite journal | last1 = Lei | year = 1990 | title = Similarity between the Mandelbrot set and Julia Sets | url = http://projecteuclid.org/euclid.cmp/1104201823| journal = Communications in Mathematical Physics | volume = 134 | issue = 3| pages = 587–617 | doi=10.1007/bf02098448| bibcode = 1990CMaPh.134..587L| s2cid = 122439436 }}</ref><ref>{{cite book |author=J. Milnor |chapter=Self-Similarity and Hairiness in the Mandelbrot Set |editor=M. C. Tangora |location=New York |pages=211–257 |title=Computers in Geometry and Topology |url=https://books.google.com/books?id=wuVJAQAAIAAJ |year=1989|publisher=Taylor & Francis|isbn=9780824780319 }})</ref> The Mandelbrot set in general is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales. These copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set.<ref>{{Cite web |title=Mandelbrot Viewer |url=https://math.hws.edu/eck/js/mandelbrot/MB.html |access-date=2025-03-01 |website=math.hws.edu}}</ref>

===Further results===
The [[Hausdorff dimension]] of the [[boundary (topology)|boundary]] of the Mandelbrot set equals 2 as determined by a result of [[Mitsuhiro Shishikura]].<ref name="shishikura"/> The fact that this is greater by a whole integer than its topological dimension, which is 1, reflects the extreme [[fractal]] nature of the Mandelbrot set boundary. Roughly speaking, Shishikura's result states that the Mandelbrot set boundary is so "wiggly" that it locally [[space-filling curve|fills space]] as efficiently as a two-dimensional planar region. Curves with Hausdorff dimension 2, despite being (topologically) 1-dimensional, are oftentimes capable of having nonzero area (more formally, a nonzero planar [[Lebesgue measure]]). Whether this is the case for the Mandelbrot set boundary is an unsolved problem.{{Citation needed|date=July 2023}}

It has been shown that the generalized Mandelbrot set in higher-dimensional hypercomplex number spaces (i.e. when the power <math>\alpha</math> of the iterated variable <math>z</math> tends to infinity) is convergent to the unit (<math>\alpha</math>−1)-sphere.<ref>{{cite journal|last1=Katunin|first1=Andrzej|last2=Fedio|first2=Kamil|title=On a Visualization of the Convergence of the Boundary of Generalized Mandelbrot Set to (n-1)-Sphere|url=https://reader.digitarium.pcss.pl/Content/295117/JAMCM_2015_1_6-Katunin_Fedio.pdf|access-date=18 May 2022|date=2015|journal=Journal of Applied Mathematics and Computational Mechanics|volume=14|issue=1|pages=63–69|doi=10.17512/jamcm.2015.1.06}}</ref>

In the [[Blum–Shub–Smale machine|Blum–Shub–Smale]] model of [[real computation]], the Mandelbrot set is not computable, but its complement is [[Recursively enumerable set|computably enumerable]]. Many simple objects (e.g., the graph of exponentiation) are also not computable in the BSS model. At present, it is unknown whether the Mandelbrot set is computable in models of real computation based on [[computable analysis]], which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.{{Citation needed|date=July 2023}}

===Relationship with Julia sets===
[[File:Julia Mandelbrot Relationship.png|thumb|A mosaic made by matching Julia sets to their values of c on the complex plane. The Mandelbrot set is a map of connected Julia sets.]]
As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the [[geometry]] of the Mandelbrot set at a given point and the structure of the corresponding [[Julia set]]. For instance, a value of c belongs to the Mandelbrot set if and only if the corresponding Julia set is connected. Thus, the Mandelbrot set may be seen as a map of the connected Julia sets.<ref>{{cite web |last=Sims |first=Karl |title=Understanding Julia and Mandelbrot Sets |url=https://www.karlsims.com/julia.html |website=karlsims.com |access-date=January 27, 2025}}</ref>{{Better source needed|date=January 2025}}

This principle is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proved that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has [[Hausdorff dimension]] two, and then transfers this information to the parameter plane.<ref name="shishikura">{{cite journal
 | last = Shishikura | first = Mitsuhiro
 | arxiv = math.DS/9201282
 | doi = 10.2307/121009
 | issue = 2
 | journal = Annals of Mathematics
 | mr = 1626737
 | pages = 225–267
 | series = Second Series
 | title = The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets
 | volume = 147
 | year = 1998| jstor = 121009
 | s2cid = 14847943
 }}.</ref> Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters.<ref name="yoccoz"/>

== Geometry ==<!--[[Douady rabbit]] links directly here.-->
For every rational number <math>\tfrac{p}{q}</math>, where ''p'' and ''q'' are [[coprime]], a hyperbolic component of period ''q'' bifurcates from the main cardioid at a point on the edge of the cardioid corresponding to an [[internal angle]] of <math>\tfrac{2\pi p}{q}</math>.<ref name="guild">{{cite web |title=Number Sequences in the Mandelbrot Set |url=https://www.youtube.com/watch?v=oNxPSP2tQEk | archive-url=https://ghostarchive.org/varchive/youtube/20211030/oNxPSP2tQEk| archive-date=2021-10-30|website=youtube.com |publisher=The Mathemagicians' Guild |date=4 June 2020}}{{cbignore}}</ref> The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the '''''p''/''q''-limb'''. Computer experiments suggest that the [[diameter]] of the limb tends to zero like <math>\tfrac{1}{q^2}</math>. The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like <math>\tfrac{1}{q}</math>.{{Citation needed|date=July 2023}}

A period-''q'' limb will have <math>q-1</math> "antennae" at the top of its limb. The period of a given bulb is determined by counting these antennas. The numerator of the rotation number, ''p'', is found by numbering each antenna counterclockwise from the limb from 1 to <math>q-1</math> and finding which antenna is the shortest.<ref name="guild" />

=== Pi in the Mandelbrot set ===
There are intriguing experiments in the Mandelbrot set that lead to the occurrence of the number <math>\pi</math>. For a parameter <math>c = -\tfrac{3}{4}+ i\varepsilon</math> with <math>\varepsilon>0</math>, verifying that <math>c</math> is not in the Mandelbrot set means iterating the sequence <math>z \mapsto z^2 + c</math> starting with <math>z=0</math>, until the sequence leaves the disk around <math>0</math> of any radius <math>R>2</math>. This is motivated by the (still open) question whether the vertical line at real part <math>-3/4</math> intersects the Mandelbrot set at points away from the real line. It turns out that the necessary number of iterations, multiplied by <math>\varepsilon</math>, converges to pi. For example, for ''<math>\varepsilon</math>'' = 0.0000001, and <math>R=2</math>, the number of iterations is 31415928 and the product is 3.1415928.<ref>{{cite book |first=Gary William |last=Flake |title=The Computational Beauty of Nature |year=1998 |page=125 |publisher=MIT Press |isbn=978-0-262-56127-3 }}</ref> This experiment was performed independently by many people in the early 1990s, if not before; for instance by David Boll.

Analogous observations have also been made at the parameters <math>c=-5/4</math> and <math>c=1/4</math> (with a necessary modification in the latter case). In 2001, Aaron Klebanoff published a (non-conceptual) proof for this phenomenon at <math>c=1/4</math><ref>{{cite journal |last=Klebanoff |first=Aaron D. |title=π in the Mandelbrot Set |journal=Fractals |volume=9 |issue=4 |pages=393–402 |year=2001 |doi=10.1142/S0218348X01000828 }}</ref>

In 2023, Paul Siewert developed, in his Bachelor thesis, a conceptual proof also for the value <math>c=1/4</math>, explaining why the number pi occurs (geometrically as half the circumference of the unit circle).<ref>Paul Siewert, Pi in the Mandelbrot set. Bachelor Thesis, Universität Göttingen, 2023</ref>

In 2025, the three high school students Thies Brockmöller, Oscar Scherz, and Nedim Srkalovic extended the theory and the conceptual proof to all the infinitely bifurcation points in the Mandelbrot set.<ref>{{cite arXiv | eprint=2505.07138 | last1=Brockmoeller | first1=Thies | last2=Scherz | first2=Oscar | last3=Srkalovic | first3=Nedim | title=Pi in the Mandelbrot set everywhere | date=2025 | class=math.DS }}</ref>

=== Fibonacci sequence in the Mandelbrot set ===
The Mandelbrot Set features a fundamental cardioid shape adorned with numerous bulbs directly attached to it.<ref name=":1">{{Cite journal |last=Devaney |first=Robert L. |date=April 1999 |title=The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence |url=http://dx.doi.org/10.2307/2589552 |journal=The American Mathematical Monthly |volume=106 |issue=4 |pages=289–302 |doi=10.2307/2589552 |jstor=2589552 |issn=0002-9890}}</ref> Understanding the arrangement of these bulbs requires a detailed examination of the Mandelbrot Set's boundary. As one zooms into specific portions with a geometric perspective, precise deducible information about the location within the boundary and the corresponding dynamical behavior for parameters drawn from associated bulbs emerges.<ref name=":2">{{Cite web |last=Devaney |first=Robert L. |date=January 7, 2019 |title=Illuminating the Mandelbrot set |url=https://math.bu.edu/people/bob/papers/mar-athan.pdf}}</ref>

The iteration of the quadratic polynomial <math>f_c(z) = z^2 + c</math>, where <math>c</math>&nbsp;is a parameter drawn from one of the bulbs attached to the main cardioid within the Mandelbrot Set, gives rise to maps featuring attracting cycles of a specified period <math>q</math>&nbsp;and a rotation number <math>p/q</math>. In this context, the attracting cycle of &nbsp;exhibits rotational motion around a central fixed point, completing an average of <math>p/q</math>&nbsp;revolutions at each iteration.<ref name=":2" /><ref>{{Cite web |last=Allaway |first=Emily |date=May 2016 |title=The Mandelbrot Set and the Farey Tree |url=https://sites.math.washington.edu/~morrow/336_16/2016papers/emily.pdf}}</ref>

The bulbs within the Mandelbrot Set are distinguishable by both their attracting cycles and the geometric features of their structure. Each bulb is characterized by an antenna attached to it, emanating from a junction point and displaying a certain number of spokes indicative of its period. For instance, the <math>2/5</math> bulb is identified by its attracting cycle with a rotation number of <math>2/5</math>. Its distinctive antenna-like structure comprises a junction point from which five spokes emanate. Among these spokes, called the principal spoke is directly attached to the <math>2/5</math> bulb, and the 'smallest' non-principal spoke is positioned approximately <math>2/5</math> of a turn counterclockwise from the principal spoke, providing a distinctive identification as a <math>2/5</math>-bulb.<ref name=":3">{{Cite web |last=Devaney |first=Robert L. |date=December 29, 1997 |title=The Mandelbrot Set and the Farey Tree |url=https://math.bu.edu/people/bob/papers/farey.pdf}}</ref> This raises the question: how does one discern which among these spokes is the 'smallest'?<ref name=":1" /><ref name=":3" /> In the theory of [[external ray]]s developed by [[Adrien Douady|Douady]] and [[John H. Hubbard|Hubbard]],<ref>{{Cite web |last1=Douady, A. |last2=Hubbard, J |date=1982 |title=Iteration des Polynomials Quadratiques Complexes |url=https://pi.math.cornell.edu/~hubbard/CR.pdf}}</ref> there are precisely two external rays landing at the root point of a satellite hyperbolic component of the Mandelbrot Set. Each of these rays possesses an external angle that undergoes doubling under the angle doubling map <math>\theta\mapsto</math> <math>2\theta</math>. According to this theorem, when two rays land at the same point, no other rays between them can intersect. Thus, the 'size' of this region is measured by determining the length of the arc between the two angles.<ref name=":2" />

If the root point of the main cardioid is the cusp at <math>c=1/4</math>, then the main cardioid is the <math>0/1</math>-bulb. The root point of any other bulb is just the point where this bulb is attached to the main cardioid. This prompts the inquiry: which is the largest bulb between the root points of the <math>0/1</math> and <math>1/2</math>-bulbs? It is clearly the <math>1/3</math>-bulb. And note that <math>1/3</math> is obtained from the previous two fractions by [[Farey sequence|Farey addition]], i.e., adding the numerators and adding the denominators

<math>\frac{0}{1}</math> <math>\oplus</math> <math>\frac{1}{2}</math><math>=</math><math>\frac{1}{3}</math>

Similarly, the largest bulb between the <math>1/3</math> and <math>1/2</math>-bulbs is the <math>2/5</math>-bulb, again given by Farey addition.

<math>\frac{1}{3}</math> <math>\oplus</math> <math>\frac{1}{2}</math><math>=</math><math>\frac{2}{5}</math>

The largest bulb between the <math>2/5</math> and <math>1/2</math>-bulb is the <math>3/7</math>-bulb, while the largest bulb between the <math>2/5</math> and <math>1/3</math>-bulbs is the <math>3/8</math>-bulb, and so on.<ref name=":2" /><ref>{{Cite web |title=The Mandelbrot Set Explorer Welcome Page |url=http://math.bu.edu/DYSYS/explorer/ |access-date=2024-02-17 |website=math.bu.edu}}</ref> The arrangement of bulbs within the Mandelbrot set follows a remarkable pattern governed by the [[Farey tree]], a structure encompassing all rationals between <math>0</math> and <math>1</math>. This ordering positions the bulbs along the boundary of the main cardioid precisely according to the [[rational number]]s in the [[unit interval]].<ref name=":3" />
[[File:Fibonacci sequence within the Mandelbrot set.png|left|thumb|Fibonacci sequence within the Mandelbrot set]]
Starting with the <math>1/3</math> bulb at the top and progressing towards the <math>1/2</math> circle, the sequence unfolds systematically: the largest bulb between <math>1/2</math> and <math>1/3</math> is <math>2/5</math>, between <math>1/3</math> and <math>2/5</math> is <math>3/8</math>, and so forth.<ref>{{Cite web |title=Maths Town |url=https://www.patreon.com/mathstown |access-date=2024-02-17 |website=Patreon}}</ref> Intriguingly, the denominators of the periods of circular bulbs at sequential scales in the Mandelbrot Set conform to the [[Fibonacci sequence|Fibonacci number sequence]], the sequence that is made by adding the previous two terms – 1, 2, 3, 5, 8, 13, 21...<ref>{{Cite journal |last1=Fang |first1=Fang |last2=Aschheim |first2=Raymond |last3=Irwin |first3=Klee |date=December 2019 |title=The Unexpected Fractal Signatures in Fibonacci Chains |journal=Fractal and Fractional |language=en |volume=3 |issue=4 |pages=49 |doi=10.3390/fractalfract3040049 |doi-access=free |issn=2504-3110|arxiv=1609.01159 }}</ref><ref>{{Cite web |title=7 The Fibonacci Sequence |url=https://math.bu.edu/DYSYS/FRACGEOM2/node7.html#SECTION00070000000000000000 |access-date=2024-02-17 |website=math.bu.edu}}</ref>

The Fibonacci sequence manifests in the number of spiral arms at a unique spot on the Mandelbrot set, mirrored both at the top and bottom. This distinctive location demands the highest number of iterations of &nbsp;for a detailed fractal visual, with intricate details repeating as one zooms in.<ref>{{Cite web |title=fibomandel angle 0.51 |url=https://www.desmos.com/calculator/oasdhfehoc |access-date=2024-02-17 |website=Desmos |language=en}}</ref>

===Image gallery of a zoom sequence===
The boundary of the Mandelbrot set shows more intricate detail the closer one looks or [[magnification|magnifies]] the image. The following is an example of an image sequence zooming to a selected ''c'' value.

The magnification of the last image relative to the first one is about 10<sup>10</sup> to 1. Relating to an ordinary [[computer monitor]], it represents a section of a Mandelbrot set with a diameter of 4 million kilometers.
{{clear}}

<gallery mode="packed">
Mandel zoom 00 mandelbrot set.jpg|Start. Mandelbrot set with continuously colored environment.
Mandel zoom 01 head and shoulder.jpg|Gap between the "head" and the "body", also called the "seahorse valley"<ref name=":7">{{Cite book |last=Lisle |first=Jason |url=https://books.google.com/books?id=h-czEAAAQBAJ |title=Fractals: The Secret Code of Creation |date=2021-07-01 |publisher=New Leaf Publishing Group |isbn=978-1-61458-780-4 |pages=28 |language=en}}</ref>
Mandel zoom 02 seehorse valley.jpg|Double-spirals on the left, "seahorses" on the right
Mandel zoom 03 seehorse.jpg|"Seahorse" upside down
</gallery>

The seahorse "body" is composed by 25 "spokes" consisting of two groups of 12 "spokes"<ref>{{Cite book |last=Devaney |first=Robert L. |url=https://books.google.com/books?id=GUpaDwAAQBAJ |title=A First Course In Chaotic Dynamical Systems: Theory And Experiment |date=2018-05-04 |publisher=CRC Press |isbn=978-0-429-97203-4 |pages=259 |language=en}}</ref> each and one "spoke" connecting to the main cardioid. These two groups can be attributed by some metamorphosis to the two "fingers" of the "upper hand" of the Mandelbrot set; therefore, the number of "spokes" increases from one "seahorse" to the next by 2; the "hub" is a [[Misiurewicz point]]. Between the "upper part of the body" and the "tail", there is a distorted copy of the Mandelbrot set, called a "satellite".

<gallery mode="packed" heights="180">
File:Mandel zoom 04 seehorse tail.jpg|The central endpoint of the "seahorse tail" is also a [[Misiurewicz point]].
File:Mandel zoom 05 tail part.jpg|Part of the "tail" – there is only one path consisting of the thin structures that lead through the whole "tail". This zigzag path passes the "hubs" of the large objects with 25 "spokes" at the inner and outer border of the "tail"; thus the Mandelbrot set is a [[Simply connected space|simply connected]] set, which means there are no islands and no loop roads around a hole.
File:Mandel zoom 06 double hook.jpg|Satellite. The two "seahorse tails" (also called ''dendritic structures'')<ref>{{Cite book |last=Kappraff |first=Jay |url=https://books.google.com/books?id=vAfBrK678_kC |title=Beyond Measure: A Guided Tour Through Nature, Myth, and Number |date=2002 |publisher=World Scientific |isbn=978-981-02-4702-7 |pages=437 |language=en}}</ref> are the beginning of a series of concentric crowns with the satellite in the center.
File:Mandel zoom 07 satellite.jpg|Each of these crowns consists of similar "seahorse tails"; their number increases with powers of 2, a typical phenomenon in the environment of satellites. The unique path to the spiral center passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head".
File:Mandel zoom 08 satellite antenna.jpg|"Antenna" of the satellite. There are several satellites of second order.
File:Mandel zoom 09 satellite head and shoulder.jpg|The "seahorse valley"<ref name=":7" /> of the satellite. All the structures from the start reappear.
File:Mandel zoom 10 satellite seehorse valley.jpg|Double-spirals and "seahorses" – unlike the second image from the start, they have appendices consisting of structures like "seahorse tails"; this demonstrates the typical linking of ''n'' + 1 different structures in the environment of satellites of the order ''n'', here for the simplest case ''n'' = 1.
File:Mandel zoom 11 satellite double spiral.jpg|Double-spirals with satellites of second order – analogously to the "seahorses", the double-spirals may be interpreted as a metamorphosis of the "antenna".
File:Mandel zoom 12 satellite spirally wheel with julia islands.jpg|In the outer part of the appendices, islands of structures may be recognized; they have a shape like [[Julia set]]s ''J<sub>c</sub>''; the largest of them may be found in the center of the "double-hook" on the right side.
File:Mandel zoom 13 satellite seehorse tail with julia island.jpg|Part of the "double-hook".
File:Mandel zoom 14 satellite julia island.jpg|Islands.
File:Mandel zoom 15 one island.jpg|A detail of one island.
File:Mandel zoom 16 spiral island.jpg|Detail of the spiral.
</gallery>
The islands in the third-to-last step seem to consist of infinitely many parts, as is the case for the corresponding Julia set <math>J_c</math>. They are connected by tiny structures, so that the whole represents a simply connected set. The tiny structures meet each other at a satellite in the center that is too small to be recognized at this magnification. The value of ''<math>c
</math>'' for the corresponding ''<math>J_c</math>'' is not the image center but, relative to the main body of the Mandelbrot set, has the same position as the center of this image relative to the satellite shown in the 6th step.

===Inner structure===
While the Mandelbrot set is typically rendered showing outside boundary detail, structure within the bounded set can also be revealed.<ref>{{Cite journal |last=Hooper |first=Kenneth J. |date=1991-01-01 |title=A note on some internal structures of the Mandelbrot Set |url=https://www.sciencedirect.com/science/article/abs/pii/009784939190082S |journal=Computers & Graphics |volume=15 |issue=2 |pages=295–297 |doi=10.1016/0097-8493(91)90082-S |issn=0097-8493}}</ref><ref>{{Cite web |last=Cunningham |first=Adam |date=December 20, 2013 |title=Displaying the Internal Structure of the Mandelbrot Set |url=https://www.acsu.buffalo.edu/~adamcunn/downloads/MandelbrotSet.pdf}}</ref><ref>{{Cite journal |last=Youvan |first=Douglas C |date=2024 |title=Shades Within: Exploring the Mandelbrot Set Through Grayscale Variations |url=https://rgdoi.net/10.13140/RG.2.2.24445.74727 |journal=Pre-print |doi=10.13140/RG.2.2.24445.74727}}</ref> For example, while calculating whether or not a given c value is bound or unbound, while it remains bound, the maximum value that this number reaches can be compared to the c value at that location. If the sum of squares method is used, the calculated number would be max:(real^2 + imaginary^2) − c:(real^2 + imaginary^2).{{Citation needed|date=March 2025}} The magnitude of this calculation can be rendered as a value on a gradient.

This produces results like the following, gradients with distinct edges and contours as the boundaries are approached. The animations serve to highlight the gradient boundaries.

<gallery mode=packed heights=160>
File:Mandelbrot full gradient.gif|Animated gradient structure inside the Mandelbrot set
File:Mandelbrot inner gradient.gif|Animated gradient structure inside the Mandelbrot set, detail
File:Mandelbrot gradient iterations.gif|Rendering of progressive iterations from 285 to approximately 200,000 with corresponding bounded gradients animated
File:Mandelbrot gradient iterations thumb.gif|Thumbnail for gradient in progressive iterations
</gallery>

==Generalizations==
{{multiple image
 | image1 = Mandelbrot Set Animation 1280x720.gif
 | image2 = Mandelbrot set from powers 0.05 to 2.webm
 | width2 = 150
 | footer = Animations of the Multibrot set for ''d'' from 0 to 5 (left) and from 0.05 to 2 (right).
}}
[[File:Quaternion Julia x=-0,75 y=-0,14.jpg|thumb|A 4D Julia set may be projected or cross-sectioned into 3D, and because of this a 4D Mandelbrot is also possible.]]

===Multibrot sets===
[[Multibrot set]]s are bounded sets found in the [[complex plane]] for members of the general monic univariate [[polynomial]] family of recursions

:<math>z \mapsto z^d + c</math>.<ref>{{Cite conference|contribution=On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets|last=Schleicher|first=Dierk|date=2004|title=Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 1|editor-last1=Lapidus|editor-first1=Michel L.|editor-last2=van Frankenhuijsen|editor-first2=Machiel|publisher=American Mathematical Society|url=https://books.google.com/books?id=uSpT729coosC|pages=477–517}}</ref>

For an [[integer]] ''d'', these sets are connectedness loci for the Julia sets built from the same formula. The full cubic connectedness locus has also been studied; here one considers the two-parameter recursion <math>z \mapsto z^3 + 3kz + c</math>, whose two [[critical point (mathematics)|critical points]] are the [[complex square root]]s of the parameter ''k''. A parameter is in the cubic connectedness locus if both critical points are stable.<ref>[[Rudy Rucker]]'s discussion of the CCM: [http://www.cs.sjsu.edu/faculty/rucker/cubic_mandel.htm CS.sjsu.edu]</ref> For general families of [[holomorphic function]]s, the ''boundary'' of the Mandelbrot set generalizes to the [[bifurcation locus]].{{Citation needed|date=July 2023}}

The [[Multibrot set]] is obtained by varying the value of the exponent ''d''. The article has a video that shows the development from ''d'' = 0 to 7, at which point there are 6 i.e. <math>(d-1)</math> lobes around the [[perimeter]]. In general, when ''d'' is a positive integer, the central region in each of these sets is always an [[epicycloid]] of <math>(d-1)</math> cusps. A similar development with negative integral exponents results in <math>(1-d)</math> clefts on the inside of a ring, where the main central region of the set is a [[hypocycloid]] of <math>(1-d)</math> cusps.{{Citation needed|date=July 2023}}

===Higher dimensions===
There is no perfect extension of the Mandelbrot set into 3D, because there is no 3D analogue of the complex numbers for it to iterate on. There is an extension of the complex numbers into 4 dimensions, the [[quaternion]]s, that creates a perfect extension of the Mandelbrot set and the Julia sets into 4 dimensions.<ref name="javier-barrallo"/> These can then be either [[cross section (geometry)|cross-sectioned]] or [[Projection mapping|projected]] into a 3D structure. The quaternion (4-dimensional) Mandelbrot set is simply a [[solid of revolution]] of the 2-dimensional Mandelbrot set (in the j-k plane), and is therefore uninteresting to look at.<ref name="javier-barrallo">{{cite web|last=Barrallo|first=Javier|date=2010|title=Expanding the Mandelbrot Set into Higher Dimensions|url=https://archive.bridgesmathart.org/2010/bridges2010-247.pdf|access-date=15 September 2021|website=BridgesMathArt}}</ref> Taking a 3-dimensional cross section at <math>d = 0\ (q = a + bi +cj + dk)</math> results in a solid of revolution of the 2-dimensional Mandelbrot set around the real axis.{{Citation needed|date=July 2023}}

===Other non-analytic mappings===
[[File:Mandelbar fractal from XaoS.PNG|left|thumb|Image of the [[Tricorn (mathematics)|Tricorn / Mandelbar fractal]]]]

The '''[[tricorn (mathematics)|tricorn]] fractal''', also called the '''Mandelbar set''', is the connectedness locus of the [[Antiholomorphic function|anti-holomorphic]] family <math>z \mapsto \bar{z}^2 + c</math>.<ref name=":6">{{Citation |last1=Inou |first1=Hiroyuki |title=Accessible hyperbolic components in anti-holomorphic dynamics |date=2022-03-23 |arxiv=2203.12156 |last2=Kawahira |first2=Tomoki}}</ref><ref>{{Citation |last1=Gauthier |first1=Thomas |title=Distribution of postcritically finite polynomials iii: Combinatorial continuity |date=2016-02-02 |arxiv=1602.00925 |last2=Vigny |first2=Gabriel}}</ref> It was encountered by [[John Milnor|Milnor]] in his study of parameter slices of real [[Cubic function|cubic polynomials]].{{Citation needed|date=March 2025}} It is not locally connected.<ref name=":6" /> This property is inherited by the connectedness locus of real cubic polynomials.{{Citation needed|date=March 2025}}

Another non-analytic generalization is the [[Burning Ship fractal]], which is obtained by iterating the following:

:<math>z \mapsto (|\Re \left(z\right)|+i|\Im \left(z\right)|)^2 + c</math>.{{Citation needed|date=March 2025}}

==Computer drawings==
{{Main|Plotting algorithms for the Mandelbrot set}}
There exist a multitude of various algorithms for plotting the Mandelbrot set via a computing device. Here, the naïve<ref>{{Cite book |last1=Jarzębowicz |first1=Aleksander |url=https://books.google.com/books?id=mQDsEAAAQBAJ |title=Software, System, and Service Engineering: S3E 2023 Topical Area, 24th Conference on Practical Aspects of and Solutions for Software Engineering, KKIO 2023, and 8th Workshop on Advances in Programming Languages, WAPL 2023, Held as Part of FedCSIS 2023, Warsaw, Poland, 17–20 September 2023, Revised Selected Papers |last2=Luković |first2=Ivan |last3=Przybyłek |first3=Adam |last4=Staroń |first4=Mirosław |last5=Ahmad |first5=Muhammad Ovais |last6=Ochodek |first6=Mirosław |date=2024-01-02 |publisher=Springer Nature |isbn=978-3-031-51075-5 |pages=142 |language=en}}</ref> "escape time algorithm" will be shown, since it is the most popular<ref>{{Cite book |last=Katunin |first=Andrzej |url=https://books.google.com/books?id=lmlQDwAAQBAJ |title=A Concise Introduction to Hypercomplex Fractals |date=2017-10-05 |publisher=CRC Press |isbn=978-1-351-80121-8 |pages=6 |language=en}}</ref> and one of the simplest algorithms.<ref>{{Cite book |last=Farlow |first=Stanley J. |url=https://books.google.com/books?id=r0QI5DMr6WAC |title=An Introduction to Differential Equations and Their Applications |date=2012-10-23 |publisher=Courier Corporation |isbn=978-0-486-13513-7 |pages=447 |language=en}}</ref> In the escape time algorithm, a repeating calculation is performed for each ''x'', ''y'' point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel.<ref>{{Cite book |last=Saha |first=Amit |url=https://books.google.com/books?id=EvWbCgAAQBAJ |title=Doing Math with Python: Use Programming to Explore Algebra, Statistics, Calculus, and More! |date=2015-08-01 |publisher=No Starch Press |isbn=978-1-59327-640-9 |pages=176 |language=en}}</ref><ref>{{Cite book |last=Crownover |first=Richard M. |url=https://books.google.com/books?id=RG3vAAAAMAAJ |title=Introduction to Fractals and Chaos |date=1995 |publisher=Jones and Bartlett |isbn=978-0-86720-464-3 |pages=201 |language=en}}</ref>

The ''x'' and ''y'' locations of each point are used as starting values in a repeating, or iterating calculation (described in detail below). The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout". If that condition is reached, the calculation is stopped, the pixel is drawn, and the next ''x'', ''y'' point is examined.

The color of each point represents how quickly the values reached the escape point. Often black is used to show values that fail to escape before the iteration limit, and gradually brighter colors are used for points that escape. This gives a visual representation of how many cycles were required before reaching the escape condition.

To render such an image, the region of the complex plane we are considering is subdivided into a certain number of [[pixel]]s. To color any such pixel, let <math>c</math> be the midpoint of that pixel. Iterate the critical point 0 under <math>f_c</math>, checking at each step whether the orbit point has a radius larger than 2. When this is the case, <math>c</math> does not belong to the Mandelbrot set, and color the pixel according to the number of iterations used to find out. Otherwise, keep iterating up to a fixed number of steps, after which we decide that our parameter is "probably" in the Mandelbrot set, or at least very close to it, and color the pixel black.

In [[pseudocode]], this algorithm would look as follows. The algorithm does not use complex numbers and manually simulates complex-number operations using two real numbers, for those who do not have a [[complex data type]]. The program may be simplified if the programming language includes complex-data-type operations.

<!-- NOTE that xtemp is necessary, otherwise y would be calculated with the new x, which would be wrong. Also note that one must plot (''x''<sub>0</sub> ''y''<sub>0</sub>), not (''x'',''y''). -->
 '''for each''' pixel (Px, Py) on the screen '''do'''
     x0 := scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.00, 0.47))
     y0 := scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1.12, 1.12))
     x := 0.0
     y := 0.0
     iteration := 0
     max_iteration := 1000
     '''while''' (x^2 + y^2 ≤ 2^2 AND iteration < max_iteration) '''do'''
         xtemp := x^2 - y^2 + x0
         y := 2*x*y + y0
         x := xtemp
         iteration := iteration + 1
     <!-- keep the following line filled with trailing whitespace to insert blank line in pseudocode block -->
 
     color := palette[iteration]
     plot(Px, Py, color)

Here, relating the pseudocode to <math>c</math>, <math>z</math> and <math>f_c</math>:
* <math>z = x + iy</math>
* <math>z^2 = x^2 +i2xy - y^2</math>
* <math>c = x_0 + i y_0</math>
and so, as can be seen in the pseudocode in the computation of ''x'' and ''y'':
* <math>x = \mathop{\mathrm{Re}} \left(z^2+c \right) = x^2-y^2 + x_0</math> and <math>y = \mathop{\mathrm{Im}} \left(z^2+c \right) = 2xy + y_0</math>.

To get colorful images of the set, the assignment of a color to each value of the number of executed iterations can be made using one of a variety of functions (linear, exponential, etc.).

=== Python code ===
Here is the code implementing the above algorithm in [[Python (programming language)|Python]]:<ref>{{Cite web |date=2018-10-03 |title=Mandelbrot Fractal Set visualization in Python |url=https://www.geeksforgeeks.org/mandelbrot-fractal-set-visualization-in-python/ |access-date=2025-03-23 |website=GeeksforGeeks |language=en-US}}</ref>{{close paraphrasing inline|date=March 2025}}

<syntaxhighlight lang="numpy">
import numpy as np
import matplotlib.pyplot as plt

# Setting parameters (these values can be changed)
x_domain, y_domain = np.linspace(-2, 2, 500), np.linspace(-2, 2, 500)
bound = 2
max_iterations = 50  # any positive integer value
colormap = "nipy_spectral"  # set to any matplotlib valid colormap

func = lambda z, p, c: z**p + c

# Computing 2D array to represent the Mandelbrot set
iteration_array = []
for y in y_domain:
    row = []
    for x in x_domain:
        z = 0
        p = 2
        c = complex(x, y)
        for iteration_number in range(max_iterations):
            if abs(z) >= bound:
                row.append(iteration_number)
                break
            else:
                try:
                    z = func(z, p, c)
                except (ValueError, ZeroDivisionError):
                    z = c
        else:
            row.append(0)

    iteration_array.append(row)

# Plotting the data
ax = plt.axes()
ax.set_aspect("equal")
graph = ax.pcolormesh(x_domain, y_domain, iteration_array, cmap=colormap)
plt.colorbar(graph)
plt.xlabel("Real-Axis")
plt.ylabel("Imaginary-Axis")
plt.show()
</syntaxhighlight>

[[File:Multibrot set of power 5.png|thumb|An image of a 2-dimensional multibrot-set represented by the equation <math>z=z^5+c</math>.]]

The value of <code>power</code> variable can be modified to generate an image of equivalent [[multibrot set]] (<math>z = z^{\text{power}}+c</math>). For example, setting <code>p = 2</code> produces the associated image.

==References in popular culture==
The Mandelbrot set is widely considered the most popular [[fractal]],<ref>Mandelbaum, Ryan F. (2018). [https://gizmodo.com/this-trippy-music-video-is-made-of-3d-fractals-1822168809 "This Trippy Music Video Is Made of 3D Fractals."] Retrieved 17 January 2019</ref><ref>Moeller, Olga de. (2018).[https://thewest.com.au/lifestyle/kids/what-are-fractals-ng-b88838072z "what are Fractals?"] Retrieved 17 January 2019.</ref> and has been referenced several times in [[popular culture]].
* The [[Jonathan Coulton]] song "Mandelbrot Set" is a tribute to both the fractal itself and to the man it is named after, Benoit Mandelbrot.<ref name="JoCopedia">{{cite web|title=Mandelbrot Set|url=http://www.jonathancoulton.com/wiki/Mandelbrot_Set|website=JoCopeda|access-date=15 January 2015}}</ref>
* [[Blue Man Group]]'s 1999 debut album ''[[Audio (album)|Audio]]'' references the Mandelbrot set in the titles of the songs "Opening Mandelbrot", "Mandelgroove", and "Klein Mandelbrot".<ref>{{cite web |title=Blue Man Group – Audio Album Reviews, Songs & More|url=https://www.allmusic.com/album/audio-mw0000672371 |website=Allmusic.com |access-date=4 July 2023 |language=en}}</ref> Their second album, ''[[The Complex (album)|The Complex]]'' (2003), closes with a [[hidden track]] titled "Mandelbrot IV".
* The second book of the ''[[Mode (book series)|Mode]]'' series by [[Piers Anthony]], ''Fractal Mode'', describes a world that is a perfect 3D model of the set.<ref name="Anthony1992">{{cite book|author=Piers Anthony|title=Fractal Mode|url=https://books.google.com/books?id=XdUyAAAACAAJ|year=1992|publisher=HarperCollins|isbn=978-0-246-13902-3}}</ref>
* The [[Arthur C. Clarke]] novel ''[[The Ghost from the Grand Banks]]'' features an artificial lake made to replicate the shape of the Mandelbrot set.<ref name="Clarke2011">{{cite book|author=Arthur C. Clarke|title=The Ghost From The Grand Banks|url=https://books.google.com/books?id=6ELsYigmXNoC|year=2011|publisher=Orion|isbn=978-0-575-12179-9}}</ref>
* Benoit Mandelbrot and the eponymous set were the subjects of the [[Google Doodle]] on 20 November 2020 (the late Benoit Mandelbrot's 96th birthday).<ref>{{cite web|url=https://www.upi.com/Top_News/US/2020/11/20/Google-honors-mathematician-Benoit-Mandelbrot-with-new-Doodle/8571605871071/ |title=Google honors mathematician Benoit Mandelbrot with new Doodle |first=Wade |last=Sheridan |website=[[United Press International]] |date=20 November 2020 |access-date=30 December 2020}}</ref>
* The American rock band [[Heart (band)|Heart]] has an image of a Mandelbrot set on the cover of their 2004 album, ''[[Jupiters Darling]]''.
* The British black metal band [[Anaal Nathrakh]] uses an image resembling the Mandelbrot set on their ''[[Eschaton (album)|Eschaton]]'' album cover art.
* The television series ''[[Dirk Gently's Holistic Detective Agency (TV series)|Dirk Gently's Holistic Detective Agency]]'' (2016) prominently features the Mandelbrot set in connection with the visions of the character Amanda. In the second season, her jacket has a large image of the fractal on the back.<ref>{{cite web |title=Hannah Marks "Amanda Brotzman" customized black leather jacket from Dirk Gently's Holistic Detective Agency |url=https://www.icollector.com/Hannah-Marks-Amanda-Brotzman-customized-black-leather-jacket-from-Dirk-Gently-s_i36007665 |website=www.icollector.com}}</ref>
* In [[Ian Stewart (mathematician)|Ian Stewart]]'s 2001 book ''[[Flatterland]]'', there is a character called the Mandelblot, who helps explain fractals to the characters and reader.<ref>{{Cite journal |last=Trout |first=Jody |date=April 2002 |title=Book Review: ''Flatterland: Like Flatland, Only More So'' |url=https://www.ams.org/notices/200204/rev-trout.pdf |journal=Notices of the AMS |volume=49 |issue=4 |pages=462–465}}</ref>
* The unfinished [[Alan Moore]] 1990 comic book series ''[[Big Numbers (comics)|Big Numbers]]'' used Mandelbrot's work on fractal geometry and chaos theory to underpin the structure of that work. Moore at one point was going to name the comic book series ''The Mandelbrot Set''.<ref>{{cite web  |title=''The Great Alan Moore Reread: Big Numbers'' by Tim Callahan|url=https://www.tor.com/2012/05/21/the-great-alan-moore-reread-big-numbers/ |website=Tor.com |date=21 May 2012 }}</ref>
* In the manga ''[[The Summer Hikaru Died]]'', Yoshiki hallucinates the Mandelbrot set when he reaches into the body of the false Hikaru.

==See also==
{{Div col|colwidth=20em}}
* [[Buddhabrot]]
* [[Collatz fractal]]
* [[Fractint]]
* [[Gilbreath permutation]]
* [[List of mathematical art software]]
* [[Mandelbox]]
* [[Mandelbulb]]
* [[Menger sponge]]
* [[Newton fractal]]
* [[Orbit portrait]]
* [[Orbit trap]]
* [[Pickover stalk]]
* [[Plotting algorithms for the Mandelbrot set]]{{Div col end}}

==References==
{{Reflist|30em}}

==Further reading==
* {{cite book |first=John W. |last=Milnor |authorlink=John W. Milnor |title=Dynamics in One Complex Variable |edition=Third |series=Annals of Mathematics Studies |volume=160 |publisher=Princeton University Press |year=2006 |isbn=0-691-12488-4 }} <br />(First appeared in 1990 as a [https://web.archive.org/web/20060424085751/http://www.math.sunysb.edu/preprints.html Stony Brook IMS Preprint], available as [http://www.arxiv.org/abs/math.DS/9201272 arXiV:math.DS/9201272] )
* {{cite book |first=Nigel |last=Lesmoir-Gordon |title=The Colours of Infinity: The Beauty, The Power and the Sense of Fractals |year=2004 |publisher=Clear Press |isbn=1-904555-05-5 }} <br />(includes a DVD featuring [[Arthur C. Clarke]] and [[David Gilmour]])
* {{cite book |first1=Heinz-Otto |last1=Peitgen |authorlink=Heinz-Otto Peitgen |first2=Hartmut |last2=Jürgens |authorlink2=Hartmut Jürgens |first3=Dietmar |last3=Saupe |authorlink3=Dietmar Saupe |title=Chaos and Fractals: New Frontiers of Science |publisher=Springer |location=New York |orig-year=1992 |year=2004 |isbn=0-387-20229-3 }}

==External links==
{{Wikibooks|Fractals }}
{{commons category}}
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* [http://vimeo.com/12185093 Video: Mandelbrot fractal zoom to 6.066 e228]
* [https://www.youtube.com/watch?v=NGMRB4O922I Relatively simple explanation of the mathematical process], by [[Holly Krieger|Dr Holly Krieger]], MIT
* [https://mandelbrot.site/ Mandelbrot Set Explorer]: Browser based Mandelbrot set viewer with a map-like interface
* [https://www.rosettacode.org/wiki/Mandelbrot_set Various algorithms for calculating the Mandelbrot set] (on [[Rosetta Code]])
* [https://github.com/pkulchenko/ZeroBraneEduPack/blob/master/fractal-samples/zplane.lua Fractal calculator written in Lua by Deyan Dobromiroiv, Sofia, Bulgaria]
{{Fractals}}

{{DEFAULTSORT:Mandelbrot Set}}
[[Category:Fractals]]
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