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{{Short description|Fractal named after mathematician Benoit Mandelbrot}}
[[File:Mandel zoom 00 mandelbrot set.jpg|322px|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 dictionary |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=[[Organization:Oxford University Press|Oxford University Press]]}}</ref><ref>{{cite Merriam-Webster|Mandelbrot set|access-date=2022-01-30}}</ref> is a two-dimensional [[Set (mathematics)|set]] with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the [[Complex plane|complex plane]] as the [[Complex number|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|absolute value]].

This set was first defined and drawn by [[Biography:Robert W. Brooks|Robert W. Brooks]] and Peter Matelski in 1978, as part of a study of [[Kleinian group]]s.<ref name=":0" /> Afterwards, in 1980, [[Biography:Benoit Mandelbrot|Benoit Mandelbrot]] obtained high-quality visualizations of the set while working at [[Company:IBM|IBM]]'s [[Organization:Thomas J. Watson Research Center|Thomas J. Watson Research Center]] in Yorktown Heights, New York.
[[File:Mandelbrot sequence new.gif|thumb|Zooming into the boundary of the Mandelbrot set]]

Images of the Mandelbrot set exhibit an infinitely complicated [[Boundary (topology)|boundary]] that reveals progressively ever-finer [[Recursion|recursive]] detail at increasing magnifications; mathematically, the boundary of the Mandelbrot set is a ''[[Fractal curve|fractal curve]]''. The "style" of this recursive detail depends on the region of the set boundary being examined. 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]]. Treating the real and [[Imaginary number|imaginary part]]s of <math>c</math> as image coordinates on the [[Complex plane|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). 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.

The Mandelbrot set has become popular outside [[Mathematics|mathematics]] both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of [[Mathematical visualization|mathematical visualization]], [[Mathematical beauty|mathematical beauty]], and motif.

==History==
[[File:Mandel.png|322px|right|thumb|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|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 [[Biography:Robert W. Brooks|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=1 May 1981|publisher=Princeton University Press|others=[[Biography:Bernard Maskit|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 [[Company:IBM|IBM]]'s [[Organization:Thomas J. Watson Research Center|Thomas J. Watson Research Center]] in Yorktown Heights, New York, [[Biography:Benoit Mandelbrot|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|parameter space]] of quadratic polynomials 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 [[Biography:Adrien Douady|Adrien Douady]] and [[Biography:John H. Hubbard|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 [[Biography:Heinz-Otto Peitgen|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|algorithm]] for computing the Mandelbrot set. The cover was created by Peitgen, Richter and Saupe at the [[Organization:University of Bremen|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|via=JSTOR}}</ref> The Mandelbrot set became prominent in the mid-1980s as a computer-graphics demo, when [[Engineering:Personal computer|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 |access-date=11 November 2015 }}</ref>

The work of Douady and Hubbard occurred during an increase in interest in [[Complex dynamics|complex dynamics]] and abstract mathematics,<ref name=rees>{{cite journal
	| last1 = Rees
	| first1 = Mary 
	| 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
	}}</ref> and the study of the Mandelbrot set has been a centerpiece of this field ever since.{{Cn|date=July 2023}}

==Formal definition==
The Mandelbrot set is the set of values of ''c'' in the [[Complex plane|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_{n+1} = z_{n}^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.<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|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|absolute value]] of <math>z_n</math> remains bounded for all <math>n > 0</math>.

For example, for ''c'' = 1, the [[Sequence|sequence]] is 0, 1, 2, 5, 26, ..., which tends to [[Infinity|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|connectedness locus]] of the family of [[Quadratic equation|quadratic]] [[Polynomial|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. In the same way, the [[Boundary (topology)|boundary]] of the Mandelbrot set can be defined as the [[Bifurcation locus|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 around the [[Origin (mathematics)|origin]]. 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|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|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>z = r\left(\frac12 - x\right),
\quad
c = \frac{r}{2}\left(1-\frac{r}{2}\right).</math>

This gives a correspondence between the entire [[Parameter space|parameter space]] of the logistic family and that of the Mandelbrot set.{{citation needed|date=April 2021}}

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. 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|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 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|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 plane algebraic curves, the ''Mandelbrot curves'', of the general type known as [[Polynomial lemniscate|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.-->{{Unreferenced section|date=July 2023}}[[File:Mandelbrot Set – Periodicities coloured.png|right|thumb|Periods of hyperbolic components]]

The ''main cardioid'' is the period 1 continent. 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]]. 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.

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. 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.

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>. 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>

[[File:Animated cycle.gif|left|thumb|200px|Attracting cycle in 2/5-bulb plotted over [[Julia set|Julia set]] (animation)]]and which contains parameters with <math>q</math>-cycles having combinatorial rotation number <math>\frac{p}{q}</math>. 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>.

[[File:Juliacycles1.png|right|thumb|300px|Attracting cycles and [[Julia set|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.

===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''.{{Cn|date=July 2023}}

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|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>.{{Cn|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 [[Biography:Adrien Douady|Adrien Douady]] and [[Biography:John H. Hubbard|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.{{Cn|date=July 2023}}

The work of Jean-Christophe Yoccoz established local connectivity of the Mandelbrot set at all finitely [[Physics: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 = http://www.math.cornell.edu/~hubbard/hubbard.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|201px|[[Self-similarity|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 points (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.{{Cn|date=July 2023}}

===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|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.{{Cn|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|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|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.{{Cn|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|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.{{Cn|date=July 2023}}

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 relatively prime, 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|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>.{{Cn|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 ===
In an attempt to demonstrate that the thickness of the ''p''/''q''-limb is zero, David Boll carried out a computer experiment in 1991, where he computed the number of iterations required for the [[Series (mathematics)|series]] to diverge for <math>z = -\tfrac{3}{4}+ i\varepsilon</math> (<math>-\tfrac{3}{4}</math> being the location thereof). As the series does not diverge for the exact value of <math>z = -\tfrac{3}{4}</math>, the number of iterations required increases with a small <math>\varepsilon</math>. It turns out that multiplying the value of <math>\varepsilon</math> with the number of iterations required yields an approximation of <math>\pi</math> that becomes better for smaller ''<math>\varepsilon</math>''. For example, for ''<math>\varepsilon</math>'' = 0.0000001, 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> In 2001, Aaron Klebanoff proved Boll's discovery.<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>

=== Fibonacci sequence in the Mandelbrot set ===
It can be shown that the Fibonacci sequence is located within the Mandelbrot set and that a relation exists between the main cardioid and the [[Farey sequence|Farey diagram]].<ref>{{Cite web |title=Fractal Foundation Online Course - Chapter 11 - FIBONACCI FRACTALS |url=https://fractalfoundation.org/OFC/OFC-11-4.html |access-date=2023-07-19 |website=fractalfoundation.org}}</ref> Upon mapping the main cardioid to a disk, the amount of antennae that extends from the next largest hyperbolic component, and that is located between the two previously selected components, follows suit with the Fibonacci sequence. The amount of antennae also correlates with the Farey diagram and the denominator amounts within the corresponding fractional values, of which relate to the distance around the disk. Both portions of these fractional values themselves can be summed together after <math>\frac{1}{3}</math> to produce the location of the next hyperbolic component within the sequence. Thus, the Fibonacci sequence of 1, 2, 3, 5, 8, 13, and 21 can be found within the Mandelbrot set.

===Image gallery of a zoom sequence===
The boundary of the Mandelbrot set shows more intricate detail the closer one looks or [[Physics: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 [[Engineering:Computer monitor|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"
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" 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">
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" 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" 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. 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). 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>
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|complex plane]] for members of the general monic univariate [[Polynomial|polynomial]] family of recursions

:<math>z \mapsto z^d + c</math>.{{Cn|date=July 2023}}

For an [[Integer|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 roots of the parameter ''k''. A parameter is in the cubic connectedness locus if both critical points are stable.<ref>[[Biography:Rudy Rucker|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|holomorphic function]]s, the ''boundary'' of the Mandelbrot set generalizes to the [[Bifurcation locus|bifurcation locus]].{{Cn|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|perimeter]]. In general, when ''d'' is a positive integer, the central region in each of these sets is always an [[Epicycloid|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|hypocycloid]] of <math>(1-d)</math> cusps.{{Cn|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|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|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.{{Cn|date=July 2023}}

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

Of particular interest is the [[Tricorn (mathematics)|tricorn]] fractal, the connectedness locus of the [[Antiholomorphic function|anti-holomorphic]] family

:<math>z \mapsto \bar{z}^2 + c</math>.

The tricorn (also sometimes called the ''Mandelbar'') was encountered by [[Biography:John Milnor|Milnor]] in his study of parameter slices of real [[Cubic function|cubic polynomials]]. It is not locally connected. This property is inherited by the connectedness locus of real cubic polynomials.

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>.

==Computer drawings==
There exist a multitude of various algorithms for plotting the Mandelbrot set via a computing device. Here, the most widely used and simplest algorithm will be demonstrated, namely, the naïve "escape time algorithm". 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.

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|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|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|complex data type]]. The program may be simplified if the programming language includes [[Complex data type|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>,&nbsp;''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*x + y*y ≤ 2*2 AND iteration < max_iteration) '''do'''
         xtemp := x*x - y*y + 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}}(z^2+c) = x^2-y^2 + x_0</math> and <math>y = \mathop{\mathrm{Im}}(z^2+c) = 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.).

==References in popular culture==
The Mandelbrot set is widely considered the most popular [[Fractal|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 [[Social:Popular culture|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>
* Multiple track titles on the 1999 debut album by the Blue Man Group, ''Audio'', refer to the Mandelbrot set. These are "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>
* The second book of the ''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 [[Biography:Arthur C. Clarke|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|date=29 September 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 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 cover art.
* The television 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 [[Biography: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 [[Biography:Alan Moore|Alan Moore]] 1990 comic book series ''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 the 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
* [[Software:Fractint|Fractint]]
* Gilbreath permutation
* [[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 |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 [[Biography:Arthur C. Clarke|Arthur C. Clarke]] and David Gilmour)
*{{cite book |first1=Heinz-Otto |last1=Peitgen |first2=Hartmut |last2=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==
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* {{Curlie|Science/Math/Chaos_and_Fractals|Chaos and Fractals}}
* [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 Dr Holly Krieger, MIT
* [https://mandelbrot.silversky.dev Mandelbrot Viewer]: Browser based Mandelbrot set renderer including a [https://mandelbrot.silversky.dev/gallery/ gallery with examples].
* [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]]
[[Category:Complex dynamics]]

{{Sourceattribution|Mandelbrot set}}