Benoit Mandelbrot

Benoit Mandelbrot

Benoit B. [n 1] Mandelbrot [n 2] (20 November 1924 – 14 October 2010) was a Polish-born, French and American mathematician with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life."[6][7] He referred to himself as a "fractalist".[8] He is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.[9]
Benoit [n 1] Mandelbrot
Benoit Mandelbrot, TED 2010.jpg
At a TED conference in 2010.
Born 20 November 1924
Warsaw, Poland
Died 14 October 2010 (aged 85)
Cambridge, Massachusetts, United States
Residence

    Poland France United States

Nationality

    Polish French American

Alma mater École Polytechnique
California Institute of Technology
University of Paris
Known for

    Mandelbrot set Chaos theory Fractals Zipf–Mandelbrot law

Spouse(s)
Aliette Kagan
married 1955–2010 (his death)
Awards
Légion d'honneur
(Chevalier 1990 · Officier 2006)
2003  Japan Prize
1993  Wolf Prize
1989  Harvey Prize
1986  Franklin Medal
1985  Barnard Medal
Scientific career
Fields

    Mathematics Aerodynamics

Institutions

    Yale University IBM

Pacific Northwest National Laboratory
Doctoral students

    L-E. Calvet Eugene Fama Ken Musgrave Murad Taqqu

Influences Johannes Kepler, Szolem Mandelbrojt
Influenced Nassim Nicholas Taleb

In 1936, while he was a child, Mandelbrot's family migrated to France. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and the United States and receiving a master's degree in aeronautics from the California Institute of Technology. He spent most of his career in both the United States and France, having dual French and American citizenship. In 1958, he began a 35-year career at IBM, where he became an IBM Fellow, and periodically took leaves of absence to teach at Harvard University. At Harvard, following the publication of his study of U.S. commodity markets in relation to cotton futures, he taught economics and applied sciences.

Because of his access to IBM's computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovering the Mandelbrot set in 1979. He showed how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess" or "chaotic", like clouds or shorelines, actually had a "degree of order."[10] His math and geometry-centered research career included contributions to such fields as statistical physics, meteorology, hydrology, geomorphology, anatomy, taxonomy, neurology, linguistics, information technology, computer graphics, economics, geology, medicine, physical cosmology, engineering, chaos theory, econophysics, metallurgy and the social sciences.[11]

Toward the end of his career, he was Sterling Professor of Mathematical Sciences at Yale University, where he was the oldest professor in Yale's history to receive tenure.[12] Mandelbrot also held positions at the Pacific Northwest National Laboratory, Université Lille Nord de France, Institute for Advanced Study and Centre National de la Recherche Scientifique. During his career, he received over 15 honorary doctorates and served on many science journals, along with winning numerous awards. His autobiography, The Fractalist: Memoir of a Scientific Maverick, was published in 2012.

Early yearsEdit

Mandelbrot was born in Warsaw during the Second Polish Republic. His family was Jewish. Although his father made his living trading clothing, the family had a strong academic tradition and his mother was a dental surgeon.[13] He was first introduced to mathematics by two of his uncles, one of whom, Szolem Mandelbrojt, was a mathematician who resided in Paris. According to Mandelbrot's autobiography, The Fractalist - Memoir of a Scientific Maverick,[14] "[t]he love of his [Szolem's] mind was mathematics".[8]:16

The family emigrated from Poland to France in 1936, when he was 11. "The fact that my parents, as economic and political refugees, joined Szolem in France saved our lives," he writes.[8]:17[15] Mandelbrot attended the Lycée Rolin in Paris until the start of World War II, when his family then moved to Tulle, France. He was helped by Rabbi David Feuerwerker, the Rabbi of Brive-la-Gaillarde, to continue his studies.[8]:62–63[16] Much of France was occupied by the Nazis at the time, and Mandelbrot recalls this period:

    Our constant fear was that a sufficiently determined foe might report us to an authority and we would be sent to our deaths. This happened to a close friend from Paris, Zina Morhange, a physician in a nearby county seat. Simply to eliminate the competition, another physician denounced her ... We escaped this fate. Who knows why?[8]:49

In 1944, Mandelbrot returned to Paris, studied at the Lycée du Parc in Lyon, and in 1945 to 1947 attended the École Polytechnique, where he studied under Gaston Julia and Paul Lévy. From 1947 to 1949 he studied at California Institute of Technology, where he earned a master's degree in aeronautics.[2] Returning to France, he obtained his PhD degree in Mathematical Sciences at the University of Paris in 1952.[13]
Research careerEdit

From 1949 to 1958, Mandelbrot was a staff member at the Centre National de la Recherche Scientifique. During this time he spent a year at the Institute for Advanced Study in Princeton, New Jersey, where he was sponsored by John von Neumann. In 1955 he married Aliette Kagan and moved to Geneva, Switzerland, and later to the Université Lille Nord de France.[17] In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York.[17] He remained at IBM for 35 years, becoming an IBM Fellow, and later Fellow Emeritus.[13]

From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in applied fields such as information theory, economics, and fluid dynamics.
States of randomness and financial marketsEdit

Mandelbrot found that price changes in financial markets did not follow a Gaussian distribution, but rather Lévy stable distributions having theoretically infinite variance. He found, for example, that cotton prices followed a Lévy stable distribution with parameter α equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger scale parameter.[18]
Developing "fractal geometry" and the Mandelbrot setEdit

As a visiting professor at Harvard University, Mandelbrot began to study fractals called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia and Pierre Fatou, Mandelbrot used a computer to plot images of the Julia sets. While investigating the topology of these Julia sets, he studied the Mandelbrot set fractal that is now named after him. In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature.[19] This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "program artifacts".
Mandelbrot speaking about the Mandelbrot set, during his acceptance speech for the Légion d'honneur in 2006

In 1975, Mandelbrot coined the term fractal to describe these structures and first published his ideas, and later translated, Fractals: Form, Chance and Dimension.[20] According to mathematics scientist Stephen Wolfram, the book was a "breakthrough" for Mandelbrot, who until then would typically "apply fairly straightforward mathematics … to areas that had barely seen the light of serious mathematics before."[10] Wolfram adds that as a result of this new research, he was no longer a "wandering scientist", and later called him "the father of fractals":

    Mandelbrot ended up doing a great piece of science and identifying a much stronger and more fundamental idea—put simply, that there are some geometric shapes, which he called "fractals", that are equally "rough" at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space.[10]

Wolfram briefly describes fractals as a form of geometric repetition, "in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you zoom in to the whole. Fern leaves and Romanesco broccoli are two examples from nature."[10] He points out an unexpected conclusion:

    One might have thought that such a simple and fundamental form of regularity would have been studied for hundreds, if not thousands, of years. But it was not. In fact, it rose to prominence only over the past 30 or so years—almost entirely through the efforts of one man, the mathematician Benoit Mandelbrot.[10]

Mandelbrot used the term "fractal" as it derived from the Latin word "fractus", defined as broken or shattered glass. Using the newly developed IBM computers at his disposal, Mandelbrot was able to create fractal images using graphic computer code, images that an interviewer described as looking like "the delirious exuberance of the 1960s psychedelic art with forms hauntingly reminiscent of nature and the human body." He also saw himself as a "would-be Kepler", after the 17th-century scientist Johannes Kepler, who calculated and described the orbits of the planets.[21]
A Mandelbrot set

Mandelbrot, however, never felt he was inventing a new idea. He describes his feelings in a documentary with science writer Arthur C. Clarke:

    Exploring this set I certainly never had the feeling of invention. I never had the feeling that my imagination was rich enough to invent all those extraordinary things on discovering them. They were there, even though nobody had seen them before. It's marvelous, a very simple formula explains all these very complicated things. So the goal of science is starting with a mess, and explaining it with a simple formula, a kind of dream of science.[22]

According to Clarke, "the Mandelbrot set is indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literally infinite complexity?" Clarke also notes an "odd coincidence:" "the name Mandelbrot, and the word "mandala"—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas."[22]

Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division.[23] He joined the Department of Mathematics at Yale, and obtained his first tenured post in 1999, at the age of 75.[24] At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences.
Fractals and the "theory of roughness"Edit

Mandelbrot created the first-ever "theory of roughness", and he saw "roughness" in the shapes of mountains, coastlines and river basins; the structures of plants, blood vessels and lungs; the clustering of galaxies. His personal quest was to create some mathematical formula to measure the overall "roughness" of such objects in nature.[8]:xi He began by asking himself various kinds of questions related to nature:

    Can geometry deliver what the Greek root of its name [geo-] seemed to promise—truthful measurement, not only of cultivated fields along the Nile River but also of untamed Earth?[8]:xii

In his paper entitled How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension published in Science in 1967 Mandelbrot discusses self-similar curves that have Hausdorff dimension that are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals.[25][26]

Mandelbrot emphasized the use of fractals as realistic and useful models for describing many "rough" phenomena in the real world. He concluded that "real roughness is often fractal and can be measured."[8]:296 Although Mandelbrot coined the term "fractal", some of the mathematical objects he presented in The Fractal Geometry of Nature had been previously described by other mathematicians. Before Mandelbrot, however, they were regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them into essential tools for the long-stalled effort to extend the scope of science to explaining non-smooth, "rough" objects in the real world. His methods of research were both old and new:

    The form of geometry I increasingly favored is the oldest, most concrete, and most inclusive, specifically empowered by the eye and helped by the hand and, today, also by the computer … bringing an element of unity to the worlds of knowing and feeling … and, unwittingly, as a bonus, for the purpose of creating beauty.[8]:292

Section of a Mandelbrot set

Fractals are also found in human pursuits, such as music, painting, architecture, and stock market prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry:

    Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
      —Mandelbrot, in his introduction to The Fractal Geometry of Nature

Mandelbrot has been called a work of art, and a visionary[27] and a maverick.[28] His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal Geometry of Nature accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics.

Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of Olbers' paradox (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox. He postulated that if the stars in the universe were fractally distributed (for example, like Cantor dust), it would not be necessary to rely on the Big Bang theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.[29]
Awards and honorsEdit

Mandelbrot's awards include the Wolf Prize for Physics in 1993, the Lewis Fry Richardson Prize of the European Geophysical Society in 2000, the Japan Prize in 2003,[30] and the Einstein Lectureship of the American Mathematical Society in 2006.

The small asteroid 27500 Mandelbrot was named in his honor. In November 1990, he was made a Chevalier in France's Legion of Honour. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory.[31] Mandelbrot was promoted to an Officer of the Legion of Honour in January 2006.[32] An honorary degree from Johns Hopkins University was bestowed on Mandelbrot in the May 2010 commencement exercises.[33]

A partial list of awards received by Mandelbrot:[34]

    2004 Best Business Book of the Year Award
    AMS Einstein Lectureship
    Barnard Medal
    Caltech Service
    Casimir Funk Natural Sciences Award
    Charles Proteus Steinmetz Medal
    Fellow, American Geophysical Union
    Fellow of the American Statistical Association[35]
    Franklin Medal
    Harvey Prize (1989)
    Honda Prize
    Humboldt Preis
    IBM Fellowship
    Japan Prize (2003)
    John Scott Award
    Légion d'honneur (Legion of Honour)
    Lewis Fry Richardson Medal
    Medaglia della Presidenza della Repubblica Italiana
    Médaille de Vermeil de la Ville de Paris
    Nevada Prize
    Member of the Norwegian Academy of Science and Letters.[36]
    Science for Art
    Sven Berggren-Priset
    Władysław Orlicz Prize
    Wolf Foundation Prize for Physics (1993)

Death and legacyEdit
Wikinews has related news: Mathematician Benoît Mandelbrot dies aged 85

Mandelbrot died from pancreatic cancer at the age of 85 in a hospice in Cambridge, Massachusetts on 14 October 2010.[1][37] Reacting to news of his death, mathematician Heinz-Otto Peitgen said: "[I]f we talk about impact inside mathematics, and applications in the sciences, he is one of the most important figures of the last fifty years."[1]

Chris Anderson, TED conference curator, described Mandelbrot as "an icon who changed how we see the world".[38] Nicolas Sarkozy, President of France at the time of Mandelbrot's death, said Mandelbrot had "a powerful, original mind that never shied away from innovating and shattering preconceived notions [… h]is work, developed entirely outside mainstream research, led to modern information theory."[39] Mandelbrot's obituary in The Economist points out his fame as "celebrity beyond the academy" and lauds him as the "father of fractal geometry".[40]

Best-selling essayist-author Nassim Nicholas Taleb, a Mandelbrot protégé and a scientific adviser at Universa Investments, has remarked that Mandelbrot's book The (Mis)Behavior of Markets is in his opinion "The deepest and most realistic finance book ever published".[9]
BibliographyEdit
in EnglishEdit

    Fractals: Form, Chance and Dimension, 1977
    The Fractal Geometry of Nature, 1982
    Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E, 1997 by Benoit B. Mandelbrot and R.E. Gomory
    Fractales, hasard et finance, 1959-1997, Nov 1, 1998
    Multifractals and 1/ƒ Noise: Wild Self-Affinity in Physics (1963–1976) (Selecta; V.N) Jan 18, 1999 by J.M. Berger and Benoit B. Mandelbrot
    Gaussian Self-Affinity and Fractals: Globality, The Earth, 1/f Noise, and R/S (Selected Works of Benoit B. Mandelbrot) Dec 14, 2001 by Benoit Mandelbrot and F.J. Damerau
    Fractals and Chaos: The Mandelbrot Set and Beyond, Jan 9, 2004
    The Misbehavior of Markets: A Fractal View of Financial Turbulence, 2006 by Benoit Mandelbrot and Richard L. Hudson
    The Fractalist: Memoir of a Scientific Maverick, 2014

In FrenchEdit

    La forme d'une vie. Mémoires (1924-2010) by Benoît Mandelbrot (Author), Johan-Frédérik Hel Guedj (Translator)

References in popular cultureEdit

In 2004, the American singer-songwriter Jonathan Coulton wrote "The Mandelbrot Set", a song dedicated to Mandelbrot and his famous fractal. In 2007, the author Laura Ruby published a sequel to "The Wall and the Wing" series named "The Chaos King". One of the main characters is named Mandelbrot and his work is referenced in the novel, specifically, the chaos theory.

In 2017, Zach Weinersmith had Mandelbrot make a cameo[41] in his webcomic, Saturday Morning Breakfast Cereal
___________________
Mandelbrot set
Initial image of a Mandelbrot set zoom sequence with a continuously colored environment
File:Progressive infinite iterations of the 'Nautilus' section of the Mandelbrot Set.ogvPlay media
Progressive infinite iterations of the "Nautilus" section of the Mandelbrot Set rendered using webGL
Mandelbrot animation based on a static number of iterations per pixel
Mandelbrot set detail

The Mandelbrot set is the set of complex numbers c {\displaystyle c} c for which the function f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c} {\displaystyle f_{c}(z)=z^{2}+c} does not diverge when iterated from z = 0 {\displaystyle z=0} z=0, i.e., for which the sequence f c ( 0 ) {\displaystyle f_{c}(0)} {\displaystyle f_{c}(0)}, f c ( f c ( 0 ) ) {\displaystyle f_{c}(f_{c}(0))} {\displaystyle f_{c}(f_{c}(0))}, etc., remains bounded in absolute value.
A zoom sequence illustrating the set of complex numbers termed the Mandelbrot set.

Its definition and name are due to Adrien Douady, in tribute to the mathematician Benoit Mandelbrot.[1] 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 determining, for each sample point c {\displaystyle c} c, whether the result of iterating the above function goes to infinity. Treating the real and imaginary parts of c {\displaystyle c} c as image coordinates ( x + y i ) {\displaystyle (x+yi)} {\displaystyle (x+yi)} on the complex plane, pixels may then be colored according to how rapidly the sequence z n 2 + c {\displaystyle z_{n}^{2}+c} {\displaystyle z_{n}^{2}+c} diverges, with the color 0 (black) usually used for points where the sequence does not diverge. If c {\displaystyle c} c is held constant and the initial value of z {\displaystyle z} z—denoted by z 0 {\displaystyle z_{0}} z_{0}—is variable instead, one obtains the corresponding Julia set for each point c {\displaystyle c} c in the parameter space of the simple function.

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

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

HistoryEdit
The first published picture of the Mandelbrot set, by Robert W. Brooks and Peter Matelski in 1978

The Mandelbrot set has its place in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. This fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups.[2] On 1 March 1980, at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York, Benoit Mandelbrot first saw a visualization of the set.[3]

Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980.[4] The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard,[1] who established many of its fundamental properties and named the set in honor of Mandelbrot.

The mathematicians Heinz-Otto Peitgen and Peter Richter became well known for promoting the set with photographs, books,[5] and an internationally touring exhibit of the German Goethe-Institut.[6][7]

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 created by Peitgen, et al.[8][9] 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.[10]

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 the mathematicians who have contributed to the understanding of this set since then is beyond the scope of this article, but such a list would notably include Mikhail Lyubich,[11][12] Curt McMullen, John Milnor, Mitsuhiro Shishikura, and Jean-Christophe Yoccoz.
Formal definitionEdit

The Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the quadratic map

    z n + 1 = z n 2 + c {\displaystyle z_{n+1}=z_{n}^{2}+c} {\displaystyle z_{n+1}=z_{n}^{2}+c}

remains bounded.[13] That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets. This can also be represented as[14]

    z n + 1 = z n 2 + c , {\displaystyle z_{n+1}=z_{n}^{2}+c,} {\displaystyle z_{n+1}=z_{n}^{2}+c,}
    c ∈ M ⟺ lim sup n → ∞ | z n + 1 | ≤ 2. {\displaystyle c\in M\iff \limsup _{n\to \infty }|z_{n+1}|\leq 2.} {\displaystyle c\in M\iff \limsup _{n\to \infty }|z_{n+1}|\leq 2.}

For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26, ..., which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = −1 gives the sequence 0, −1, 0, −1, 0, ..., which is bounded, and so −1 belongs to the Mandelbrot set.

The Mandelbrot set M {\displaystyle M} M is defined by a family of complex quadratic polynomials

    P c : C → C {\displaystyle P_{c}:\mathbb {C} \to \mathbb {C} } P_{c}:\mathbb {C} \to \mathbb {C}

given by

    P c : z ↦ z 2 + c , {\displaystyle P_{c}:z\mapsto z^{2}+c,} P_{c}:z\mapsto z^{2}+c,

where c {\displaystyle c} c is a complex parameter. For each c {\displaystyle c} c, one considers the behavior of the sequence

    ( 0 , P c ( 0 ) , P c ( P c ( 0 ) ) , P c ( P c ( P c ( 0 ) ) ) , … ) {\displaystyle (0,P_{c}(0),P_{c}(P_{c}(0)),P_{c}(P_{c}(P_{c}(0))),\ldots )} (0,P_{c}(0),P_{c}(P_{c}(0)),P_{c}(P_{c}(P_{c}(0))),\ldots )

obtained by iterating P c ( z ) {\displaystyle P_{c}(z)} P_{c}(z) starting at critical point z = 0 {\displaystyle z=0} z=0, which either escapes to infinity or stays within a disk of some finite radius. The Mandelbrot set is defined as the set of all points c {\displaystyle c} c such that the above sequence does not escape to infinity.
A mathematician's depiction of the Mandelbrot set M. A point c is colored black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of c, respectively.

More formally, if P c n ( z ) {\displaystyle P_{c}^{n}(z)} P_{c}^{n}(z) denotes the nth iterate of P c ( z ) {\displaystyle P_{c}(z)} P_{c}(z) (i.e. P c ( z ) {\displaystyle P_{c}(z)} P_{c}(z) composed with itself n times), the Mandelbrot set is the subset of the complex plane given by

    M = { c ∈ C : ∃ s ∈ R , ∀ n ∈ N , | P c n ( 0 ) | ≤ s } . {\displaystyle M=\left\{c\in \mathbb {C} :\exists s\in \mathbb {R} ,\forall n\in \mathbb {N} ,|P_{c}^{n}(0)|\leq s\right\}.} M=\left\{c\in \mathbb {C} :\exists s\in \mathbb {R} ,\forall n\in \mathbb {N} ,|P_{c}^{n}(0)|\leq s\right\}.

As explained below, it is in fact possible to simplify this definition by taking s = 2 {\displaystyle s=2} s=2.

Mathematically, the Mandelbrot set is just a set of complex numbers. A given complex number c either belongs to M or it does not. A picture of the Mandelbrot set can be made by coloring all the points c {\displaystyle c} c that belong to M black, and all other points white. The more colorful pictures usually seen are generated by coloring points not in the set according to which term in the sequence | P c n ( 0 ) | {\displaystyle |P_{c}^{n}(0)|} |P_{c}^{n}(0)| is the first term with an absolute value greater than a certain cutoff value, usually 2. See the section on computer drawings below for more details.

The Mandelbrot set can also be defined as the connectedness locus of the family of polynomials P c ( z ) {\displaystyle P_{c}(z)} P_{c}(z). That is, it is the subset of the complex plane consisting of those parameters c {\displaystyle c} c for which the Julia set of P c {\displaystyle P_{c}} P_{c} is connected.

P c n ( 0 ) {\displaystyle P_{c}^{n}(0)} P_{c}^{n}(0) is a polynomial in c and its leading terms settle down as n grows large enough. These terms are given by the Catalan numbers. The polynomials P c n ( 0 ) {\displaystyle P_{c}^{n}(0)} P_{c}^{n}(0) are bounded by the generating function for the Catalan numbers and tend to it as n goes to infinity.
Basic propertiesEdit

The Mandelbrot set is a compact set, since it is closed and contained in the closed disk of radius 2 around the origin. More specifically, a point c {\displaystyle c} c belongs to the Mandelbrot set if and only if

    | P c n ( 0 ) | ≤ 2 {\displaystyle |P_{c}^{n}(0)|\leq 2} |P_{c}^{n}(0)|\leq 2 for all n ≥ 0. {\displaystyle n\geq 0.} {\displaystyle n\geq 0.}

In other words, if the absolute value of P c n ( 0 ) {\displaystyle P_{c}^{n}(0)} P_{c}^{n}(0) ever becomes larger than 2, the sequence will escape to infinity.
Correspondence between the Mandelbrot set and the bifurcation diagram of the logistic map

The intersection of M {\displaystyle M} M 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 family,

    z ↦ λ z ( 1 − z ) , λ ∈ [ 1 , 4 ] . {\displaystyle z\mapsto \lambda z(1-z),\quad \lambda \in [1,4].} {\displaystyle z\mapsto \lambda z(1-z),\quad \lambda \in [1,4].}

The correspondence is given by

    c = λ 2 ( 1 − λ 2 ) . {\displaystyle c={\frac {\lambda }{2}}\left(1-{\frac {\lambda }{2}}\right).} c={\frac {\lambda }{2}}\left(1-{\frac {\lambda }{2}}\right).

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

As of October 2012, the area of the Mandelbrot is estimated to be 1.5065918849 ± 0.0000000028.[15]

Douady and Hubbard have shown that the Mandelbrot set is connected. In fact, 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 {\displaystyle M} M. Upon further experiments, he revised his conjecture, deciding that M {\displaystyle M} M should be connected.
External rays of wakes near the period 1 continent in the Mandelbrot set

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 {\displaystyle M} 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.[16]

The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of parameters c {\displaystyle c} c for which the dynamics changes abruptly under small changes of c . {\displaystyle c.} 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 p0 = z, pn+1 = pn2 + z, and then interpreting the set of points |pn(z)| = 2 in the complex plane as a curve in the real Cartesian plane of degree 2n+1 in x and y. These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.
Other propertiesEdit
Main cardioid and period bulbsEdit
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 c {\displaystyle c} c for which P c {\displaystyle P_{c}} P_{c} has an attracting fixed point. It consists of all parameters of the form

    c = μ 2 ( 1 − μ 2 ) {\displaystyle c={\frac {\mu }{2}}\left(1-{\frac {\mu }{2}}\right)} c={\frac {\mu }{2}}\left(1-{\frac {\mu }{2}}\right)

for some μ {\displaystyle \mu } \mu in the open unit disk.

To the left of the main cardioid, attached to it at the point c = − 3 / 4 {\displaystyle c=-3/4} c=-3/4, a circular-shaped bulb is visible. This bulb consists of those parameters c {\displaystyle c} c for which P c {\displaystyle P_{c}} P_{c} has an 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 p q {\displaystyle {\tfrac {p}{q}}} {\tfrac {p}{q}}, with p and q coprime, there is such a bulb that is tangent at the parameter

    c p q = e 2 π i p q 2 ( 1 − e 2 π i p q 2 ) . {\displaystyle c_{\frac {p}{q}}={\frac {e^{2\pi i{\frac {p}{q}}}}{2}}\left(1-{\frac {e^{2\pi i{\frac {p}{q}}}}{2}}\right).} c_{\frac {p}{q}}={\frac {e^{2\pi i{\frac {p}{q}}}}{2}}\left(1-{\frac {e^{2\pi i{\frac {p}{q}}}}{2}}\right).

Attracting cycle in 2/5-bulb plotted over Julia set (animation)

This bulb is called the p q {\displaystyle {\tfrac {p}{q}}} {\tfrac {p}{q}}-bulb of the Mandelbrot set. It consists of parameters that have an attracting cycle of period q {\displaystyle q} q and combinatorial rotation number p q {\displaystyle {\tfrac {p}{q}}} {\tfrac {p}{q}}. More precisely, the q {\displaystyle q} q periodic Fatou components containing the attracting cycle all touch at a common point (commonly called the α {\displaystyle \alpha } \alpha -fixed point). If we label these components U 0 , … , U q − 1 {\displaystyle U_{0},\dots ,U_{q-1}} U_{0},\dots ,U_{q-1} in counterclockwise orientation, then P c {\displaystyle P_{c}} P_{c} maps the component U j {\displaystyle U_{j}} U_{j} to the component U j + p ( mod ⁡ q ) {\displaystyle U_{j+p\,(\operatorname {mod} q)}} U_{j+p\,(\operatorname {mod} q)}.
Attracting cycles and Julia sets for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs
Cycle periods and antennae

The change of behavior occurring at c p q {\displaystyle c_{\frac {p}{q}}} c_{\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 p q {\displaystyle {\tfrac {p}{q}}} {\tfrac {p}{q}}-bulb, the attracting fixed point turns into a repelling fixed point (the α {\displaystyle \alpha } \alpha -fixed point), and the period q-cycle becomes attracting.
Hyperbolic componentsEdit

All the bulbs we encountered in the previous section were interior components of the Mandelbrot set in which the maps P c {\displaystyle P_{c}} P_{c} have an attracting periodic cycle. Such components are called hyperbolic components.

It is conjectured that these are the only interior regions of M {\displaystyle M} M. 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.[17][18] 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 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 P c ( z ) {\displaystyle P_{c}(z)} P_{c}(z) has a super-attracting cycle – that is, that the attraction is infinite (see the image 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 P c {\displaystyle P_{c}} P_{c}n ( 0 ) = 0 {\displaystyle (0)=0} (0)=0 for some n. If we call this polynomial Q n ( c ) {\displaystyle Q^{n}(c)} Q^{n}(c) (letting it depend on c instead of z), we have that Q n + 1 ( c ) = Q n ( c ) 2 + c {\displaystyle Q^{n+1}(c)=Q^{n}(c)^{2}+c} Q^{n+1}(c)=Q^{n}(c)^{2}+c and that the degree of Q n ( c ) {\displaystyle Q^{n}(c)} Q^{n}(c) is 2 n − 1 {\displaystyle 2^{n-1}} 2^{n-1}. We can therefore construct the centers of the hyperbolic components by successively solving the equations Q n ( c ) = 0 , n = 1 , 2 , 3 , . . . {\displaystyle Q^{n}(c)=0,n=1,2,3,...} Q^{n}(c)=0,n=1,2,3,.... The number of new centers produced in each step is given by Sloane's OEIS A000740.
Local connectivityEdit
Topological model of Mandelbrot set without mini Mandelbrot sets and Misiurewicz points (Cactus model)
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 renormalizable parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies.[19] Since then, local connectivity has been proved at many other points of M {\displaystyle M} M, but the full conjecture is still open.
Self-similarityEdit
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 ratio δ {\displaystyle \delta } \delta .
Self-similarity around Misiurewicz point −0.1011 + 0.9563i.

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.[20][21]
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 resultsEdit

The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro Shishikura.[22] 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 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.

The occurrence of π in the Mandelbrot set was discovered by David Boll in 1991.[23] He found that when looking at the pinch points of the Mandelbrot set, the number of iterations needed for the point (−3/4, ε) before escaping, multiplied by ε, was equal to π. Based on this initial finding, Aaron Klebanoff developed a further test near another pinch point (1/4 + ε, 0) in the Mandelbrot set and found that the number of iterations times the square root of ε was equal to π.
A zoom into the Mandelbrot set illustrating a Julia "island" and the corresponding Julia set of the form f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c\,} f_{c}(z)=z^{2}+c\,, in which c is the center of the Mandelbrot set zoom-in
Relationship with Julia setsEdit

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 proves 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.[22] Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters.[19] Adrien Douady phrases this principle as:

        Plough in the dynamical plane, and harvest in parameter space.

GeometryEdit
Components on main cardioid for periods 8–14 with antennae 7–13

For every rational number p q {\displaystyle {\tfrac {p}{q}}} {\tfrac {p}{q}}, 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 1 q 2 {\displaystyle {\tfrac {1}{q^{2}}}} {\displaystyle {\tfrac {1}{q^{2}}}}. The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like 1 q {\displaystyle {\tfrac {1}{q}}} {\tfrac {1}{q}}.

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.

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 = − 3 4 + i ϵ {\displaystyle -{\tfrac {3}{4}}+i\epsilon } {\displaystyle -{\tfrac {3}{4}}+i\epsilon } ( − 3 4 {\displaystyle -{\tfrac {3}{4}}} {\displaystyle -{\tfrac {3}{4}}} being the location thereof). As the series doesn't diverge for the exact value of z = − 3 4 {\displaystyle -{\tfrac {3}{4}}} {\displaystyle -{\tfrac {3}{4}}}, 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.[24]
Image gallery of a zoom sequenceEdit

The Mandelbrot set shows more intricate detail the closer one looks or 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 1010 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.

    Start. Mandelbrot set with continuously colored environment.

    Gap between the "head" and the "body", also called the "seahorse valley"

    Double-spirals on the left, "seahorses" on the right

    "Seahorse" upside down

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.

    The central endpoint of the "seahorse tail" is also a Misiurewicz point.

    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.

    Satellite. The two "seahorse tails" are the beginning of a series of concentric crowns with the satellite in the center. Open this location in an interactive viewer.

    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".

    "Antenna" of the satellite. Several satellites of second order may be recognized.

    The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.

    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.

    Double-spirals with satellites of second order – analogously to the "seahorses", the double-spirals may be interpreted as a metamorphosis of the "antenna"

    In the outer part of the appendices, islands of structures may be recognized; they have a shape like Julia sets Jc; the largest of them may be found in the center of the "double-hook" on the right side

    Part of the "double-hook"

    Islands

    Detail of one island

    Detail of the spiral

The islands above seem to consist of infinitely many parts like Cantor sets, as is[clarification needed] actually the case for the corresponding Julia set Jc. 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 Jc 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.
GeneralizationsEdit
Animation of the Mandelbrot set for d from 0 to 5
File:Mandelbrot set from powers 0.05 to 2.webmPlay media
Animation of the set for d from 0.05 to 2

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

    z ↦ z d + c .   {\displaystyle z\mapsto z^{d}+c.\ } z\mapsto z^{d}+c.\

For 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 ↦ z 3 + 3 k z + c {\displaystyle z\mapsto z^{3}+3kz+c} z\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.[25] For general families of holomorphic functions, 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.

In 2000, D. Rochon used the bicomplex numbers, a commutative generalization of the complex numbers, to give a new version of the Mandelbrot set in dimensions three and four. From his article arises the so-called Tetrabrot[26].
Animation of the Tetrabrot set

The Multibrot sets can also be generalized to the bicomplex and tricomplex numbers. These tricomplex Multibrot sets[27] include the Tetrabrot as a specific 3D slice.
Other, non-analytic, mappingsEdit

Of particular interest is the tricorn fractal, the connectedness locus of the anti-holomorphic family

    z ↦ z ¯ 2 + c . {\displaystyle z\mapsto {\bar {z}}^{2}+c.} {\displaystyle z\mapsto {\bar {z}}^{2}+c.}

The tricorn (also sometimes called the Mandelbar set) was encountered by Milnor in his study of parameter slices of real 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 mapping

    z ↦ ( | ℜ ( z ) | + i | ℑ ( z ) | ) 2 + c . {\displaystyle z\mapsto (|\Re \left(z\right)|+i|\Im \left(z\right)|)^{2}+c.} {\displaystyle z\mapsto (|\Re \left(z\right)|+i|\Im \left(z\right)|)^{2}+c.}

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. A similar development with negative exponents results in (1 − d) clefts on the inside of a ring.
Computer drawingsEdit
Still image of a movie of increasing magnification on 0.001643721971153 − 0.822467633298876i
Still image of an animation of increasing magnification

There are many programs used to generate the Mandelbrot set and other fractals, some of which are described in fractal-generating software. These programs use a variety of algorithms to determine the color of individual pixels and achieve efficient computation.
Escape time algorithmEdit

The simplest algorithm for generating a representation of the Mandelbrot set is known as the "escape time" algorithm. A repeating calculation is performed for each x, y point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel.

The x and y locations of each point are used as starting values in a repeating, or iterating calculation (described in detail below). The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout". If that condition is reached, the calculation is stopped, the pixel is drawn, and the next x, y point is examined. For some starting values, escape occurs quickly, after only a small number of iterations. For starting values very close to but not in the set, it may take hundreds or thousands of iterations to escape. For values within the Mandelbrot set, escape will never occur. The programmer or user must choose how much iteration, or "depth", they wish to examine. The higher the maximal number of iterations, the more detail and subtlety emerge in the final image, but the longer time it will take to calculate the fractal image.

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

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

To render such an image, the region of the complex plane we are considering is subdivided into a certain number of pixels. To color any such pixel, let c {\displaystyle c} c be the midpoint of that pixel. We now iterate the critical point 0 under P c {\displaystyle P_{c}} P_{c}, checking at each step whether the orbit point has modulus larger than 2. When this is the case, we know that c {\displaystyle c} c 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.

For each pixel (Px, Py) on the screen, do:
{
  x0 = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
  y0 = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))
  x = 0.0
  y = 0.0
  iteration = 0
  max_iteration = 1000
  while (x*x + y*y < 2*2  AND  iteration < max_iteration) {
    xtemp = x*x - y*y + x0
    y = 2*x*y + y0
    x = xtemp
    iteration = iteration + 1
  }
  color = palette[iteration]
  plot(Px, Py, color)
}

Here, relating the pseudocode to c {\displaystyle c} c, z {\displaystyle z} z and P c {\displaystyle P_{c}} P_{c}:

    z = x + i y   {\displaystyle z=x+iy\ } z=x+iy\
    z 2 = x 2 + i 2 x y − y 2   {\displaystyle z^{2}=x^{2}+i2xy-y^{2}\ } z^{2}=x^{2}+i2xy-y^{2}\
    c = x 0 + i y 0   {\displaystyle c=x_{0}+iy_{0}\ } c=x_{0}+iy_{0}\

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

    x = R e ( z 2 + c ) = x 2 − y 2 + x 0 {\displaystyle x={\mathop {\mathrm {Re} }}(z^{2}+c)=x^{2}-y^{2}+x_{0}} x={\mathop {\mathrm {Re} }}(z^{2}+c)=x^{2}-y^{2}+x_{0} and y = I m ( z 2 + c ) = 2 x y + y 0 .   {\displaystyle y={\mathop {\mathrm {Im} }}(z^{2}+c)=2xy+y_{0}.\ } y={\mathop {\mathrm {Im} }}(z^{2}+c)=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.). One practical way, without slowing down calculations, is to use the number of executed iterations as an entry to a look-up color palette table initialized at startup. If the color table has, for instance, 500 entries, then the color selection is n mod 500, where n is the number of iterations.
Histogram coloringEdit

A more accurate coloring method involves using a histogram, which keeps track of how many pixels reached each iteration number, from 1 to n. This method will equally distribute colors to the same overall area, and, importantly, is independent of the maximal number of iterations chosen.

First, create an array of size n. For each pixel, which took i iterations, find the ith element and increment it. This creates the histogram during computation of the image. Then, when finished, perform a second "rendering" pass over each pixel, utilizing the completed histogram. If you had a continuous color palette ranging from 0 to 1, you could find the normalized color of each pixel as follows, using the variables from above.

total = 0
for (i = 0; i < max_iterations; i += 1) {
  total += histogram[i]
}

hue = 0.0;
for (i = 0; i < iteration; i += 1) {
  hue += histogram[i] / total // Must be floating-point division.
}

color = palette[hue]

This method may be combined with the smooth coloring method below for more aesthetically pleasing images.
Continuous (smooth) coloringEdit
This image was rendered with the escape time algorithm. There are very obvious "bands" of color
This image was rendered with the normalized iteration count algorithm. The bands of color have been replaced by a smooth gradient. Also, the colors take on the same pattern that would be observed if the escape time algorithm were used.

The escape time algorithm is popular for its simplicity. However, it creates bands of color, which, as a type of aliasing, can detract from an image's aesthetic value. This can be improved using an algorithm known as "normalized iteration count",[28][29] which provides a smooth transition of colors between iterations. The algorithm associates a real number ν {\displaystyle \nu } \nu with each value of z by using the connection of the iteration number with the potential function. This function is given by

    ϕ ( z ) = lim n → ∞ ( log ⁡ | z n | / P n ) , {\displaystyle \phi (z)=\lim _{n\to \infty }(\log |z_{n}|/P^{n}),} {\displaystyle \phi (z)=\lim _{n\to \infty }(\log |z_{n}|/P^{n}),}

where zn is the value after n iterations and P is the power for which z is raised to in the Mandelbrot set equation (zn+1 = znP + c, P is generally 2).

If we choose a large bailout radius N (e.g., 10100), we have that

    log ⁡ | z n | / P n = log ⁡ ( N ) / P ν ( z ) {\displaystyle \log |z_{n}|/P^{n}=\log(N)/P^{\nu (z)}} {\displaystyle \log |z_{n}|/P^{n}=\log(N)/P^{\nu (z)}}

for some real number ν ( z ) {\displaystyle \nu (z)} \nu (z), and this is

    ν ( z ) = n − log P ⁡ ( log ⁡ | z n | / log ⁡ ( N ) ) , {\displaystyle \nu (z)=n-\log _{P}(\log |z_{n}|/\log(N)),} {\displaystyle \nu (z)=n-\log _{P}(\log |z_{n}|/\log(N)),}

and as n is the first iteration number such that |zn| > N, the number we subtract from n is in the interval [0, 1).

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

For example, modifying the above pseudocode and also using the concept of linear interpolation would yield

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

Distance estimatesEdit

One can compute the distance from point c (in exterior or interior) to nearest point on the boundary of the Mandelbrot set.[30]
Exterior distance estimationEdit

The proof of the connectedness of the Mandelbrot set in fact gives a formula for the uniformizing map of the complement of M {\displaystyle M} M (and the derivative of this map). By the Koebe 1/4 theorem, one can then estimate the distance between the midpoint of our pixel and the Mandelbrot set up to a factor of 4.

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

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

Exterior distance estimate may be used to color whole complement of Mandelbrot set

The distance estimate b of a pixel c (a complex number) from the Mandelbrot set is given by

    b = lim n → ∞ 2 ⋅ | P c n ( c ) | ⋅ ln ⁡ | P c n ( c ) | | ∂ ∂ c P c n ( c ) | , {\displaystyle b=\lim _{n\to \infty }2\cdot {\frac {|{P_{c}^{n}(c)|\cdot \ln |{P_{c}^{n}(c)}}|}{|{\frac {\partial }{\partial {c}}}P_{c}^{n}(c)|}},} {\displaystyle b=\lim _{n\to \infty }2\cdot {\frac {|{P_{c}^{n}(c)|\cdot \ln |{P_{c}^{n}(c)}}|}{|{\frac {\partial }{\partial {c}}}P_{c}^{n}(c)|}},}

where

    P c ( z ) {\displaystyle P_{c}(z)\,} P_{c}(z)\, stands for complex quadratic polynomial
    P c n ( c ) {\displaystyle P_{c}^{n}(c)} P_{c}^{n}(c) stands for n iterations of P c ( z ) → z {\displaystyle P_{c}(z)\to z} P_{c}(z)\to z or z 2 + c → z {\displaystyle z^{2}+c\to z} z^{2}+c\to z, starting with z = c {\displaystyle z=c} z=c: P c 0 ( c ) = c {\displaystyle P_{c}^{0}(c)=c} P_{c}^{0}(c)=c, P c n + 1 ( c ) = P c n ( c ) 2 + c {\displaystyle P_{c}^{n+1}(c)=P_{c}^{n}(c)^{2}+c} P_{c}^{n+1}(c)=P_{c}^{n}(c)^{2}+c;
    ∂ ∂ c P c n ( c ) {\displaystyle {\frac {\partial }{\partial {c}}}P_{c}^{n}(c)} {\frac {\partial }{\partial {c}}}P_{c}^{n}(c) is the derivative of P c n ( c ) {\displaystyle P_{c}^{n}(c)} P_{c}^{n}(c) with respect to c. This derivative can be found by starting with ∂ ∂ c P c 0 ( c ) = 1 {\displaystyle {\frac {\partial }{\partial {c}}}P_{c}^{0}(c)=1} {\frac {\partial }{\partial {c}}}P_{c}^{0}(c)=1 and then ∂ ∂ c P c n + 1 ( c ) = 2 ⋅ P c n ( c ) ⋅ ∂ ∂ c P c n ( c ) + 1 {\displaystyle {\frac {\partial }{\partial {c}}}P_{c}^{n+1}(c)=2\cdot {}P_{c}^{n}(c)\cdot {\frac {\partial }{\partial {c}}}P_{c}^{n}(c)+1} {\frac {\partial }{\partial {c}}}P_{c}^{n+1}(c)=2\cdot {}P_{c}^{n}(c)\cdot {\frac {\partial }{\partial {c}}}P_{c}^{n}(c)+1. This can easily be verified by using the chain rule for the derivative.

The idea behind this formula is simple: When the equipotential lines for the potential function ϕ ( z ) {\displaystyle \phi (z)} \phi (z) lie close, the number | ϕ ′ ( z ) | {\displaystyle |\phi '(z)|} |\phi '(z)| is large, and conversely, therefore the equipotential lines for the function ϕ ( z ) / | ϕ ′ ( z ) | {\displaystyle \phi (z)/|\phi '(z)|} \phi (z)/|\phi '(z)| should lie approximately regularly.

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

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

The distance estimation can be used for drawing of the boundary of the Mandelbrot set, see the article Julia set.
Interior distance estimationEdit
Pixels colored according to the estimated interior distance

It is also possible to estimate the distance of a limitly periodic (i.e., inner) point to the boundary of the Mandelbrot set. The estimate is given by

    b = 1 − | ∂ ∂ z P c p ( z 0 ) | 2 | ∂ ∂ c ∂ ∂ z P c p ( z 0 ) + ∂ ∂ z ∂ ∂ z P c p ( z 0 ) ∂ ∂ c P c p ( z 0 ) 1 − ∂ ∂ z P c p ( z 0 ) | , {\displaystyle b={\frac {1-\left|{{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0})}\right|^{2}}{\left|{{\frac {\partial }{\partial {c}}}{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0})+{\frac {\partial }{\partial {z}}}{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0}){\frac {{\frac {\partial }{\partial {c}}}P_{c}^{p}(z_{0})}{1-{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0})}}}\right|}},} {\displaystyle b={\frac {1-\left|{{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0})}\right|^{2}}{\left|{{\frac {\partial }{\partial {c}}}{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0})+{\frac {\partial }{\partial {z}}}{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0}){\frac {{\frac {\partial }{\partial {c}}}P_{c}^{p}(z_{0})}{1-{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0})}}}\right|}},}

where

    p {\displaystyle p} p is the period,
    c {\displaystyle c} c is the point to be estimated,
    P c ( z ) {\displaystyle P_{c}(z)} P_{c}(z) is the complex quadratic polynomial P c ( z ) = z 2 + c {\displaystyle P_{c}(z)=z^{2}+c} P_{c}(z)=z^{2}+c
    P c p ( z 0 ) {\displaystyle P_{c}^{p}(z_{0})} P_{c}^{p}(z_{0}) is the p {\displaystyle p} p-fold iteration of P c ( z ) → z {\displaystyle P_{c}(z)\to z} P_{c}(z)\to z, starting with P c 0 ( z ) = z 0 {\displaystyle P_{c}^{0}(z)=z_{0}} P_{c}^{0}(z)=z_{0}
    z 0 {\displaystyle z_{0}} z_{0} is any of the p {\displaystyle p} p points that make the attractor of the iterations of P c ( z ) → z {\displaystyle P_{c}(z)\to z} P_{c}(z)\to z starting with P c 0 ( z ) = c {\displaystyle P_{c}^{0}(z)=c} P_{c}^{0}(z)=c; z 0 {\displaystyle z_{0}} z_{0} satisfies z 0 = P c p ( z 0 ) {\displaystyle z_{0}=P_{c}^{p}(z_{0})} z_{0}=P_{c}^{p}(z_{0}),
    ∂ ∂ c ∂ ∂ z P c p ( z 0 ) {\displaystyle {\frac {\partial }{\partial {c}}}{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0})} {\frac {\partial }{\partial {c}}}{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0}), ∂ ∂ z ∂ ∂ z P c p ( z 0 ) {\displaystyle {\frac {\partial }{\partial {z}}}{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0})} {\frac {\partial }{\partial {z}}}{\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0}), ∂ ∂ c P c p ( z 0 ) {\displaystyle {\frac {\partial }{\partial {c}}}P_{c}^{p}(z_{0})} {\frac {\partial }{\partial {c}}}P_{c}^{p}(z_{0}) and ∂ ∂ z P c p ( z 0 ) {\displaystyle {\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0})} {\frac {\partial }{\partial {z}}}P_{c}^{p}(z_{0}) are various derivatives of P c p ( z ) {\displaystyle P_{c}^{p}(z)} P_{c}^{p}(z), evaluated at z 0 {\displaystyle z_{0}} z_{0}.

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

There are two practical problems with the interior distance estimate: first, we need to find z 0 {\displaystyle z_{0}} z_{0} precisely, and second, we need to find p {\displaystyle p} p precisely. The problem with z 0 {\displaystyle z_{0}} z_{0} is that the convergence to z 0 {\displaystyle z_{0}} z_{0} by iterating P c ( z ) {\displaystyle P_{c}(z)} P_{c}(z) requires, theoretically, an infinite number of operations. The problem with any given p {\displaystyle p} p is that, sometimes, due to rounding errors, a period is falsely identified to be an integer multiple of the real period (e.g., a period of 86 is detected, while the real period is only 43=86/2). In such case, the distance is overestimated, i.e., the reported radius could contain points outside the Mandelbrot set.
3D view: smallest absolute value of the orbit of the interior points of the Mandelbrot set
OptimizationsEdit
Cardioid / bulb checkingEdit

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

    p = ( x − 1 4 ) 2 + y 2 {\displaystyle p={\sqrt {\left(x-{\frac {1}{4}}\right)^{2}+y^{2}}}} p={\sqrt {\left(x-{\frac {1}{4}}\right)^{2}+y^{2}}},
    x < p − 2 p 2 + 1 4 {\displaystyle x<p-2p^{2}+{\frac {1}{4}}} x<p-2p^{2}+{\frac {1}{4}},
    ( x + 1 ) 2 + y 2 < 1 16 {\displaystyle (x+1)^{2}+y^{2}<{\frac {1}{16}}} (x+1)^{2}+y^{2}<{\frac {1}{16}},

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

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

    q = ( x − 1 4 ) 2 + y 2 , {\displaystyle q=\left(x-{\frac {1}{4}}\right)^{2}+y^{2},} {\displaystyle q=\left(x-{\frac {1}{4}}\right)^{2}+y^{2},}
    q ( q + ( x − 1 4 ) ) < 1 4 y 2 . {\displaystyle q\left(q+\left(x-{\frac {1}{4}}\right)\right)<{\frac {1}{4}}y^{2}.} q\left(q+\left(x-{\frac {1}{4}}\right)\right)<{\frac {1}{4}}y^{2}.

3rd- and higher-order buds do not have equivalent tests, because they are not perfectly circular.[31] However, it is possible to find whether the points are within circles inscribed within these higher-order bulbs, preventing many, though not all, of the points in the bulb from being iterated.
Periodicity checkingEdit

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

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

However, checking against only one previous iteration can detect many periods with little performance overhead. For example, within the while loop of the pseudocode above, make the following modifications.

  while (x*x + y*y < 2*2  AND  iteration < max_iteration) {
    xtemp = x*x - y*y + x0
    ytemp = 2*x*y + y0
    if (x == xtemp  AND  y == ytemp) {
      iteration = max_iteration
      break
    }
    x = xtemp
    y = ytemp
    iteration = iteration + 1
  }

Border tracing / edge checkingEdit
Edge detection using Sobel filter of hyperbolic components of Mandelbrot set

It can be shown that if a solid shape can be drawn on the Mandelbrot set, with all the border colors being the same, then the shape can be filled in with that color. This is a result of the Mandelbrot set being simply connected. Boundary tracing works by following the lemniscates of the various iteration levels (colored bands) all around the set, and then filling the entire band at once. This can be a good speed increase, because it means that large numbers of points can be skipped.[32]

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

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

Very highly magnified images require more than the standard 64–128 or so bits of precision that most hardware floating-point units provide, requiring renderers to use slow "bignum" or "arbitrary-precision" math libraries to calculate. However, this can be sped up by the exploitation of perturbation theory. Given

    z n + 1 = z n 2 + c {\displaystyle z_{n+1}=z_{n}^{2}+c} z_{n+1}=z_{n}^{2}+c

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

    ( z n + ϵ ) 2 + ( c + δ ) = z n 2 + 2 z n ϵ + ϵ 2 + c + δ , {\displaystyle (z_{n}+\epsilon )^{2}+(c+\delta )=z_{n}^{2}+2z_{n}\epsilon +\epsilon ^{2}+c+\delta ,} {\displaystyle (z_{n}+\epsilon )^{2}+(c+\delta )=z_{n}^{2}+2z_{n}\epsilon +\epsilon ^{2}+c+\delta ,}

or

    z n + 1 + 2 z n ϵ + ϵ 2 + δ , {\displaystyle z_{n+1}+2z_{n}\epsilon +\epsilon ^{2}+\delta ,} {\displaystyle z_{n+1}+2z_{n}\epsilon +\epsilon ^{2}+\delta ,}

so if one defines

    ϵ n + 1 = 2 z n ϵ n + ϵ n 2 + δ , {\displaystyle \epsilon _{n+1}=2z_{n}\epsilon _{n}+\epsilon _{n}^{2}+\delta ,} {\displaystyle \epsilon _{n+1}=2z_{n}\epsilon _{n}+\epsilon _{n}^{2}+\delta ,}

one can calculate a single point (e.g. the center of an image) using high-precision arithmetic (z), giving a reference orbit, and then compute many points around it in terms of various initial offsets delta plus the above iteration for epsilon, where epsilon-zero is set to 0. For most iterations, epsilon does not need more than 16 significant figures, and consequently hardware floating-point may be used to get a mostly accurate image.[33] There will often be some areas where the orbits of points diverge enough from the reference orbit that extra precision is needed on those points, or else additional local high-precision-calculated reference orbits are needed. By measuring the orbit distance between the reference point and the point calculated with low precision, it can be detected that it is not possible to calculate the point correctly, and the calculation can be stopped. These incorrect points can later be re-calculated e.g. from another closer reference point.

Further, it is possible to approximate the starting values for the low-precision points with a truncated Taylor series, which often enables a significant amount of iterations to be skipped.[34] Renderers implementing these techniques are publicly available and offer speedups for highly magnified images by around two orders of magnitude.[35]
References in popular cultureEdit

    The Jonathan Coulton song "Mandelbrot Set" is a tribute to both the fractal itself and to its father Benoit Mandelbrot.[36]
    The second book of the Mode series by Piers Anthony, Fractal Mode, describes a world that is a perfect 3D model of the set.[37]
    The Arthur C. Clarke novel The Ghost from the Grand Banks features an artificial lake made to replicate the shape of the Mandelbrot set.[38]
    The South Korean heavy metal singer Norazo made a music video Ni pal za ya (your fortune), which starts with hypnotic video including Mandelbrot set.[39]
    The album Jupiters Darling by American rock band Heart prominently features a Mandelbrot set on the cover. The set is rotated so that the cusp is on the top, resembling a heart.
    In the movie American Ultra, the fictional character Mike Howell (played by actor Jesse Eisenberg) is activated as a sleeper agent after hearing a coded message including the phrase "Mandelbrot set is in motion."