View source for Mandelbrot set

{{short description|Fractal named after mathematician Benoit Mandelbrot}}
{{Use dmy dates|date=October 2016}}
[[File:Mandel zoom 00 mandelbrot set.jpg|322px|thumb|The Mandelbrot set (black) within a continuously colored environment|alt=]]<!-- The sequence \, is inserted in MATH items to ensure consistency of representation.
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[[File:Progressive infinite iterations of the 'Nautilus' section of the Mandelbrot Set.ogv|thumb|Progressive infinite iterations of the "Nautilus" section of the Mandelbrot Set rendered using webGL]]
[[File:Animation of the growth of the Mandelbrot set as you iterate towards infinity.gif|thumb|Mandelbrot animation based on a static number of iterations per pixel]]
[[File:Mandelbrot set image.png|thumb|Mandelbrot set detail]]
The '''Mandelbrot set''' is the [[set (mathematics)|set]] of [[complex number]]s <math>c</math> for which the function <math>f_c(z)=z^2+c</math> does not [[diverge (stability theory)|diverge]] when [[Iteration|iterated]] from <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.
[[File:Mandelbrot sequence new.gif|thumb|Zooming into the Mandelbrot set]]
 
Its definition and name are due to [[Adrien Douady]], in tribute to the [[mathematician]] [[Benoit Mandelbrot]].<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> The set is connected to a [[Julia set]], and related Julia sets produce similarly complex [[fractal]] shapes.

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]] (in practice &mdash; whether it leaves some predetermined bounded neighborhood of 0 after a predetermined number of iterations). 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 coloured according to how soon the sequence <math>|f_c(0)|, |f_c(f_c(0))|,\dotsc</math> crosses an arbitrarily chosen threshold, with a special color (usually black) used for the values of <math>c</math> for which the sequence has not crossed the threshold after the predetermined number of iterations (this is necessary to clearly distinguish the Mandelbrot set image from the image of its complement). If <math>c</math> is held constant and the initial value of <math>z</math>—denoted by <math>z_0</math>—is variable instead, one obtains the [[Julia set#Quadratic polynomials|corresponding Julia set]] for each point <math>c</math> in the [[parameter space]] of the simple function.

Images of the Mandelbrot set exhibit an elaborate and infinitely complicated [[Boundary (topology)|boundary]] that reveals progressively ever-finer [[Recursion|recursive]] detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the [[fractal]] property of [[self-similarity]] applies to the entire set, and not just to its parts.

The Mandelbrot set has become popular outside [[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]] and [[mathematical beauty]].

==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 place in [[complex dynamics]], a field first investigated by the [[French mathematicians]] [[Pierre Fatou]] and [[Gaston Julia]] at the beginning of the 20th century. This 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>Robert Brooks and Peter Matelski, ''The dynamics of 2-generator subgroups of PSL(2,C)'', in {{cite book|url=http://www.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=[[Bernard Maskit]]|isbn=0-691-08267-7|editor=Irwin Kra|location=|pages=}}</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 saw a visualization of the set.<ref name="bf">{{cite web |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 |accessdate=1 January 2009 |year=2008 |work=Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 12, No. 1 |publisher=Society for Chaos Theory in Psychology & Life Sciences }}</ref>

Mandelbrot studied the [[parameter space]] of [[quadratic polynomial]]s in an article that appeared in 1980.<ref>Benoit Mandelbrot, ''Fractal aspects of the iteration of <math>z\mapsto\lambda z(1-z)</math> for complex <math>\lambda, z</math>'', ''Annals of the New York Academy of Sciences'' '''357''', 249/259</ref> The mathematical study of the Mandelbrot set really began with work by the mathematicians [[Adrien Douady]] and [[John H. Hubbard]],<ref name="John H. Hubbard 1985"/> 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,<ref>{{cite book |title=The Beauty of Fractals |last=Peitgen |first=Heinz-Otto |authorlink= |author2=Richter Peter |year=1986 |publisher=Springer-Verlag |location=Heidelberg |isbn=0-387-15851-0 |pages= |title-link=The Beauty of Fractals }}</ref> and an internationally touring exhibit of the German [[Goethe-Institut]].<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 |authorlink= |year=1987 |publisher=Cardinal |location=London |isbn= |pages=229 |title-link=Chaos: Making a New Science }}</ref>

The cover article of the August 1985 ''[[Scientific American]]'' introduced a wide audience to the [[algorithm]] for computing the Mandelbrot set. The cover featured an image located at [https://mandelbrot-svelte.netlify.com/#{%22pos%22:{%22x%22:-0.909,%22y%22:-0.275},%22zoom%22:10000} -0.909 + -0.275] and was created by Peitgen, et al.<ref>{{cite book |title=A computer microscope zooms in for a close look at the most complicated object in mathematics |last=Dewdney |first=A. K. |authorlink= |year=1985 |publisher=Scientific American |location=http://www.nature.com/scientificamerican/journal/v253/n2/pdf/scientificamerican0885-16.pdf |pages=16–24}} {{subscription required}}</ref><ref>{{cite book |title=Fractals: The Patterns of Chaos |author=John Briggs |year=1992 |page=80}}</ref> The Mandelbrot set became prominent in the mid-1980s as a computer [[Demo (computer programming)|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 coincided with a huge increase in interest in complex dynamics and [[abstract mathematics]], and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of all who have contributed to the understanding of this set since then is long but would include [[Mikhail Lyubich]],<ref>{{cite journal
  | author = Lyubich, Mikhail
  | title = Six Lectures on Real and Complex Dynamics
  | version =
  | publisher =
  | date = May–June 1999
  | url = http://citeseer.ist.psu.edu/cache/papers/cs/28564/http:zSzzSzwww.math.sunysb.eduzSz~mlyubichzSzlectures.pdf/
  | format =
  | accessdate = 2007-04-04 }}</ref><ref>{{cite journal
  | last = Lyubich
  | first = Mikhail
  | authorlink = Mikhail Lyubich
  | title = Regular and stochastic dynamics in the real quadratic family
  | journal = Proceedings of the National Academy of Sciences of the United States of America
  | volume = 95
  | issue =24
  | pages = 14025–14027
  | date=November 1998
  | url = http://www.pnas.org/cgi/reprint/95/24/14025.pdf
  | doi = 10.1073/pnas.95.24.14025
  | id =
  | accessdate = 2007-04-04
  | pmid = 9826646
  | pmc = 24319 | bibcode =1998PNAS...9514025L
  }}</ref> [[Curtis T. McMullen|Curt McMullen]], [[John Milnor]], [[Mitsuhiro Shishikura]] and [[Jean-Christophe Yoccoz]].

==Formal definition==
The Mandelbrot set is the set of values of ''c'' in the [[complex plane]] for which the [[Orbit (dynamics)|orbit]] of 0 under [[Iterated function|iteration]] of the [[quadratic map]]

:<math>z_{n+1} = z_n^2 + c</math>

remains [[Bounded sequence|bounded]].<ref>{{cite web|url=http://math.bu.edu/DYSYS/explorer/def.html|title=Mandelbrot Set Explorer: Mathematical Glossary|accessdate=2007-10-07}}</ref> That is, a complex number ''c'' is part of the Mandelbrot set if, when starting with ''z''<sub>0</sub> = 0 and applying the iteration repeatedly, the [[absolute value]] of ''z''<sub>''n''</sub> remains bounded however large ''n'' gets. This can also be represented as<ref>{{cite web|url=http://mrob.com/pub/muency/escaperadius.html|title=Escape Radius, Mu-ENCY (the Encyclopedia of the Mandelbrot Set) at MROB|accessdate=2015-10-21}}</ref>

:<math>z_{n+1} = z_n^2 + c,</math>
:<math>c \in M \iff \limsup_{n \to \infty}|z_{n+1}| \leq 2.</math>
 
For example, letting ''c'' = 1 gives the sequence 0, 1, 2, 5, 26, ..., which tends to [[infinity]]. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, ''c'' = −1 gives the sequence 0, −1, 0, −1, 0, ..., which is bounded, and so −1 belongs to the Mandelbrot set.

The Mandelbrot set <math>M</math> is defined by a family of [[complex quadratic polynomial]]s

:<math>P_c:\mathbb C\to\mathbb C</math>

given by

:<math>P_c: z\mapsto z^2 + c,</math>

where <math>c</math> is a complex parameter. For each <math>c</math>, one considers the behavior of the [[sequence]]
:<math>(0, P_c(0), P_c(P_c(0)), P_c(P_c(P_c(0))), \ldots)</math>

obtained by [[iterated function|iterating]] <math>P_c(z)</math> starting at [[critical point (mathematics)|critical point]] <math>z = 0</math>, which either escapes to infinity or stays within a [[Disk (mathematics)|disk]] of some finite [[radius]]. The Mandelbrot set is defined as the set of all points <math>c</math> such that the above sequence does ''not'' escape to infinity.

[[File:Mandelset hires.png|right|thumb|322px|A mathematician's depiction of the Mandelbrot set ''M''. A point ''c'' is colored black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of ''c'', respectively.]]
More formally, if <math>P_c^n(z)</math> denotes the ''n''th iterate of <math>P_c(z)</math> (i.e. <math>P_c(z)</math> [[function composition|composed]] with itself ''n'' times), the Mandelbrot set is the [[subset]] of the [[complex plane]] given by

:<math>M = \left\{c\in \mathbb C \mid (\exists s\in \mathbb R^+)(\forall n\in \mathbb N)(|P_c^n(0)| \le s) \right\}.</math>

As explained below, it is in fact possible to simplify this definition by taking <math>s=2</math>.

Mathematically, the Mandelbrot set is just a [[Set (mathematics)|set]] of complex numbers. A given complex number ''c'' either belongs to ''M'' or it does not. A picture of the Mandelbrot set can be made by coloring all the points <math>c</math> that belong to ''M'' black, and all other points white. The more colorful pictures usually seen are generated by coloring points not in the set according to which term in the sequence <math>|P_c^n(0)|</math> is the first term with an absolute value greater than a certain cutoff value, usually 2. The list of colors used is always predefined by the program used or created by the user, the next color in the list is chosen when the iteration count rises. See the section on [[#Computer drawings|computer drawings]] below for more details.

The Mandelbrot set can also be defined as the [[connectedness locus]] of the family of [[polynomial]]s <math>P_c(z)</math>. That is, it is the subset of the complex plane consisting of those parameters <math>c</math> for which the [[Julia set]] of <math>P_c</math> is [[connected space|connected]].

<math>P_c^n(0)</math> is a polynomial in c and its leading terms settle down as <math>n</math> grows large enough. These terms are given by the [[Catalan numbers]]. The polynomials <math>P_c^n(0)</math> are bounded by the [[generating function]] for the Catalan numbers and tend to it as <math>n</math> goes to infinity.

==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]]. More specifically, a point <math>c</math> belongs to the Mandelbrot set if and only if
:<math>|P_c^n(0)|\leq 2</math> for all <math>n\geq 0.</math>

In other words, if the [[absolute value]] of <math>P_c^n(0)</math> ever becomes larger than 2, the sequence will escape to infinity.

[[File:Verhulst-Mandelbrot-Bifurcation.jpg|thumb|Correspondence between the Mandelbrot set and the [[bifurcation diagram]] of the [[logistic map]]]]
The [[intersection (set theory)|intersection]] of <math>M</math> with the real axis is precisely the interval [−2, 1/4]. 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>

In fact, this gives a correspondence between the entire [[parameter space]] of the logistic family and that of the Mandelbrot set.

Douady and Hubbard have shown that the Mandelbrot set is [[connected space|connected]]. In fact, 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. There also exists a [[Topology|topological]] proof to the connectedness that 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|website=|archive-url=|archive-date=|access-date=}}</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 exactly the [[bifurcation locus]] of the quadratic family; that is, the set of parameters <math>c</math> for which the dynamics changes abruptly under 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 ''p''<sub>0</sub> = ''z'', ''p''<sub>''n''+1</sub> = ''p''<sub>''n''</sub><sup>2</sup> + ''z'', and then interpreting the set of points |''p''<sub>''n''</sub>(''z'')| = 2 in the complex plane as a curve in the real [[Cartesian coordinate system|Cartesian plane]] of degree 2<sup>''n''+1</sup> in ''x'' and ''y''.  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]]

Upon looking at a picture of the Mandelbrot set, one immediately notices the large [[cardioid]]-shaped region in the center. This ''main cardioid''
is the region of parameters <math>c</math> for which <math>P_c</math> has an [[Periodic points of complex quadratic mappings|attracting fixed point]]. It consists of all parameters of the form
:<math> c = \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-shaped '''bulb''' is visible. This bulb consists of those parameters <math>c</math> for which <math>P_c</math> has an [[Periodic points of complex quadratic mappings|attracting cycle of period 2]]. This set of parameters is an actual circle, namely that of radius 1/4 around −1.

There are infinitely many other bulbs tangent to the main cardioid: for every rational number <math>\tfrac{p}{q}</math>, with ''p'' and ''q'' [[coprime]], there is such a bulb that is tangent at the parameter

:<math> c_{\frac{p}{q}} = \frac{e^{2\pi i\frac pq}}2\left(1-\frac{e^{2\pi i\frac pq}}2\right).</math>

[[File:Animated cycle.gif|left|thumb|200px|Attracting cycle in 2/5-bulb plotted over [[Julia set]] (animation)]]This bulb is called the ''<math>\tfrac{p}{q}</math>-bulb'' of the Mandelbrot set. It consists of parameters that have an attracting cycle of period <math>q</math> and combinatorial rotation number <math>\tfrac{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>P_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]]s for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs]]

[[File:Mandel rays.jpg|thumb|right|425px|Cycle periods and antennae]]

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.

{{clear|left}}

===Hyperbolic components===
All the bulbs we encountered in the previous section were interior components of
the Mandelbrot set in which the maps <math>P_c</math> have an attracting periodic cycle. Such components are called ''hyperbolic components''.

It is conjectured that these are the ''only'' interior regions of <math>M</math>. This problem, known as ''density of hyperbolicity'', may be the most important open problem in the field of complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components.<ref>''Exploring the Mandelbrot set. The Orsay Notes'' by Adrien Douady and John H. Hubbard. page 12</ref><ref>Wolf Jung, March 2002, [http://www.mndynamics.com/papers/thesis.pdf Homeomorphisms on Edges of the Mandelbrot Set by Wolf Jung]</ref>
For ''real'' quadratic polynomials, this question was answered positively 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. However, such a component ''can'' be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).

Each of the hyperbolic components has a ''center'', which is a point ''c'' such that the inner Fatou domain for <math>P_c(z)</math> has a super-attracting cycle – that is, that the attraction is infinite (see the image [[:commons:File:Centers8.png|here]]). This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. We therefore have that <math>P_c</math><sup>n</sup><math>(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>. We can therefore construct the centers of the hyperbolic components by successively solving the equations <math>Q^{n}(c) = 0, n = 1, 2, 3, ...</math>. The number of new centers produced in each step is given by Sloane's {{oeis|A000740}}.

===Local connectivity===
[[File:Cactus model of Mandelbrot set.svg|right|thumb|Topological model of Mandelbrot set without mini Mandelbrot sets and Misiurewicz points (Cactus model)]]
[[File:Lavaurs-12.png|right|thumb|Thurston model of Mandelbrot set (abstract Mandelbrot set)]]

It is conjectured that the Mandelbrot set is [[locally connected]]. This famous 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.

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">{{citation
 | 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:Mandelbrot zoom.gif|right|thumb|201px|[[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 from (−1,&nbsp;0) to (−1.31,&nbsp;0) while the view magnifies from 0.5&nbsp;×&nbsp;0.5 to 0.12&nbsp;×&nbsp;0.12 to approximate the [[Feigenbaum constants|Feigenbaum ratio]] <math>\delta</math>.]]
[[File:Mandelzoom1.jpg|left|thumb|280px|Self-similarity around Misiurewicz point −0.1011&nbsp;+&nbsp;0.9563i.]] 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}}</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}})</ref>
[[File:Blue Mandelbrot Zoom.jpg|left|thumb|340px|Quasi-self-similarity in the Mandelbrot set]]The Mandelbrot set in general is not strictly self-similar but it is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales.

The little copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set.

===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"/> It is not known whether the boundary of the Mandelbrot set has positive planar [[Lebesgue measure]].

In the [[Blum-Shub-Smale]] model of [[real computation]], the Mandelbrot set is not computable, but its complement is [[Recursively enumerable set|computably enumerable]]. However, 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.[[File:Relationship between Mandelbrot sets and Julia sets.PNG|3220x|left|thumb|A zoom into the Mandelbrot set illustrating a Julia "island" and a similar Julia set.]]

===Relationship with 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 point is in the Mandelbrot set exactly when the corresponding Julia set is connected.

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">{{citation
 | 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
 }}.</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"/> [[Adrien Douady]] phrases this principle as:

:{{quote|Plough in the dynamical plane, and harvest in parameter space.}}

==Geometry==
<!--[[Douady rabbit]] links directly here.-->
[[File:Unrolled main cardioid of Mandelbrot set for periods 8-14.png|thumbnail|right|Components on main cardioid for periods 8–14 with antennae 7–13]]

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. 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 ''[[Jean-Christophe Yoccoz|Yoccoz-inequality]]'', which states that the size tends to zero like <math>\tfrac{1}{q}</math>.

A period-''q'' limb will have ''q''&nbsp;−&nbsp;1 "antennae" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas.

=== 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 to diverge for z = <math>-\tfrac{3}{4} + i\epsilon</math> (<math>-\tfrac{3}{4}</math> being the location thereof). As the series doesn't diverge for the exact value of z = <math>-\tfrac{3}{4}</math>, the number of iterations required increases with a small ε. It turns out that multiplying the value of ε with the number of iterations required yields an approximation of π that becomes better for smaller ε. For example, for ε = 0.0000001 the number of iterations is 31415928 and the product is 3.1415928.<ref>Gary William Flake, ''The Computational Beauty of Nature'', 1998. p.&nbsp;125. {{isbn|978-0-262-56127-3}}.</ref>

=== Fibonacci sequence in the Mandelbrot set ===
It can be shown that the [[Fibonacci Sequence|Fibonacci sequence]] is located within the Mandelbrot Set and that a relation exists between the main cardioid and the [[Farey sequence|Farey Diagram]]. Upon mapping the main cardioid to a disk, one can notice that 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 Mandelbrot set shows more intricate detail the closer one looks or [[magnification|magnifies]] the image, usually called "zooming in". The following example of an image sequence zooming to a selected ''c'' value gives an impression of the infinite richness of different geometrical structures and explains some of their typical rules.

The magnification of the last image relative to the first one is about 10<sup>10</sup> to 1. Relating to an ordinary monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers. Its border would show an astronomical number of different fractal structures.
{{Clear}}

<gallery mode="packed">
File:Mandel zoom 00 mandelbrot set.jpg|Start. Mandelbrot set with continuously colored environment.
File:Mandel zoom 01 head and shoulder.jpg|Gap between the "head" and the "body", also called the "seahorse valley"
File:Mandel zoom 02 seehorse valley.jpg|Double-spirals on the left, "seahorses" on the right
File: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 kind of 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 so-called [[Misiurewicz point]]. Between the "upper part of the body" and the "tail" a distorted small copy of the Mandelbrot set called satellite may be recognized.

<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. [https://mandelbrot-svelte.netlify.com/#{%22pos%22:{%22x%22:-0.743904874255535,%22y%22:-0.1317119067802009},%22zoom%22:7502494.442311305} Open this location in an interactive viewer.]
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. Several satellites of second order may be recognized.
File:Mandel zoom 09 satellite head and shoulder.jpg|The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.
File:Mandel zoom 10 satellite seehorse valley.jpg|Double-spirals and "seahorses" – unlike the 2nd 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|Detail of one island
File:Mandel zoom 16 spiral island.jpg|Detail of the spiral. [https://guciek.github.io/web_mandelbrot.html#-0.7436439049875745;0.13182591455433018;2.2351741790771485e-13;7000 Open this location in an interactive viewer.]
</gallery>
The islands above seem to consist of infinitely many parts like [[Cantor set]]s, as is{{Clarify|date=May 2010}} actually the case for the corresponding Julia set ''J<sub>c</sub>''. However, 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 ''c'' for the corresponding ''J<sub>c</sub>'' is not that of 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 zoom step.

==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>

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]], which is a natural object to study even when the connectedness locus is not useful.

The [[Multibrot set]] is obtained by varying the value of the exponent ''d''. The [[Multibrot set|article]] has a video that shows the development from ''d'' = 0 to 7, at which point there are 6 i.e. (''d'' − 1) lobes around the perimeter. A similar development with negative exponents results in (1 − ''d'') clefts on the inside of a ring.

===Higher dimensions===
There is no perfect extension of the Mandelbrot set into 3D. This is because there is no 3D analogue of the complex numbers for it to iterate on. However, there is an extension of the complex numbers into 4 dimensions, called the [[quaternion]]s, that creates a perfect extension of the Mandelbrot set and the Julia sets into 4 dimensions.<ref>http://archive.bridgesmathart.org/2010/bridges2010-247.pdf retrieved August 19, 2018</ref> These can then be either [[cross section (geometry)|cross-sectioned]] or [[Projection mapping|projected]] into a 3D structure.

===Other, non-analytic, mappings===
[[File:Mandelbar fractal from XaoS.PNG|left|thumb|Image of the Tricorn / Mandelbar fractal]]
[[File:BurningShip01.png|thumb|Image of the burning ship fractal]]

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

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

The tricorn (also sometimes called the ''Mandelbar'') was encountered by [[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==
[[File:Fractal-zoom-1-03-Mandelbrot Buzzsaw.png|right|thumbnail|Still image of [//upload.wikimedia.org/wikipedia/commons/0/07/Fractal-zoom-1-03-Mandelbrot_Buzzsaw.ogv a movie of increasing magnification] on 0.001643721971153 − 0.822467633298876''i'']]
[[File:Mandelbrot sequence new still.png|right|thumbnail|Still image of [//upload.wikimedia.org/wikipedia/commons/5/51/Mandelbrot_sequence_new.webm an animation of increasing magnification]]]
There are many programs used to generate the Mandelbrot set and other fractals, some of which are described in [[fractal-generating software]]. These programs use a variety of algorithms to determine the color of individual pixels and achieve efficient computation.

===Escape time algorithm===
<!--[[Spatial anti-aliasing]] links directly here.-->
The simplest algorithm for generating a representation of the Mandelbrot set is known as 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. For some starting values, escape occurs quickly, after only a small number of iterations. For starting values very close to but not in the set, it may take hundreds or thousands of iterations to escape. For values within the Mandelbrot set, escape will never occur. The programmer or user must choose how many iterations–or how much "depth"–they wish to examine. The higher the maximal number of iterations, the more detail and subtlety emerge in the final image, but the longer time it will take to calculate the fractal image.

Escape conditions can be simple or complex. Because no complex number with a real or imaginary part greater than 2 can be part of the set, a common bailout is to escape when either coefficient exceeds 2. A more computationally complex method that detects escapes sooner, is to compute distance from the origin using the [[Pythagorean theorem]], i.e., to determine the [[absolute value]], or ''modulus'', of the complex number. If this value exceeds 2, or equivalently, when the sum of the squares of the real and imaginary parts exceed 4, the point has reached escape. More computationally intensive rendering variations include the [[Buddhabrot]] method, which finds escaping points and plots their iterated coordinates.

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. We now iterate the critical point 0 under <math>P_c</math>, checking at each step whether the orbit point has modulus larger than 2. When this is the case, we know that <math>c</math> does not belong to the Mandelbrot set, and we color our pixel according to the number of iterations used to find out. Otherwise, we 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>,&nbsp;''y''<sub>0</sub>), not (''x'',''y''). -->
<pre>For each pixel (Px, Py) on the screen, do:
{
  x0 = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
  y0 = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))
  x = 0.0
  y = 0.0
  iteration = 0
  max_iteration = 1000
  while (x*x + y*y <= 2*2  AND  iteration < max_iteration) {
    xtemp = x*x - y*y + x0
    y = 2*x*y + y0
    x = xtemp
    iteration = iteration + 1
  }
  color = palette[iteration]
  plot(Px, Py, color)
}
</pre>
Here, relating the pseudocode to <math>c</math>, <math>z</math> and <math>P_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>

The above code uses an unoptimized inner while loop for clarity. In the unoptimized version, one must perform five multiplications per iteration. To reduce the number of multiplications the following code for the inner while loop may be used instead :

<pre>
rsquare = 0
isquare = 0
zsquare = 0

while (rsquare + isquare <= 4  AND  iteration < max_iteration) {
  x = rsquare - isquare + x0
  y = zsquare - rsquare - isquare + y0
  rsquare = x*x
  isquare = y*y
  zsquare = (x + y)*(x + y)
  iteration = iteration + 1
}
</pre>

The above code works via some algebraic simplification of the complex multiplication:

* <math>(iy + x)^2 = -y^2 + 2iyx + x^2\ </math>
* <math>= x^2 - y^2 + 2iyx\ </math>

Using the above identity, the number of multiplications can be reduced to three instead of five.

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.). One practical way, without slowing down calculations, is to use the number of executed iterations as an entry to a [[Lookup table|look-up color palette table]] initialized at startup. If the color table has, for instance, 500 entries, then the color selection is ''n''&nbsp;mod&nbsp;500, where ''n'' is the number of iterations.

====Histogram coloring====
<!-- {{Unreferenced section|date=October 2018}} -->
{{More citations needed section|talk="Histogram coloring" section|date=June 2019}}
A more complex coloring method involves using a [[histogram]] which pairs each pixel with said pixel's maximum iteration count before escape / bailout <!--, from 1 to ''n''-->. This method will equally distribute colors to the same overall area, and, importantly, is independent of the maximum number of iterations chosen.<ref>{{cite web |url=http://www.fractalforums.com/programming/newbie-how-to-map-colors-in-the-mandelbrot-set/msg3465/#msg3465 |title=Newbie: How to map colors in the Mandelbrot set? |author= |date=May 2007 |website=www.fractalforums.com |publisher= |access-date=June 2019 |quote=}}</ref>

This algorithm has four passes. The first pass involves calculating the iteration counts associated with each pixel (but without any pixels being plotted). These are stored in an array which we'll call IterationCounts[x][y], where x and y are the x and y coordinates of said pixel on the screen respectively.

<small>
{| style="{{float right}} width: 30%; border: 1px solid black; text-align: center;"
|-
|[[File:Mandelbrot-no-histogram-coloring-10000-iterations.png|thumb|upright=0.65|center]]
|[[File:Mandelbrot-no-histogram-coloring-1000-iterations.png|thumb|upright=0.65|center]]
|[[File:Mandelbrot-no-histogram-coloring-100-iterations.png|thumb|upright=0.65|center]]
|-
|[[File:Mandelbrot-histogram-10000-iterations.png|thumb|upright=0.65|center]]
|[[File:Mandelbrot-histogram-1000-iterations.png|thumb|upright=0.65|center]]
|[[File:Mandelbrot-histogram-100-iterations.png|thumb|upright=0.65|center]]
|-
| rowspan=2 colspan=3 style="text-align: center;" |The top row is a series of plots using the escape time algorithm for 10000, 1000 and 100 maximum iterations per pixel respectively. The bottom row uses the same maximum iteration values but utilizes the histogram coloring method. Notice how little the coloring changes per different maximum iteration counts for the histogram coloring method plots.
|}
</small>

The first step of the second pass is to create an array of size ''n'', which is the maximum iteration count. We'll call that array NumIterationsPerPixel<!--[''i'']--> . Next, one must iterate over the array of pixel-iteration count pairs, IterationCounts[][], and retrieve each pixel's saved iteration count, ''i'', via eg. ''i'' = IterationCounts[x][y]. After each pixel's iteration count ''i'' is retrieved, it is necessary to index the NumIterationsPerPixel by ''i'' and increment the indexed value (which is initially zero) -- eg. NumIterationsPerPixel[''i''] = NumIterationsPerPixel[''i''] + 1 .<!-- This creates the histogram during computation of the image. -->

<pre>
for (x = 0; x < width; x++) {
    for (y = 0; y < height; y++) {
        i = IterationCounts[x][y];
        NumIterationsPerPixel[i]++;
    }
}
</pre>

<!--The third pass "renders" each pixel by using the accumulated histogram data--><!-- (Also explained further in the following code sample)-->The third pass iterates through the NumIterationsPerPixel array and adds up all the stored values, saving them in ''total''. The array index represents the number of pixels that reached that iteration count before bailout. <!-- After this step has been completed, the ''total'' value is stored, which is used later. --><!-- Next, each pixel is consulted to determine its maximum iteration count.--><!-- as shown in the following code. -->

<pre>
total = 0;
for (i = 0; i < max_iterations; i++) {
   total += NumIterationsPerPixel[i];
}
</pre>

After this, the fourth pass begins and all the values in the IterationCounts array are indexed, and, for each iteration count ''i'', associated with each pixel, the count is added to a global sum of all the iteration counts from 1 to ''i'' in the NumIterationsPerPixel array. <!--and summed, from 1 to ''i'', up to and including the maximum iteration count of the associated pixel, ''i''--><!--, as shown in the following code-->. This value is then normalized by dividing the sum by the ''total'' value computed earlier.

<pre>
hue[][] = 0.0;
for (x = 0; x < width; x++) {
    for (y = 0; y < height; y++) {
        iteration = IterationCounts[x][y];
        for (i = 0; i <= iteration; i++) {
            hue[x][y] += NumIterationsPerPixel[i] / total; /* Must be floating-point division. */
        }
    }
}

...

color = palette[hue[m,n]]

...

</pre>

Finally, the computed value is used, eg. as an index to a color palette.

If one has a continuous color palette ranging from 0 to 1 for each color component, then the normalized color of each pixel can be computed as follows, using the variables from above.

This method may be combined with the smooth coloring method below for more aesthetically pleasing images.

====Continuous (smooth) coloring====
{{multiple image| direction = vertical
| width = 180
| image1 = Escape Time Algorithm bands.png
| caption1 = This image was rendered with the escape time algorithm. There are very obvious "bands" of color
| image2 = Normalized Iteration Count Algorithm 1.png
| caption2 = This image was rendered with the normalized iteration count algorithm. The bands of color have been replaced by a smooth gradient. Also, the colors take on the same pattern that would be observed if the escape time algorithm were used.
}}

The escape time algorithm is popular for its simplicity. However, it creates bands of color, which, as a type of [[aliasing]], can detract from an image's aesthetic value. This can be improved using an algorithm known as "normalized iteration count",<ref>{{cite journal|last=García|first=Francisco|author2=Ángel Fernández |author3=Javier Barrallo |author4=Luis Martín |title=Coloring Dynamical Systems in the Complex Plane|url=http://math.unipa.it/~grim/Jbarrallo.PDF|accessdate=2008-01-21}}</ref><ref>{{cite web|url=http://linas.org/art-gallery/escape/escape.html|title=Renormalizing the Mandelbrot Escape|author=Linas Vepstas}}</ref> which provides a smooth transition of colors between iterations. The algorithm associates a real number <math>\nu</math> with each value of ''z'' by using the connection of the iteration number with the [[Julia set|potential function]]. This function is given by

: <math>\phi(z) = \lim_{n \to \infty} (\log|z_n|/P^{n}),</math>

where ''z''<sub>''n''</sub> is the value after ''n'' iterations and ''P'' is the power for which ''z'' is raised to in the Mandelbrot set equation (''z''<sub>''n''+1</sub>&nbsp;=&nbsp;''z''<sub>''n''</sub><sup>''P''</sup>&nbsp;+&nbsp;''c'', ''P'' is generally 2).

If we choose a large bailout radius ''N'' (e.g., 10<sup>100</sup>), we have that

: <math>\log|z_n|/P^{n} = \log(N)/P^{\nu(z)}</math>

for some real number <math>\nu(z)</math>, and this is

: <math>\nu(z) = n - \log_P (\log|z_n|/\log(N)),</math>

and as ''n'' is the first iteration number such that |''z''<sub>''n''</sub>| > ''N'', the number we subtract from ''n'' is in the interval <nowiki>[0,&nbsp;1)</nowiki>.

For the coloring we must have a cyclic scale of colors (constructed mathematically, for instance) and containing ''H'' colors numbered from 0 to ''H''&nbsp;−&nbsp;1 (''H'' =&nbsp;500, for instance). We multiply the real number <math>\nu(z)</math> by a fixed real number determining the density of the colors in the picture, take the integral part of this number modulo&nbsp;''H'', and use it to look up the corresponding color in the color table.

For example, modifying the above pseudocode and also using the concept of [[linear interpolation]] would yield
<pre>For each pixel (Px, Py) on the screen, do:
{
  x0 = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
  y0 = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))
  x = 0.0
  y = 0.0
  iteration = 0
  max_iteration = 1000
  // Here N=2^8 is chosen as a reasonable bailout radius.
  while ( x*x + y*y <= (1 << 16)  AND  iteration < max_iteration ) {
    xtemp = x*x - y*y + x0
    y = 2*x*y + y0
    x = xtemp
    iteration = iteration + 1
  }
  // Used to avoid floating point issues with points inside the set.
  if ( iteration < max_iteration ) {
    // sqrt of inner term removed using log simplification rules.
    log_zn = log( x*x + y*y ) / 2
    nu = log( log_zn / log(2) ) / log(2)
    // Rearranging the potential function.
    // Dividing log_zn by log(2) instead of log(N = 1<<8)
    // because we want the entire palette to range from the
    // center to radius 2, NOT our bailout radius.
    iteration = iteration + 1 - nu
  }
  color1 = palette[floor(iteration)]
  color2 = palette[floor(iteration) + 1]
  // iteration % 1 = fractional part of iteration.
  color = linear_interpolate(color1, color2, iteration % 1)
  plot(Px, Py, color)
}
</pre>

===Distance estimates===
One can compute the [[distance]] from point ''c'' (in [[exterior (topology)|exterior]] or [[interior (topology)|interior]]) to nearest point on the [[boundary (topology)|boundary]] of the Mandelbrot set.<ref>{{cite web | url=http://www.moleculardensity.net/buddhabrot/appendix/2 | title=Interior and exterior distance bounds for the Mandelbrot | author=Albert Lobo Cusidó}}</ref>

====Exterior distance estimation====
The proof of the [[Mandelbrot set#Basic properties|connectedness]] of the Mandelbrot set in fact gives a formula for the [[Uniform isomorphism|uniformizing map]] of the [[complement (set theory)|complement]] of <math>M</math> (and the [[derivative]] of this map). By the [[Koebe quarter theorem]], one can then estimate the distance between the midpoint of our [[pixel]] and the Mandelbrot set up to a factor of 4.

In other words, provided that the maximal number of iterations is sufficiently high, one obtains a picture of the Mandelbrot set with the following properties:

# Every pixel that contains a point of the Mandelbrot set is colored black.
# Every pixel that is colored black is close to the Mandelbrot set.

[[File:Demm 2000 Mandelbrot set.jpg|thumb|Exterior distance estimate may be used to color whole complement of Mandelbrot set]]
The distance estimate ''b'' of a pixel ''c'' (a complex number) from the Mandelbrot set is given by

: <math>b=\lim_{n \to \infty} 2 \cdot \frac{|{P_c^n(c)| \cdot \ln|{P_c^n(c)}}|}{|\frac{\partial}{\partial{c}} P_c^n(c)|},</math>
where 
* <math>P_c(z) \,</math> stands for [[complex quadratic polynomial]]
* <math>P_c^n(c)</math> stands for ''n'' iterations of <math>P_c(z) \to z</math> or <math>z^2 + c \to z</math>, starting with <math>z=c</math>:<!--
 --> <math>P_c^{ 0}(c) = c</math>,<!--
 --> <math>P_c^{ n+1}(c) = P_c^n(c)^2 + c</math>;
* <math>\frac{\partial}{\partial{c}} P_c^n(c)</math> is the derivative of<!--
 --> <math>P_c^n(c)</math> with respect to c. This derivative can be found by starting with<!--
 --> <math>\frac{\partial}{\partial{c}} P_c^{ 0}(c) = 1</math><!--
 --> and then <math>\frac{\partial}{\partial{c}} P_c^{ n+1}(c) = 2\cdot{}P_c^n(c)\cdot\frac{\partial}{\partial{c}} P_c^n(c) + 1</math>.<!--
 --> This can easily be verified by using the chain rule for the derivative.

The idea behind this formula is simple: When the [[equipotential]] lines for the potential function <math>\phi(z)</math> lie close, the number <math>|\phi'(z)|</math> is large, and conversely, therefore the equipotential lines for the function <math>\phi(z)/|\phi'(z)|</math> should lie approximately regularly.

From a mathematician's point of view, this formula only works in limit where ''n'' goes to infinity, but very reasonable estimates can be found with just a few additional iterations after the main loop exits.

Once ''b'' is found, by the Koebe 1/4-theorem, we know that there is no point of the Mandelbrot set with distance from ''c'' smaller than ''b/4''.

The distance estimation can be used for drawing of the boundary of the Mandelbrot set, see the article [[Julia set]].

====Interior distance estimation====
[[File:Mandelbrot Interior 600.png|thumb|Pixels colored according to the estimated interior distance]]
It is also possible to estimate the distance of a limitly periodic (i.e., inner) point to the boundary of the Mandelbrot set. The estimate is given by

: <math>b=\frac{1-\left|{\frac{\partial}{\partial{z}}P_c^p(z_0)}\right|^2}
  {\left|{\frac{\partial}{\partial{c}}\frac{\partial}{\partial{z}}P_c^p(z_0) +
        \frac{\partial}{\partial{z}}\frac{\partial}{\partial{z}}P_c^p(z_0)
        \frac{\frac{\partial}{\partial{c}}P_c^p(z_0)}
             {1-\frac{\partial}{\partial{z}}P_c^p(z_0)}} \right|},</math>
where
* <math>p</math> is the period,
* <math>c</math> is the point to be estimated,
* <math>P_c(z)</math> is the [[complex quadratic polynomial]] <math>P_c(z)=z^2 + c</math>
* <math>P_c^p(z_0)</math> is the <math>p</math>-fold iteration of <math>P_c(z) \to z</math>, starting with <math>P_c^{ 0}(z) = z_0</math>
* <math>z_0</math> is any of the <math>p</math> points that make the [[attractor]] of the iterations of <math>P_c(z) \to z</math> starting with<!--
 --> <math>P_c^{ 0}(z) = c</math>; <math>z_0</math> satisfies <math>z_0 = P_c^p(z_0)</math>,
* <math>\frac{\partial}{\partial{c}}\frac{\partial}{\partial{z}}P_c^p(z_0)</math>,<!--
 --> <math>\frac{\partial}{\partial{z}}\frac{\partial}{\partial{z}}P_c^p(z_0)</math>,<!--
 --> <math>\frac{\partial}{\partial{c}}P_c^p(z_0)</math> and<!--
 --> <math>\frac{\partial}{\partial{z}}P_c^p(z_0)</math> are various derivatives of<!--
 --> <math>P_c^p(z)</math>, evaluated at <math>z_0</math>.

Analogous to the exterior case, once ''b'' is found, we know that all points within the distance of ''b''/4 from ''c'' are inside the Mandelbrot set.

There are two practical problems with the interior distance estimate: first, we need to find <math>z_0</math> precisely, and second, we need to find <math>p</math> precisely.
The problem with <math>z_0</math> is that the convergence to <math>z_0</math> by iterating <math>P_c(z)</math> requires, theoretically, an infinite number of operations.
The problem with any given <math>p</math> is that, sometimes, due to rounding errors, a period is falsely identified to be an integer multiple of the real period (e.g., a period of 86 is detected, while the real period is only 43=86/2). In such case, the distance is overestimated, i.e., the reported radius could contain points outside the Mandelbrot set.
[[File:MandelbrotOrbitInfimum.png|thumb|3D view: smallest absolute value of the orbit of the interior points of the Mandelbrot set]]

===Optimizations===

====Cardioid / bulb checking====
One way to improve calculations is to find out beforehand whether the given point lies within the cardioid or in the period-2 bulb. Before passing the complex value through the escape time algorithm, first check that:

:<math> p = \sqrt{  \left(x - \frac{1}{4}\right)^2 + y^2} </math>,
:<math> x \leq p - 2p^2 + \frac{1}{4} </math>,
:<math> (x+1)^2 + y^2 \leq \frac{1}{16} </math>,

where ''x'' represents the real value of the point and ''y'' the imaginary value. The first two equations determine that the point is within the cardioid, the last the period-2 bulb.

The cardioid test can equivalently be performed without the square root:

:<math> q = \left(x - \frac{1}{4}\right)^2 + y^2, </math>
:<math> q \left(q + \left(x - \frac{1}{4}\right)\right) \leq \frac{1}{4}y^2. </math>

3rd- and higher-order buds do not have equivalent tests, because they are not perfectly circular.<ref>
{{cite web
|url=http://linas.org/art-gallery/bud/bud.html
|title=Mandelbrot Bud Maths}}</ref> However, it is possible to find whether the points are within circles inscribed within these higher-order bulbs, preventing many, though not all, of the points in the bulb from being iterated.

====Periodicity checking====
To prevent having to do huge numbers of iterations for points in the set, one can perform periodicity checking. Check whether a point reached in iterating a pixel has been reached before. If so, the pixel cannot diverge and must be in the set.

Periodicity checking is, of course, a trade-off. The need to remember points costs memory and ''data management'' instructions, whereas it saves ''computational'' instructions.

However, checking against only one previous iteration can detect many periods with little performance overhead. For example, within the while loop of the pseudocode above, make the following modifications.
<pre>
  while (x*x + y*y <= 2*2  AND  iteration < max_iteration) {
    xtemp = x*x - y*y + x0
    ytemp = 2*x*y + y0
    if (x == xtemp  AND  y == ytemp) {  // do not work, because 1) only the body has a cycle period of 1, 2) end cycles will be reached after millions or billions of iterations, 3) rounding errors end up in longer final cycles
      iteration = max_iteration
      break
    }
    x = xtemp
    y = ytemp
    iteration = iteration + 1
  }
</pre>

====Border tracing / edge checking====
[[File:Mandelbrot DEM Sobel.png|thumbnail|right|Edge detection using Sobel filter of hyperbolic components of Mandelbrot set]]
It can be shown that if a solid shape can be drawn on the Mandelbrot set, with all the border colors being the same, then the shape can be filled in with that color. This is a result of the Mandelbrot set being simply connected. Boundary tracing works by following the [[polynomial lemniscate|lemniscates]] of the various iteration levels (colored bands) all around the set, and then filling the entire band at once. This can be a good speed increase, because it means that large numbers of points can be skipped.<ref>{{cite web|url=http://www.reocities.com/CapeCanaveral/5003/mandel.htm|title=Boundary Tracing Method|archiveurl=https://web.archive.org/web/20150220012221/http://www.reocities.com/CapeCanaveral/5003/mandel.htm|archivedate=2015-02-20}}</ref>

A similar method operating on the same principle uses rectangles instead of arbitrary border shapes. It is usually faster than boundary tracing because it requires fewer calculations to work out the rectangle. It is inefficient, however, because boundaries are not rectangular, and so some areas can be missed. This issue can be minimized by creating a recursive algorithm that, if a rectangle border fails, will subdivide it into four smaller rectangles and test those, and either fill each or subdivide again and repeat the process.

However, this only works using discrete colors in the escape time algorithm. It will not work for smooth/continuous coloring.

==== Symmetry utilization ====
The horizontal symmetry of the Mandelbrot set allows for portions of the rendering process to be skipped upon the presence of the real axis in the final image. However, regardless of the portion that gets mirrored, the same number of points will be rendered.

==== Advanced bailout method ====
Simple programs and scripts generally tend to set the escape value to two. This process can be improved by utilizing the distance from the origin and the point being rendered with the Pythagorean theorem by summing the squares of the real and imaginary portions of <math>z</math> and then escaping if the value is larger or equal to four. The result of this optimization is a faster rendering of the image.

====Perturbation theory and series approximation====
Very highly magnified images require more than the standard 64–128 or so bits of precision that most hardware [[floating-point unit]]s provide, requiring renderers to use slow "BigNum" or "[[Arbitrary-precision arithmetic|arbitrary-precision]]" math libraries to calculate. However, this can be sped up by the exploitation of [[perturbation theory]]. Given

:<math> z_{n+1} = z_n^2 + c </math>

as the iteration, and a small epsilon and delta, it is the case that

:<math> (z_n + \epsilon)^2 + (c + \delta) = z_n^2 + 2z_n\epsilon + \epsilon^2 + c + \delta, </math>

or

:<math> z_{n+1} + 2z_n\epsilon + \epsilon^2 + \delta, </math>

so if one defines

:<math> \epsilon_{n+1} = 2z_n\epsilon_n + \epsilon_n^2 + \delta, </math>

one can calculate a single point (e.g. the center of an image) using high-precision arithmetic (''z''), giving a ''reference orbit'', and then compute many points around it in terms of various initial offsets delta plus the above iteration for epsilon, where epsilon-zero is set to 0. For most iterations, epsilon does not need more than 16 significant figures, and consequently hardware floating-point may be used to get a mostly accurate image.<ref>
{{cite web
|url=http://www.fractalforums.com/announcements-and-news/superfractalthing-arbitrary-precision-mandelbrot-set-rendering-in-java/
|title=Superfractalthing - Arbitrary Precision Mandelbrot Set Rendering in Java}}</ref> There will often be some areas where the orbits of points diverge enough from the reference orbit that extra precision is needed on those points, or else additional local high-precision-calculated reference orbits are needed. By measuring the orbit distance between the reference point and the point calculated with low precision, it can be detected that it is not possible to calculate the point correctly, and the calculation can be stopped. These incorrect points can later be re-calculated e.g. from another closer reference point.

Further, it is possible to approximate the starting values for the low-precision points with a truncated [[Taylor series]], which often enables a significant amount of iterations to be skipped.<ref>
{{cite journal
|author=K. I. Martin
|url=http://www.superfractalthing.co.nf/sft_maths.pdf
|title=Superfractalthing Maths}}</ref>
Renderers implementing these techniques are [[Kalles Fraktaler|publicly available]] and offer speedups for highly magnified images by around two orders of magnitude.<ref>
{{cite web
|url=http://www.chillheimer.de/kallesfraktaler/
|title=Kalles Fraktaler 2}}</ref>

An alternate explanation of the above:

For the central point in the disc <math> c </math> and its iterations <math> z_n </math>, and an arbitrary point in the disc <math> c + \delta </math> and its iterations <math> z'_n </math>, it is possible to define the following iterative relationship:

:<math> z'_{n} = z_{n} + \epsilon_{n} </math>

With <math> \epsilon_{1} = \delta </math>. Successive iterations of <math> \epsilon_n </math> can be found using the following:

:<math> z'_{n+1} = {z'_n}^2 + (c + \delta) </math>

:<math> z'_{n+1} = (z_n + \epsilon_n)^2 + c + \delta </math>

:<math> z'_{n+1} = {z_n}^2 + c + 2z_n\epsilon_n + {\epsilon_n}^2 + \delta</math>

:<math> z'_{n+1} = z_{n+1} + 2z_n\epsilon_n + {\epsilon_n}^2 + \delta</math>

Now from the original definition:
:<math> z'_{n+1} = z_{n+1} + \epsilon_{n+1} </math>,

It follows that:

:<math> \epsilon_{n+1} = 2z_n\epsilon_n + {\epsilon_n}^2 + \delta </math>

As the iterative relationship relates an arbitrary point to the central point by a very small change <math> \delta </math>, then most of the iterations of <math> \epsilon_n </math> are also small and can be calculated using floating point hardware.

However, for every arbitrary point in the disc it is possible to calculate a value for a given <math> \epsilon_{n} </math> without having to iterate through the sequence from <math> \epsilon_0 </math>, by expressing <math> \epsilon_n </math> as a power series of <math> \delta </math>.

:<math> \epsilon_n = A_{n}\delta + B_{n}\delta^2 + C_{n}\delta^3 + \dotsc </math>

With <math> A_{1} = 1, B_{1} = 0, C_{1} = 0, \dotsc </math>.

Now given the iteration equation of <math> \epsilon </math>, it is possible to calculate the coefficients of the power series for each <math> \epsilon_n </math>:

:<math> \epsilon_{n+1} = 2z_n\epsilon_n + {\epsilon_n}^2 + \delta </math>

:<math> \epsilon_{n+1} = 2z_n(A_n\delta + B_n\delta^2 + C_n\delta^3 + \dotsc) + (A_n\delta + B_n\delta^2 + C_n\delta^3 + \dotsc)^2 + \delta </math>

:<math> \epsilon_{n+1} = (2z_nA_n+1)\delta + (2z_nB_n + {A_n}^2)\delta^2 + (2z_nC_n + 2A_nB_n)\delta^3  + \dotsc </math>

Therefore it follows that:

:<math> A_{n+1} = 2z_nA_n + 1 </math>
:<math> B_{n+1} = 2z_nB_n + {A_n}^2 </math>
:<math> C_{n+1} = 2z_nC_n + 2A_nB_n </math>
:<math> \vdots </math>

The coefficients in the power series can be calculated as iterative series using only values from the central point's iterations <math> z </math>, and do not change for any arbitrary point in the disc. If <math> \delta </math> is very small, <math> \epsilon_n </math> should be calculable to sufficient accuracy using only a few terms of the power series. As the Mandelbrot Escape Contours are 'continuous' over the complex plane, if a points escape time has been calculated, then the escape time of that points neighbours should be similar. Interpolation of the neighbouring points should provide a good estimation of where to start in the <math> \epsilon_n </math> series.

Further, separate interpolation of both real axis points and imaginary axis points should provide both an upper and lower bound for the point being calculated. If both results are the same (i.e. both escape or dot not escape) then the difference <math> \Delta n </math> can be used to recuse until both an upper and lower bound can be established. If floating point hardware can be used to iterate the <math> \epsilon </math> series, then there exists a relation between how many iterations can be achieved in the time it takes to use BigNum software to compute a given <math> \epsilon_n </math>. If the difference between the bounds is greater than the number of iterations, it is possible to perform binomial search using BigNum software, successively halving the gap until it becomes more time efficient to find the escape value using floating point hardware.

==Syntax examples==
===Processing code===
<syntaxhighlight lang="java">
// There are innumerable ways to color a Mandelbrot Fractal, this is one.

void setup() {
  size(800, 800);
  colorMode(HSB, 1);
}

void draw() {
  loadPixels();
  int maxN = 100;
  for (int x = 0; x < width; x++) {
    for (int y = 0; y < height; y++) {
      float a = map(x, 0, width, -2, 2);
      float b = map(y, 0, height, -2, 2);
      
      float A = a;
      float B = b;
      int n = 0;
      while (n < maxN) {
        float ab = a * a - b * b;
        float bb = 2 * a * b;
        a = ab + A; b = bb + B;
        if (a * a + b * b > 4.0) break;
        n++;
      }
      float saturation = sqrt(float(n) / maxN);
      int pix = x + y * width;
      if (n == maxN) pixels[pix] = color(0);
      else pixels[pix] = color(0.6, saturation, 1);
    }
  }
  updatePixels();
}
</syntaxhighlight>

===P5.JS code===
<syntaxhighlight lang="javascript">
function setup() {
    createCanvas(800,800);
}

function draw() {
    loadPixels();
    for (let x = 0; x < width; x++) {
        for (let y = 0; y < height; y++) {
            let a = map(x, 0, width, -2, 2),
                b = map(y, 0, height, -2, 2),
                A = a,
                B = b;

            let maxN = 1000;
            for (var n = 0; n < maxN; n++) {
                let ab = a * a - b * b,
                    bb = 2 * a * b;
                a = ab + A, b = bb + B;
                if (a * a + b * b > 4) break;
            }
            let pix = (x + y * width) * 4;
            let brightness = map(sqrt(n / maxN), 0, 1, 0, 255);
            if (n == maxN) brightness = 0;
            pixels[pix] = brightness;
            pixels[pix + 1] = brightness;
            pixels[pix + 2] = brightness * 2;
            pixels[pix + 3] = 255;
        }
    }
    updatePixels();
}
</syntaxhighlight>

===JavaScript code===
<syntaxhighlight lang="javascript">
let map = (n, start1, end1, start2, end2) => {
    return ((n - start1) / (end1 - start1)) * (end2 - start2) + start2;
};
let canv = document.createElement('canvas');
canv.width = 800;
canv.height = 800;
document.body.appendChild(canv);
let ctx = canv.getContext('2d');
let data = ctx.createImageData(canv.width, canv.height);
let pixels = data.data;

for (let x = 0; x < canv.width; x++) {
    for (let y = 0; y < canv.height; y++) {
        let a = map(x, 0, canv.width, -2, 2),
            b = map(y, 0, canv.height, -2, 2),
            A = a,
            B = b;
        
        let maxN = 1000;
        for (var n = 0; n < maxN; n++) {
            let ab = a * a - b * b,
                bb = 2 * a * b;
            a = ab + A, b = bb + B;
            if (a * a + b * b > 4) break;
        }
        let pix = (x + y * canv.width) * 4;
        let brightness = map(Math.sqrt(n / maxN), 0, 1, 0, 255);
        if (n == maxN) brightness = 0;
        pixels[pix] = brightness;
        pixels[pix + 1] = brightness;
        pixels[pix + 2] = brightness * 2; // I like the bluish tint.
        pixels[pix + 3] = 255;
    }
}
ctx.putImageData(data, 0, 0);
</syntaxhighlight>

==References in popular culture==
The Mandelbrot set is considered by many 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 Fratals?"] 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 its discoverer Benoit Mandelbrot.<ref name="JoCopedia">{{cite web|title=Mandelbrot Set|url=http://www.jonathancoulton.com/wiki/Mandelbrot_Set|website=JoCopeda|accessdate=15 January 2015}}</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 [[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>

==See also==

* [[Buddhabrot]]
* [[Collatz fractal]]
* [[Fractint]]
* [[Gilbreath permutation]]
* [[Mandelbox]]
* [[Mandelbulb]]
* [[Menger Sponge]]
* [[Newton fractal]]
* [[Orbit portrait]]
* [[Orbit trap]]
* [[Pickover stalk]]

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

==Further reading==
* [[John W. Milnor]], ''Dynamics in One Complex Variable'' (Third Edition), Annals of Mathematics Studies 160, (Princeton University Press, 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] )
* Nigel Lesmoir-Gordon, ''The Colours of Infinity: The Beauty, The Power and the Sense of Fractals'', {{isbn|1-904555-05-5}} <br />(includes a DVD featuring [[Arthur C. Clarke]] and [[David Gilmour]])
* [[Heinz-Otto Peitgen]], [[Hartmut Jürgens]], [[Dietmar Saupe]], ''Chaos and Fractals: New Frontiers of Science'' (Springer, New York, 1992, 2004), {{isbn|0-387-20229-3}}

==External links==
{{Wikibooks|Fractals }}
{{commons}}
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* {{Curlie|Science/Math/Chaos_and_Fractals|Chaos and Fractals}}
* [http://classes.yale.edu/Fractals/MandelSet/welcome.html The Mandelbrot Set and Julia Sets by Michael Frame, Benoit Mandelbrot, and Nial Neger]
* [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-svelte.netlify.com Mandelbrot set images online rendering]
* [https://github.com/pkulchenko/ZeroBraneEduPack/blob/master/fractal-samples/zplane.lua Fractal calculator written in Lua by Deyan Dobromiroiv, Sofia, Bulgaria]

{{Fractal software}}
{{Fractals}}

{{DEFAULTSORT:Mandelbrot Set}}
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