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创建页面，内容为“{{short description|Fractal named after mathematician Benoit Mandelbrot}} {{Use dmy dates|date=February 2020}} File:Mandel zoom 00 mandelbrot set.jpg|322px|thumb|T…”
{{short description|Fractal named after mathematician Benoit Mandelbrot}}
{{Use dmy dates|date=February 2020}}
[[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.
-- Please don't remove it -->
[[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 is credited to [[Adrien Douady]] who named it 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 – 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. In other words, the boundary of the Mandelbrot set is a ''[[fractal|fractal curve]]''. 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 origin 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|access-date=1 July 2019|archive-url=https://web.archive.org/web/20190728201429/http://www.math.harvard.edu/archive/118r_spring_05/docs/brooksmatelski.pdf|archive-date=28 July 2019|url-status=dead}}</ref> On 1 March 1980, at [[IBM]]'s [[Thomas J. Watson Research Center]] in [[Yorktown Heights, New York|Yorktown Heights]], [[New York (state)|New York]], [[Benoit Mandelbrot]] first 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]] (1985),<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 (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 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 magazine |title= Computer Recreations, August 1985; A computer microscope zooms in for a look at the most complex object in mathematics |last=Dewdney |first=A. K. |year=1985 |magazine=Scientific American |url=https://www.scientificamerican.com/media/inline/blog/File/Dewdney_Mandelbrot.pdf}}</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 =
| date = May–June 1999
| url = http://citeseer.ist.psu.edu/cache/papers/cs/28564/http:zSzzSzwww.math.sunysb.eduzSz~mlyubichzSzlectures.pdf/
| 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
| 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> Thus, a complex number ''c'' is a member 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 for all ''n''>0.
For example, for ''c''=1, the sequence is 0, 1, 2, 5, 26, ..., which tends to [[infinity]], so 1 is not an element of the Mandelbrot set. On the other hand, for ''c''=−1, the sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set.
[[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.]]
The Mandelbrot set can also be defined as the [[connectedness locus]] of a family of [[polynomial]]s.
==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]]]]
[[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 where the set is finite]]
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}}</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. 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, 0) to (−1.31, 0) while the view magnifies from 0.5 × 0.5 to 0.12 × 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 + 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 + 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'' − 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. 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.
===3D images of Mandelbrot and Julia sets===
{{More citations needed section|talk=3D images|date=January 2020}} <!-- Once enough citations have been added to this sectionb then the preceding template can be removed -->
In addition to creating two dimensional images of the Mandelbrot set, various techniques can be used to render Mandelbrot and Julia sets as 3D [[Heightmap]] images, where each pixel in a 2D image is given a height value, and the resulting image is rendered as a 3D graphic.
The simplest approach to 3D rendering uses the iteration value for each pixel as a height value. This produces images with distinct "steps" in the height value.
[[File: Mandelbrot set 3D integer iterations.jpg|thumb|none|Mandelbrot set rendered in 3D using integer iterations]]
If instead you use the fractional iteration value (also known as the potential function<ref>[[Julia set#The potential function and the real iteration number|potential function]])</ref> to calculate the height value for each point, you avoid steps in the resulting image. However, images rendered in 3D using fractional iteration data still look rather bumpy and visually noisy.
[[File: Mandelbrot set 3D fractional iterations.jpg|thumb|none| Mandelbrot set rendered in 3D using fractional iteration values]]
An alternative approach is to use Distance Estimate<ref>{{cite book |title=The Science of Fractal Images |last=Peitgen |first=Heinz-Otto |author2=Saupe Dietmar |year=1988 |publisher=Springer-Verlag |location=New York |isbn=0-387-96608-0 |pages=121, 196–197 |title-link=The Beauty of Fractals }}</ref> (DE) data for each point to calculate a height value. Non-linear mapping of distance estimate value using an exponential function can produce visually pleasing images. Images plotted using DE data are often visually striking, and more importantly, the 3D shape makes it easy to visualize the thin "tendrils" that connect points of the set. Color plates 29 and 30 on page 121 of "The Science of Fractal Images" show a 2D and 3D image plotted using External Distance Estimates.
[[File: Mandelbrot set 3D Distance Estimates.jpg |thumb|none| Mandelbrot set rendered in 3D using Distance Estimates]]
Below is a 3D version of the "Image gallery of a zoom sequence" gallery above, rendered as height maps using Distance Estimate data, and using similar cropping and coloring.
<gallery mode="packed">
File:Mandel zoom 00 mandelbrot set 3D.jpg|[[:File: Mandel zoom 00 mandelbrot set.jpg|Zoom 00]]. Start. Mandelbrot set with continuously colored environment.
File:Mandel zoom 01 head and shoulder 3D.jpg|[[:File: Mandel zoom 01 head and shoulder.jpg|Zoom 01]]. Gap between the "head" and the "body", also called the "seahorse valley"
File:Mandel zoom 02 seehorse valley 3D.jpg|[[:File: Mandel zoom 02 seehorse valley.jpg|Zoom 02]]. Double-spirals on the left, "seahorses" on the right
File:Mandel zoom 03 seehorse 3D.jpg|[[:File: Mandel zoom 03 seehorse.jpg|Zoom 03]]. "Seahorse" upside down
</gallery>
<gallery mode="packed">
File:Mandel zoom 04 seehorse tail 3D.jpg|[[:File: Mandel zoom 04 seehorse tail.jpg|Zoom 04]]. A "seahorse tail".
File:Mandel zoom 05 tail part 3D.jpg|[[:File: Mandel zoom 05 tail part.jpg|Zoom 05]]. Part of the "tail".
File:Mandel zoom 06 double hook 3D.jpg|[[:File: Mandel zoom 06 double hook.jpg|Zoom 06]]. Satellite with twin "Seahorse tails."
File:Mandel zoom 07 satellite 3D.jpg|[[:File: Mandel zoom 07 satellite.jpg|Zoom 07]]. Satellite closeup.
File:Mandel zoom 08 satellite antenna 3D.jpg|[[:File: Mandel zoom 08 satellite antenna.jpg|Zoom 08]]. "Antenna" of the satellite. Several satellites of second order may be recognized.
File:Mandel zoom 09 satellite head and shoulder 3D.jpg|[[:File: Mandel zoom 09 satellite head and shoulder.jpg|Zoom 09]]. The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.
File:Mandel zoom 10 satellite seehorse valley 3D.jpg|[[:File: Mandel zoom 10 satellite seehorse valley.jpg|Zoom 10]]. Double-spirals and "seahorses"
File:Mandel zoom 11 satellite double spiral 3D.jpg|[[:File: Mandel zoom 11 satellite double spiral.jpg|Zoom 11]]. Double-spirals with satellites of second order.
File:Mandel zoom 12 satellite spirally wheel with julia islands 3D.jpg|[[:File: Mandel zoom 12 satellite spirally wheel with julia islands.jpg|Zoom 12]].
File:Mandel zoom 13 satellite seehorse tail with julia island 3D.jpg|[[:File: Mandel zoom 13 satellite seehorse tail with julia island.jpg|Zoom 13]]. Part of the "double-hook"
File:Mandel zoom 14 satellite julia island 3D.jpg|[[:File: Mandel zoom 14 satellite julia island.jpg|Zoom 14]]. Islands
File:Mandel zoom 15 one island 3D.jpg|[[:File: Mandel zoom 15 one island.jpg|Zoom 15]]. Detail of one island
File:Mandel zoom 16 spiral island 3D.jpg|[[:File: Mandel zoom 16 spiral island.jpg|Zoom 16]]. Detail of the spiral.
</gallery>
The image below is similar to "zoom 5", above, but is an attempt to create a 3D version of the image "Map 44" from page 85 of the book "The Beauty of Fractals"<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 |pages=[https://archive.org/details/beautyoffractals0000peit/page/85 85] |title-link=The Beauty of Fractals }}</ref> using a visually similar color scheme that shows the details of the plot in 3D.
[[File:A 3D version of the Mandelbrot set plot "Map 44" from the book "The Beauty of Fractals".jpg|thumb|none|A 3D version of the Mandelbrot set plot "Map 44" from the book "The Beauty of Fractals]]
==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 19 August 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==
{{Main|Plotting algorithms for the Mandelbrot set}}
[[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 and algorithms 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. -->
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".<!-- One of the simplest algorithms used to plot the Mandelbrot set on a computer is called the "escape time algorithm". --><!--[[Spatial anti-aliasing]] links directly here.-->
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]]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>, ''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 (-2.5, 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) '''do'''
xtemp := x×x - y×y + x0
y := 2×x×y + y0
x := xtemp
iteration := iteration + 1
color := palette[iteration]
plot(Px, Py, color)
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>
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 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://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]
{{Fractal software}}
{{Fractals}}
{{DEFAULTSORT:Mandelbrot Set}}
[[Category:Fractals]]
[[Category:Articles containing video clips]]
[[Category:Articles with example pseudocode]]
[[Category:Complex dynamics]]
{{Use dmy dates|date=February 2020}}
[[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.
-- Please don't remove it -->
[[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 is credited to [[Adrien Douady]] who named it 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 – 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. In other words, the boundary of the Mandelbrot set is a ''[[fractal|fractal curve]]''. 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 origin 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|access-date=1 July 2019|archive-url=https://web.archive.org/web/20190728201429/http://www.math.harvard.edu/archive/118r_spring_05/docs/brooksmatelski.pdf|archive-date=28 July 2019|url-status=dead}}</ref> On 1 March 1980, at [[IBM]]'s [[Thomas J. Watson Research Center]] in [[Yorktown Heights, New York|Yorktown Heights]], [[New York (state)|New York]], [[Benoit Mandelbrot]] first 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]] (1985),<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 (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 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 magazine |title= Computer Recreations, August 1985; A computer microscope zooms in for a look at the most complex object in mathematics |last=Dewdney |first=A. K. |year=1985 |magazine=Scientific American |url=https://www.scientificamerican.com/media/inline/blog/File/Dewdney_Mandelbrot.pdf}}</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 =
| date = May–June 1999
| url = http://citeseer.ist.psu.edu/cache/papers/cs/28564/http:zSzzSzwww.math.sunysb.eduzSz~mlyubichzSzlectures.pdf/
| 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
| 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> Thus, a complex number ''c'' is a member 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 for all ''n''>0.
For example, for ''c''=1, the sequence is 0, 1, 2, 5, 26, ..., which tends to [[infinity]], so 1 is not an element of the Mandelbrot set. On the other hand, for ''c''=−1, the sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set.
[[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.]]
The Mandelbrot set can also be defined as the [[connectedness locus]] of a family of [[polynomial]]s.
==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]]]]
[[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 where the set is finite]]
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}}</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. 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, 0) to (−1.31, 0) while the view magnifies from 0.5 × 0.5 to 0.12 × 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 + 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 + 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'' − 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. 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.
===3D images of Mandelbrot and Julia sets===
{{More citations needed section|talk=3D images|date=January 2020}} <!-- Once enough citations have been added to this sectionb then the preceding template can be removed -->
In addition to creating two dimensional images of the Mandelbrot set, various techniques can be used to render Mandelbrot and Julia sets as 3D [[Heightmap]] images, where each pixel in a 2D image is given a height value, and the resulting image is rendered as a 3D graphic.
The simplest approach to 3D rendering uses the iteration value for each pixel as a height value. This produces images with distinct "steps" in the height value.
[[File: Mandelbrot set 3D integer iterations.jpg|thumb|none|Mandelbrot set rendered in 3D using integer iterations]]
If instead you use the fractional iteration value (also known as the potential function<ref>[[Julia set#The potential function and the real iteration number|potential function]])</ref> to calculate the height value for each point, you avoid steps in the resulting image. However, images rendered in 3D using fractional iteration data still look rather bumpy and visually noisy.
[[File: Mandelbrot set 3D fractional iterations.jpg|thumb|none| Mandelbrot set rendered in 3D using fractional iteration values]]
An alternative approach is to use Distance Estimate<ref>{{cite book |title=The Science of Fractal Images |last=Peitgen |first=Heinz-Otto |author2=Saupe Dietmar |year=1988 |publisher=Springer-Verlag |location=New York |isbn=0-387-96608-0 |pages=121, 196–197 |title-link=The Beauty of Fractals }}</ref> (DE) data for each point to calculate a height value. Non-linear mapping of distance estimate value using an exponential function can produce visually pleasing images. Images plotted using DE data are often visually striking, and more importantly, the 3D shape makes it easy to visualize the thin "tendrils" that connect points of the set. Color plates 29 and 30 on page 121 of "The Science of Fractal Images" show a 2D and 3D image plotted using External Distance Estimates.
[[File: Mandelbrot set 3D Distance Estimates.jpg |thumb|none| Mandelbrot set rendered in 3D using Distance Estimates]]
Below is a 3D version of the "Image gallery of a zoom sequence" gallery above, rendered as height maps using Distance Estimate data, and using similar cropping and coloring.
<gallery mode="packed">
File:Mandel zoom 00 mandelbrot set 3D.jpg|[[:File: Mandel zoom 00 mandelbrot set.jpg|Zoom 00]]. Start. Mandelbrot set with continuously colored environment.
File:Mandel zoom 01 head and shoulder 3D.jpg|[[:File: Mandel zoom 01 head and shoulder.jpg|Zoom 01]]. Gap between the "head" and the "body", also called the "seahorse valley"
File:Mandel zoom 02 seehorse valley 3D.jpg|[[:File: Mandel zoom 02 seehorse valley.jpg|Zoom 02]]. Double-spirals on the left, "seahorses" on the right
File:Mandel zoom 03 seehorse 3D.jpg|[[:File: Mandel zoom 03 seehorse.jpg|Zoom 03]]. "Seahorse" upside down
</gallery>
<gallery mode="packed">
File:Mandel zoom 04 seehorse tail 3D.jpg|[[:File: Mandel zoom 04 seehorse tail.jpg|Zoom 04]]. A "seahorse tail".
File:Mandel zoom 05 tail part 3D.jpg|[[:File: Mandel zoom 05 tail part.jpg|Zoom 05]]. Part of the "tail".
File:Mandel zoom 06 double hook 3D.jpg|[[:File: Mandel zoom 06 double hook.jpg|Zoom 06]]. Satellite with twin "Seahorse tails."
File:Mandel zoom 07 satellite 3D.jpg|[[:File: Mandel zoom 07 satellite.jpg|Zoom 07]]. Satellite closeup.
File:Mandel zoom 08 satellite antenna 3D.jpg|[[:File: Mandel zoom 08 satellite antenna.jpg|Zoom 08]]. "Antenna" of the satellite. Several satellites of second order may be recognized.
File:Mandel zoom 09 satellite head and shoulder 3D.jpg|[[:File: Mandel zoom 09 satellite head and shoulder.jpg|Zoom 09]]. The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.
File:Mandel zoom 10 satellite seehorse valley 3D.jpg|[[:File: Mandel zoom 10 satellite seehorse valley.jpg|Zoom 10]]. Double-spirals and "seahorses"
File:Mandel zoom 11 satellite double spiral 3D.jpg|[[:File: Mandel zoom 11 satellite double spiral.jpg|Zoom 11]]. Double-spirals with satellites of second order.
File:Mandel zoom 12 satellite spirally wheel with julia islands 3D.jpg|[[:File: Mandel zoom 12 satellite spirally wheel with julia islands.jpg|Zoom 12]].
File:Mandel zoom 13 satellite seehorse tail with julia island 3D.jpg|[[:File: Mandel zoom 13 satellite seehorse tail with julia island.jpg|Zoom 13]]. Part of the "double-hook"
File:Mandel zoom 14 satellite julia island 3D.jpg|[[:File: Mandel zoom 14 satellite julia island.jpg|Zoom 14]]. Islands
File:Mandel zoom 15 one island 3D.jpg|[[:File: Mandel zoom 15 one island.jpg|Zoom 15]]. Detail of one island
File:Mandel zoom 16 spiral island 3D.jpg|[[:File: Mandel zoom 16 spiral island.jpg|Zoom 16]]. Detail of the spiral.
</gallery>
The image below is similar to "zoom 5", above, but is an attempt to create a 3D version of the image "Map 44" from page 85 of the book "The Beauty of Fractals"<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 |pages=[https://archive.org/details/beautyoffractals0000peit/page/85 85] |title-link=The Beauty of Fractals }}</ref> using a visually similar color scheme that shows the details of the plot in 3D.
[[File:A 3D version of the Mandelbrot set plot "Map 44" from the book "The Beauty of Fractals".jpg|thumb|none|A 3D version of the Mandelbrot set plot "Map 44" from the book "The Beauty of Fractals]]
==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 19 August 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==
{{Main|Plotting algorithms for the Mandelbrot set}}
[[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 and algorithms 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. -->
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".<!-- One of the simplest algorithms used to plot the Mandelbrot set on a computer is called the "escape time algorithm". --><!--[[Spatial anti-aliasing]] links directly here.-->
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]]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>, ''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 (-2.5, 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) '''do'''
xtemp := x×x - y×y + x0
y := 2×x×y + y0
x := xtemp
iteration := iteration + 1
color := palette[iteration]
plot(Px, Py, color)
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>
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 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://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]
{{Fractal software}}
{{Fractals}}
{{DEFAULTSORT:Mandelbrot Set}}
[[Category:Fractals]]
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[[Category:Complex dynamics]]