Mandelbrot set#Distance estimates

File:Mandel.png and Peter Matelski in 1978]]

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

Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980.{{cite journal |first=Benoit |last=Mandelbrot |title=Fractal aspects of the iteration of $z\backslash mapsto\backslash lambda\; z(1-z)$ for complex $\backslash lambda,\; z$ |journal=Annals of the New York Academy of Sciences |volume=357 |issue=1 |pages=249–259 |year=1980 |doi=10.1111/j.1749-6632.1980.tb29690.x |s2cid=85237669 }} The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard (1985),Adrien Douady and John H. Hubbard, Etude dynamique des polynômes complexes, Prépublications mathémathiques d'Orsay 2/4 (1984 / 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),{{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 }} and an internationally touring exhibit of the German Goethe-Institut (1985).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.{{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 }}

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

The work of Douady and Hubbard occurred during an increase in interest in complex dynamics and abstract mathematics,{{cite journal

| last1 = Rees

| first1 = Mary

| author-link = Mary Rees

| date = January 2016

| title = One hundred years of complex dynamics

| journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

| volume = 472

| issue = 2185

| pages =

| doi = 10.1098/rspa.2015.0453

| pmid = 26997888

| pmc = 4786033

| bibcode = 2016RSPSA.47250453R

}} and the topological and geometric study of the Mandelbrot set remains a key topic in the field of complex dynamics.{{Cite book |last=Schleicher |first=Dierk |url=https://books.google.com/books?id=Ek3rBgAAQBAJ |title=Complex Dynamics: Families and Friends |date=2009-11-03 |publisher=CRC Press |isbn=978-1-4398-6542-2 |pages=xii |language=en}}

The Mandelbrot set is the uncountable set of values of c in the complex plane for which the orbit of the critical point $z\; =\; 0$ under iteration of the quadratic map

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

remains bounded.{{cite web|url=http://math.bu.edu/DYSYS/explorer/def.html|title=Mandelbrot Set Explorer: Mathematical Glossary|access-date=7 October 2007}} Thus, a complex number c is a member of the Mandelbrot set if, when starting with $z\_0\; =\; 0$ and applying the iteration repeatedly, the absolute value of $z\_n$ 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.

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

The Mandelbrot set is a compact set, since it is closed and contained in the closed disk of radius 2 centred on zero. A point $c$ belongs to the Mandelbrot set if and only if $|z\_n|\backslash leq\; 2$ for all $n\backslash geq\; 0$. In other words, the absolute value of $z\_n$ must remain at or below 2 for $c$ to be in the Mandelbrot set, $M$, and if that absolute value exceeds 2, the sequence will escape to infinity. Since $c=z\_1$, it follows that $|c|\backslash leq\; 2$, establishing that $c$ will always be in the closed disk of radius 2 around the origin.{{cite web|url=https://mrob.com/pub/muency/escaperadius.html|title=Escape Radius|access-date=17 January 2024}}

File:Verhulst-Mandelbrot-Bifurcation.jpg of the quadratic map]]

File:Logistic Map Bifurcations Underneath Mandelbrot Set.gif

The intersection of $M$ with the real axis is the interval $\backslash left[-2,\backslash frac\{1\}\{4\}\backslash right]$. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family,

:$x\_\{n+1\}\; =\; r\; x\_n(1-x\_n),\backslash quad\; r\backslash in[1,4].$

The correspondence is given by

:$r\; =\; 1+\backslash sqrt\{1-\; 4\; c\},$

\quad

c = \frac{r}{2}\left(1-\frac{r}{2}\right),

\quad

z_n = r\left(\frac{1}{2} - x_n\right).

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

Douady and Hubbard showed that the Mandelbrot set is connected. They constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk. Mandelbrot had originally conjectured that the Mandelbrot set is disconnected. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of $M$. Upon further experiments, he revised his conjecture, deciding that $M$ should be connected. A topological proof of the connectedness was discovered in 2001 by Jeremy Kahn.{{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}}

File:Wakes near the period 1 continent in the Mandelbrot set.png

The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of $M$, gives rise to external rays of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle.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]

The boundary of the Mandelbrot set is the bifurcation locus of the family of quadratic polynomials. In other words, the boundary of the Mandelbrot set is the set of all parameters $c$ for which the dynamics of the quadratic map $z\_n=z\_\{n-1\}^2+c$ exhibits sensitive dependence on $c,$ i.e. changes abruptly under arbitrarily small changes of $c.$ It can be constructed as the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates. The Mandelbrot curves are defined by setting $p\_0=z,\backslash \; p\_\{n+1\}=p\_n^2+z$, and then interpreting the set of points $|p\_n(z)|\; =\; 2$ in the complex plane as a curve in the real Cartesian plane of degree $2^\{n+1\}$in x and y.{{Cite web |last=Weisstein |first=Eric W. |title=Mandelbrot Set Lemniscate |url=https://mathworld.wolfram.com/MandelbrotSetLemniscate.html |access-date=2023-07-17 |website=Wolfram Mathworld |language=en}} Each curve $n\; >\; 0$ is the mapping of an initial circle of radius 2 under $p\_n$. These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.

File:Mandelbrot Set – Periodicities coloured.png

The main cardioid is the period 1 continent.{{Cite book |last1=Brucks |first1=Karen M. |url=https://books.google.com/books?id=p-amwZp0R-0C |title=Topics from One-Dimensional Dynamics |last2=Bruin |first2=Henk |date=2004-06-28 |publisher=Cambridge University Press |isbn=978-0-521-54766-6 |pages=264 |language=en}} It is the region of parameters $c$ for which the map $f\_c(z)\; =\; z^2\; +\; c$ has an attracting fixed point.{{Cite book |last=Devaney |first=Robert |url=https://books.google.com/books?id=YEIPEAAAQBAJ |title=An Introduction To Chaotic Dynamical Systems |date=2018-03-09 |publisher=CRC Press |isbn=978-0-429-97085-6 |pages=147 |language=en}} It consists of all parameters of the form $c(\backslash mu)\; :=\; \backslash frac\backslash mu2\backslash left(1-\backslash frac\backslash mu2\backslash right)$ for some $\backslash mu$ in the open unit disk.{{Cite book |last1=Ivancevic |first1=Vladimir G. |url=https://books.google.com/books?id=mbtCAAAAQBAJ |title=High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction |last2=Ivancevic |first2=Tijana T. |date=2007-02-06 |publisher=Springer Science & Business Media |isbn=978-1-4020-5456-3 |pages=492–493 |language=en}}{{close paraphrasing inline|date=March 2025}}

To the left of the main cardioid, attached to it at the point $c=-3/4$, a circular bulb, the period-2 bulb is visible.{{close paraphrasing inline|date=March 2025}} The bulb consists of $c$ for which $f\_c$ has an attracting cycle of period 2. It is the filled circle of radius 1/4 centered around −1.{{close paraphrasing inline|date=March 2025}}

File:Animated cycle.gif (animation)]]

More generally, for every positive integer $q>2$, there are $\backslash phi(q)$ circular bulbs tangent to the main cardioid called period-q bulbs (where $\backslash phi$ denotes the Euler phi function), which consist of parameters $c$ for which $f\_c$ has an attracting cycle of period $q$.{{Citation needed|date=March 2025}} More specifically, for each primitive $q$th root of unity $r=e^\{2\backslash pi\; i\backslash frac\{p\}\{q\}\}$ (where ), there is one period-q bulb called the $\backslash frac\{p\}\{q\}$ bulb, which is tangent to the main cardioid at the parameter $c\_\{\backslash frac\{p\}\{q\}\}\; :=\; c(r)\; =\; \backslash frac\{r\}2\backslash left(1-\backslash frac\{r\}2\backslash right),$ and which contains parameters with $q$-cycles having combinatorial rotation number $\backslash frac\{p\}\{q\}$.{{Cite book |last1=Devaney |first1=Robert L. |url=https://books.google.com/books?id=4XrHCQAAQBAJ |title=Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets: The Mathematics Behind the Mandelbrot and Julia Sets |last2=Branner |first2=Bodil |date=1994 |publisher=American Mathematical Soc. |isbn=978-0-8218-0290-8 |pages=18–19 |language=en}} More precisely, the $q$ periodic Fatou components containing the attracting cycle all touch at a common point (commonly called the $\backslash alpha$-fixed point). If we label these components $U\_0,\backslash dots,U\_\{q-1\}$ in counterclockwise orientation, then $f\_c$ maps the component $U\_j$ to the component $U\_\{j+p\backslash ,(\backslash operatorname\{mod\}\; q)\}$.{{close paraphrasing inline|date=March 2025}}

File:Juliacycles1.pngs for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs]]

The change of behavior occurring at $c\_\{\backslash frac\{p\}\{q\}\}$ is known as a bifurcation: the attracting fixed point "collides" with a repelling period-q cycle. As we pass through the bifurcation parameter into the $\backslash tfrac\{p\}\{q\}$-bulb, the attracting fixed point turns into a repelling fixed point (the $\backslash alpha$-fixed point), and the period-q cycle becomes attracting.{{close paraphrasing inline|date=March 2025}}

Bulbs that are interior components of the Mandelbrot set in which the maps $f\_c$ have an attracting periodic cycle are called hyperbolic components.{{cite thesis |last=Redona |first=Jeffrey Francis |title=The Mandelbrot set |year=1996 |type=Masters of Arts in Mathematics

|publisher=Theses Digitization Project |url=https://scholarworks.lib.csusb.edu/etd-project/1166}}

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

Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. Such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).

File:Centers8.png

Each of the hyperbolic components has a center, which is a point c such that the inner Fatou domain for $f\_c(z)$ has a super-attracting cycle—that is, that the attraction is infinite. This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. Therefore, $f\_c^n(0)\; =\; 0$ for some n. If we call this polynomial $Q^\{n\}(c)$ (letting it depend on c instead of z), we have that $Q^\{n+1\}(c)\; =\; Q^\{n\}(c)^\{2\}\; +\; c$ and that the degree of $Q^\{n\}(c)$ is $2^\{n-1\}$. Therefore, constructing the centers of the hyperbolic components is possible by successively solving the equations $Q^\{n\}(c)\; =\; 0,\; n\; =\; 1,\; 2,\; 3,\; ...$.{{Citation needed|date=July 2023}} The number of new centers produced in each step is given by Sloane's {{oeis|A000740}}.

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

The work of Jean-Christophe Yoccoz established local connectivity of the Mandelbrot set at all finitely renormalizable parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies.{{cite book

| last = Hubbard | first = J. H.

| contribution = Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz

| contribution-url = https://pi.math.cornell.edu/~hubbard/Yoccoz.pdf

| location = Houston, TX

| mr = 1215974

| pages = 467–511

| publisher = Publish or Perish

| title = Topological methods in modern mathematics (Stony Brook, NY, 1991)

| year = 1993}}. Hubbard cites as his source a 1989 unpublished manuscript of Yoccoz. Since then, local connectivity has been proved at many other points of $M$, but the full conjecture is still open.

File:Self-Similarity-Zoom.gif in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-x direction. The display center pans left from the fifth to the seventh round feature (−1.4002, 0) to (−1.4011, 0) while the view magnifies by a factor of 21.78 to approximate the square of the Feigenbaum ratio.]]

The Mandelbrot set is self-similar under magnification in the neighborhoods of the Misiurewicz points. It is also conjectured to be self-similar around generalized Feigenbaum points (e.g., −1.401155 or −0.1528 + 1.0397i), in the sense of converging to a limit set.{{cite journal | last1 = Lei | year = 1990 | title = Similarity between the Mandelbrot set and Julia Sets | url = http://projecteuclid.org/euclid.cmp/1104201823| journal = Communications in Mathematical Physics | volume = 134 | issue = 3| pages = 587–617 | doi=10.1007/bf02098448| bibcode = 1990CMaPh.134..587L| s2cid = 122439436 }}{{cite book |author=J. Milnor |chapter=Self-Similarity and Hairiness in the Mandelbrot Set |editor=M. C. Tangora |location=New York |pages=211–257 |title=Computers in Geometry and Topology |url=https://books.google.com/books?id=wuVJAQAAIAAJ |year=1989|publisher=Taylor & Francis|isbn=9780824780319 }}) The Mandelbrot set in general is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales. These copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set.{{Cite web |title=Mandelbrot Viewer |url=https://math.hws.edu/eck/js/mandelbrot/MB.html |access-date=2025-03-01 |website=math.hws.edu}}

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

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

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

File:Julia Mandelbrot Relationship.png

As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set. For instance, a value of c belongs to the Mandelbrot set if and only if the corresponding Julia set is connected. Thus, the Mandelbrot set may be seen as a map of the connected Julia sets.{{cite web |last=Sims |first=Karl |title=Understanding Julia and Mandelbrot Sets |url=https://www.karlsims.com/julia.html |website=karlsims.com |access-date=January 27, 2025}}{{Better source needed|date=January 2025}}

This principle is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proved that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff dimension two, and then transfers this information to the parameter plane.{{cite journal

| last = Shishikura | first = Mitsuhiro

| arxiv = math.DS/9201282

| doi = 10.2307/121009

| issue = 2

| journal = Annals of Mathematics

| mr = 1626737

| pages = 225–267

| series = Second Series

| title = The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets

| volume = 147

| year = 1998| jstor = 121009

| s2cid = 14847943

}}. Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters.

== Geometry ==

For every rational number $\backslash tfrac\{p\}\{q\}$, where p and q are coprime, a hyperbolic component of period q bifurcates from the main cardioid at a point on the edge of the cardioid corresponding to an internal angle of $\backslash tfrac\{2\backslash pi\; p\}\{q\}$.{{cite web |title=Number Sequences in the Mandelbrot Set |url=https://www.youtube.com/watch?v=oNxPSP2tQEk | archive-url=https://ghostarchive.org/varchive/youtube/20211030/oNxPSP2tQEk| archive-date=2021-10-30|website=youtube.com |publisher=The Mathemagicians' Guild |date=4 June 2020}}{{cbignore}} 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 $\backslash tfrac\{1\}\{q^2\}$. The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like $\backslash tfrac\{1\}\{q\}$.{{Citation needed|date=July 2023}}

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

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

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

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

In 2025, the three high school students Thies Brockmöller, Oscar Scherz, and Nedim Srkalovic extended the theory and the conceptual proof to all the infinitely bifurcation points in the Mandelbrot set. https://arxiv.org/abs/2505.07138

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

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

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

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

$\backslash frac\{0\}\{1\}$ $\backslash oplus$ $\backslash frac\{1\}\{2\}$$=$$\backslash frac\{1\}\{3\}$

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

$\backslash frac\{1\}\{3\}$ $\backslash oplus$ $\backslash frac\{1\}\{2\}$$=$$\backslash frac\{2\}\{5\}$

The largest bulb between the $2/5$ and $1/2$-bulb is the $3/7$-bulb, while the largest bulb between the $2/5$ and $1/3$-bulbs is the $3/8$-bulb, and so on.{{Cite web |title=The Mandelbrot Set Explorer Welcome Page |url=http://math.bu.edu/DYSYS/explorer/ |access-date=2024-02-17 |website=math.bu.edu}} The arrangement of bulbs within the Mandelbrot set follows a remarkable pattern governed by the Farey tree, a structure encompassing all rationals between $0$ and $1$. This ordering positions the bulbs along the boundary of the main cardioid precisely according to the rational numbers in the unit interval.

File:Fibonacci sequence within the Mandelbrot set.png

Starting with the $1/3$ bulb at the top and progressing towards the $1/2$ circle, the sequence unfolds systematically: the largest bulb between $1/2$ and $1/3$ is $2/5$, between $1/3$ and $2/5$ is $3/8$, and so forth.{{Cite web |title=Maths Town |url=https://www.patreon.com/mathstown |access-date=2024-02-17 |website=Patreon}} Intriguingly, the denominators of the periods of circular bulbs at sequential scales in the Mandelbrot Set conform to the Fibonacci number sequence, the sequence that is made by adding the previous two terms – 1, 2, 3, 5, 8, 13, 21...{{Cite journal |last1=Fang |first1=Fang |last2=Aschheim |first2=Raymond |last3=Irwin |first3=Klee |date=December 2019 |title=The Unexpected Fractal Signatures in Fibonacci Chains |journal=Fractal and Fractional |language=en |volume=3 |issue=4 |pages=49 |doi=10.3390/fractalfract3040049 |doi-access=free |issn=2504-3110|arxiv=1609.01159 }}{{Cite web |title=7 The Fibonacci Sequence |url=https://math.bu.edu/DYSYS/FRACGEOM2/node7.html#SECTION00070000000000000000 |access-date=2024-02-17 |website=math.bu.edu}}

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

The boundary of the Mandelbrot set shows more intricate detail the closer one looks or magnifies the image. The following is an example of an image sequence zooming to a selected c value.

The magnification of the last image relative to the first one is about 10¹⁰ to 1. Relating to an ordinary computer monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers.

{{clear}}

Mandel zoom 00 mandelbrot set.jpg|Start. Mandelbrot set with continuously colored environment.

Mandel zoom 01 head and shoulder.jpg|Gap between the "head" and the "body", also called the "seahorse valley"{{Cite book |last=Lisle |first=Jason |url=https://books.google.com/books?id=h-czEAAAQBAJ |title=Fractals: The Secret Code of Creation |date=2021-07-01 |publisher=New Leaf Publishing Group |isbn=978-1-61458-780-4 |pages=28 |language=en}}

Mandel zoom 02 seehorse valley.jpg|Double-spirals on the left, "seahorses" on the right

Mandel zoom 03 seehorse.jpg|"Seahorse" upside down

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

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 set, which means there are no islands and no loop roads around a hole.

File:Mandel zoom 06 double hook.jpg|Satellite. The two "seahorse tails" (also called dendritic structures){{Cite book |last=Kappraff |first=Jay |url=https://books.google.com/books?id=vAfBrK678_kC |title=Beyond Measure: A Guided Tour Through Nature, Myth, and Number |date=2002 |publisher=World Scientific |isbn=978-981-02-4702-7 |pages=437 |language=en}} are the beginning of a series of concentric crowns with the satellite in the center.

File:Mandel zoom 07 satellite.jpg|Each of these crowns consists of similar "seahorse tails"; their number increases with powers of 2, a typical phenomenon in the environment of satellites. The unique path to the spiral center passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head".

File:Mandel zoom 08 satellite antenna.jpg|"Antenna" of the satellite. There are several satellites of second order.

File:Mandel zoom 09 satellite head and shoulder.jpg|The "seahorse valley" of the satellite. All the structures from the start reappear.

File:Mandel zoom 10 satellite seehorse valley.jpg|Double-spirals and "seahorses" – unlike the second image from the start, they have appendices consisting of structures like "seahorse tails"; this demonstrates the typical linking of n + 1 different structures in the environment of satellites of the order n, here for the simplest case n = 1.

File:Mandel zoom 11 satellite double spiral.jpg|Double-spirals with satellites of second order – analogously to the "seahorses", the double-spirals may be interpreted as a metamorphosis of the "antenna".

File:Mandel zoom 12 satellite spirally wheel with julia islands.jpg|In the outer part of the appendices, islands of structures may be recognized; they have a shape like Julia sets J_c; the largest of them may be found in the center of the "double-hook" on the right side.

File:Mandel zoom 13 satellite seehorse tail with julia island.jpg|Part of the "double-hook".

File:Mandel zoom 14 satellite julia island.jpg|Islands.

File:Mandel zoom 15 one island.jpg|A detail of one island.

File:Mandel zoom 16 spiral island.jpg|Detail of the spiral.

The islands in the third-to-last step seem to consist of infinitely many parts, as is the case for the corresponding Julia set $J\_c$. 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 $J\_c$'' is not the image center but, relative to the main body of the Mandelbrot set, has the same position as the center of this image relative to the satellite shown in the 6th step.

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

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

File:Mandelbrot full gradient.gif|Animated gradient structure inside the Mandelbrot set

File:Mandelbrot inner gradient.gif|Animated gradient structure inside the Mandelbrot set, detail

File:Mandelbrot gradient iterations.gif|Rendering of progressive iterations from 285 to approximately 200,000 with corresponding bounded gradients animated

File:Mandelbrot gradient iterations thumb.gif|Thumbnail for gradient in progressive iterations

{{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

Multibrot sets are bounded sets found in the complex plane for members of the general monic univariate polynomial family of recursions

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

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 $z\; \backslash mapsto\; z^3\; +\; 3kz\; +\; c$, whose two critical points are the complex square roots of the parameter k. A parameter is in the cubic connectedness locus if both critical points are stable.Rudy Rucker's discussion of the CCM: [http://www.cs.sjsu.edu/faculty/rucker/cubic_mandel.htm CS.sjsu.edu] For general families of holomorphic functions, the boundary of the Mandelbrot set generalizes to the bifurcation locus.{{Citation needed|date=July 2023}}

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

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

File:Mandelbar fractal from XaoS.PNG]]

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

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

:$z\; \backslash mapsto\; (|\backslash Re\; \backslash left(z\backslash right)|+i|\backslash Im\; \backslash left(z\backslash right)|)^2\; +\; c$.{{Citation needed|date=March 2025}}

{{Main|Plotting algorithms for the Mandelbrot set}}

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

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 pixels. To color any such pixel, let $c$ be the midpoint of that pixel. Iterate the critical point 0 under $f\_c$, checking at each step whether the orbit point has a radius larger than 2. When this is the case, $c$ does not belong to the Mandelbrot set, and color the pixel according to the number of iterations used to find out. Otherwise, keep iterating up to a fixed number of steps, after which we decide that our parameter is "probably" in the Mandelbrot set, or at least very close to it, and color the pixel black.

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

for each pixel (Px, Py) on the screen do

x0 := scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.00, 0.47))

y0 := scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1.12, 1.12))

x := 0.0

y := 0.0

iteration := 0

max_iteration := 1000

while (x^2 + y^2 ≤ 2^2 AND iteration < max_iteration) do

xtemp := x^2 - y^2 + x0

y := 2*x*y + y0

x := xtemp

iteration := iteration + 1

color := palette[iteration]

plot(Px, Py, color)

Here, relating the pseudocode to $c$, $z$ and $f\_c$:

• $z\; =\; x\; +\; iy$
• $z^2\; =\; x^2\; +i2xy\; -\; y^2$
• $c\; =\; x\_0\; +\; i\; y\_0$

and so, as can be seen in the pseudocode in the computation of x and y:

• $x\; =\; \backslash mathop\{\backslash mathrm\{Re\}\}\; \backslash left(z^2+c\; \backslash right)\; =\; x^2-y^2\; +\; x\_0$ and $y\; =\; \backslash mathop\{\backslash mathrm\{Im\}\}\; \backslash left(z^2+c\; \backslash right)\; =\; 2xy\; +\; y\_0$.

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.).

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

import numpy as np

import matplotlib.pyplot as plt

1. setting parameters (these values can be changed)

xDomain, yDomain = np.linspace(-2, 2, 500), np.linspace(-2, 2, 500)

bound = 2

max_iterations = 50 # any positive integer value

colormap = "nipy_spectral" # set to any matplotlib valid colormap

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

1. computing 2-d array to represent the mandelbrot-set

iterationArray = []

for y in yDomain:

row = []

for x in xDomain:

z = 0

p = 2

c = complex(x, y)

for iterationNumber in range(max_iterations):

if abs(z) >= bound:

row.append(iterationNumber)

break

else:

try:

z = func(z, p, c)

except(ValueError, ZeroDivisionError):

z = c

else:

row.append(0)

iterationArray.append(row)

1. plotting the data

ax = plt.axes()

ax.set_aspect("equal")

graph = ax.pcolormesh(xDomain, yDomain, iterationArray, cmap=colormap)

plt.colorbar(graph)

plt.xlabel("Real-Axis")

plt.ylabel("Imaginary-Axis")

plt.show()

File:Multibrot set of power 5.png

The value of power variable can be modified to generate an image of equivalent multibrot set ($z\; =\; z^\{\backslash text\{power\}\}+c$). For example, setting p = 2 produces the associated image.

The Mandelbrot set is widely considered the most popular fractal,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 2019Moeller, Olga de. (2018).[https://thewest.com.au/lifestyle/kids/what-are-fractals-ng-b88838072z "what are Fractals?"] Retrieved 17 January 2019. and has been referenced several times in popular culture.

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

{{Div col|colwidth=20em}}

• Buddhabrot
• Collatz fractal
• Fractint
• Gilbreath permutation
• List of mathematical art software
• Mandelbox
• Mandelbulb
• Menger sponge
• Newton fractal
• Orbit portrait
• Orbit trap
• Pickover stalk
• Plotting algorithms for the Mandelbrot set{{Div col end}}

{{Reflist|30em}}

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

{{Wikibooks|Fractals }}

{{commons category}}

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

{{Fractals}}

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