Mandelbrot Set
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The Mandelbrot set () is the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of
Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's
Thomas J. Watson Research Center The Thomas J. Watson Research Center is the headquarters for IBM Research. The center comprises three sites, with its main laboratory in Yorktown Heights, New York, U.S., 38 miles (61 km) north of New York City, Albany, New York and wit ...
in
Yorktown Heights, New York Yorktown Heights is a census-designated place (CDP) in the town of Yorktown in Westchester County, New York, United States. The population was 1,781 at the 2010 census. History Yorktown Heights is in the town of Yorktown, New York, in northern ...
. Images of the Mandelbrot set exhibit an elaborate and infinitely complicated
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
that reveals progressively ever-finer
recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
detail at increasing magnifications; mathematically, one would say that the boundary of the Mandelbrot set is a ''
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
curve''. The "style" of this recursive detail depends on the region of the set boundary being examined. Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point c, whether the sequence f_c(0), f_c(f_c(0)),\dotsc goes to infinity. Treating the
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and
imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s of c as
image coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
s on the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, pixels may then be coloured according to how soon the sequence , f_c(0), , , f_c(f_c(0)), ,\dotsc crosses an arbitrarily chosen threshold (the threshold has to be at least 2, as -2 is the complex number with the largest magnitude within the set, but otherwise the threshold is arbitrary). If c is held constant and the initial value of z is varied instead, one obtains the corresponding
Julia set In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
for the point c. The Mandelbrot set has become popular outside
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of
mathematical visualization Mathematical phenomena can be understood and explored via visualization. Classically this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century), while today it ...
, mathematical beauty, and motif.


History

The Mandelbrot set has its origin in
complex dynamics Complex dynamics is the study of dynamical systems defined by Iterated function, iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Mo ...
, a field first investigated by the French mathematicians
Pierre Fatou Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him. Biography P ...
and
Gaston Julia Gaston Maurice Julia (3 February 1893 – 19 March 1978) was a French Algerian mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are ...
at the beginning of the 20th century. This fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of
Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s.Robert Brooks and Peter Matelski, ''The dynamics of 2-generator subgroups of PSL(2,C)'', in On 1 March 1980, at IBM's
Thomas J. Watson Research Center The Thomas J. Watson Research Center is the headquarters for IBM Research. The center comprises three sites, with its main laboratory in Yorktown Heights, New York, U.S., 38 miles (61 km) north of New York City, Albany, New York and wit ...
in
Yorktown Heights Yorktown Heights is a census-designated place (CDP) in the town of Yorktown in Westchester County, New York, United States. The population was 1,781 at the 2010 census. History Yorktown Heights is in the town of Yorktown, New York, in northern ...
, New York, Benoit Mandelbrot first saw a visualization of the set. Mandelbrot studied the
parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for th ...
of
quadratic polynomial In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
s in an article that appeared in 1980. The mathematical study of the Mandelbrot set really began with work by the mathematicians
Adrien Douady Adrien Douady (; 25 September 1935 – 2 November 2006) was a French mathematician. Douady was a student of Henri Cartan at the École normale supérieure, and initially worked in homological algebra. His thesis concerned deformations of complex ...
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 In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
. The mathematicians
Heinz-Otto Peitgen Heinz-Otto Peitgen (born April 30, 1945 in Bruch, Nümbrecht near Cologne) is a German mathematician and was President of Jacobs University from January 1, 2013 to December 31, 2013. Peitgen contributed to the study of fractals, chaos theory, an ...
and Peter Richter became well known for promoting the set with photographs, books (1986), and an internationally touring exhibit of the German
Goethe-Institut The Goethe-Institut (, GI, en, Goethe Institute) is a non-profit German cultural association operational worldwide with 159 institutes, promoting the study of the German language abroad and encouraging international cultural exchange and ...
(1985). The cover article of the August 1985 ''
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
'' introduced a wide audience to the
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
for computing the Mandelbrot set. The cover was created by Peitgen, Richter and Saupe at the
University of Bremen The University of Bremen (German: ''Universität Bremen'') is a public university in Bremen, Germany, with approximately 23,500 people from 115 countries. It is one of 11 institutions which were successful in the category "Institutional Strategi ...
. The Mandelbrot set became prominent in the mid-1980s as a computer graphics demo, when
personal computer A personal computer (PC) is a multi-purpose microcomputer whose size, capabilities, and price make it feasible for individual use. Personal computers are intended to be operated directly by an end user, rather than by a computer expert or tec ...
s became powerful enough to plot and display the set in high resolution. The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics and abstract mathematics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of all who have contributed to the understanding of this set since then is long but would include
Jean-Christophe Yoccoz Jean-Christophe Yoccoz (29 May 1957 – 3 September 2016) was a French mathematician. He was awarded a Fields Medal in 1994, for his work on dynamical systems. Biography Yoccoz attended the Lycée Louis-le-Grand, during which time he was a silv ...
, Mitsuhiro Shishikura, Curt McMullen,
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
and
Mikhail Lyubich Mikhail Lyubich (born 25 February 1959 in Kharkiv, Ukraine) is a mathematician who made important contributions to the fields of holomorphic dynamics and chaos theory. Lyubich graduated from Kharkiv University with a master's degree in 1980, and ...
.


Formal definition

The Mandelbrot set is the set of values of ''c'' in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
for which the
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
of the critical point z = 0 under
iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
of the quadratic map :z_ = z_^2 + c remains bounded. Thus, a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
''c'' is a member of the Mandelbrot set if, when starting with z_0 = 0 and applying the iteration repeatedly, the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of z_n remains bounded for all n > 0. For example, for ''c'' = 1, the
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
is 0, 1, 2, 5, 26, ..., which tends to
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
, 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 In one-dimensional complex dynamics, the connectedness locus of a parameterized family of one-variable holomorphic functions is a subset of the parameter space which consists of those parameters for which the corresponding Julia set is connected. ...
of the family of quadratic
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s, while its
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
can be defined as the
bifurcation locus In complex dynamics, the bifurcation locus of a parameterized family of one-variable holomorphic functions informally is a locus of those parameterized points for which the dynamical behavior changes drastically under a small perturbation of the pa ...
of this quadratic family.


Basic properties

The Mandelbrot set is a
compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
, since it is closed and contained in the
closed disk In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usu ...
of radius 2 around the
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. More specifically, a point c belongs to the Mandelbrot set if and only if , z_n, \leq 2 for all n\geq 0. In other words, the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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, \leq 2, establishing that c will always be in the closed disk of radius 2 around the origin. The intersection of M with the real axis is precisely the interval 2,\frac/math>. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family, :x_ = r x_n(1-x_n),\quad r\in ,4 The correspondence is given by :z = r\left(\frac12 - x\right), \quad c = \frac\left(1-\frac\right). In fact, this gives a correspondence between the entire
parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for th ...
of the logistic family and that of the Mandelbrot set. Douady and Hubbard have shown that the Mandelbrot set is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
. 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. Upon further experiments, he revised his conjecture, deciding that M should be connected. There also exists a
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
proof to the connectedness that was discovered in 2001 by
Jeremy Kahn Jeremy Adam Kahn (born October 26, 1969) is an American mathematician. He works on hyperbolic geometry, Riemann surfaces and complex dynamics. Education Kahn grew up in New York City and attended Hunter College High School. He was a child prod ...
. 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 ray An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set. Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray. External rays are used in complex analysis, particularly ...
s of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle. As mentioned earlier in the article, the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
of the Mandelbrot set is the
bifurcation locus In complex dynamics, the bifurcation locus of a parameterized family of one-variable holomorphic functions informally is a locus of those parameterized points for which the dynamical behavior changes drastically under a small perturbation of the pa ...
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_^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 lemniscate In mathematics, a polynomial lemniscate or ''polynomial level curve'' is a plane algebraic curve of degree 2n, constructed from a polynomial ''p'' with complex coefficients of degree ''n''. For any such polynomial ''p'' and positive real number ' ...
s. The Mandelbrot curves are defined by setting p_0=z,\ p_=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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
of degree 2^in ''x'' and ''y''. 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.


Other properties


Main cardioid and period bulbs

Upon looking at a picture of the Mandelbrot set, one immediately notices the large
cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal ...
region in the center. This ''main cardioid'' is the region of parameters c for which the map :f_c(z) = z^2 + c has an attracting fixed point. It consists of all parameters of the form : c(\mu) := \frac\mu2\left(1-\frac\mu2\right) for some \mu in the
open unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
. To the left of the main cardioid, attached to it at the point c=-3/4, a circular bulb called the ''period-2 bulb'' is visible. The reason for the name is that the bulb consists of precisely those parameters c for which f_c has an attracting cycle of period 2. It is in fact the filled circle of radius 1/4 centered around −1. More generally, for every positive integer q>2, there are \phi(q) circular bulbs tangent to the main cardioid called ''period-q bulbs'' (where \phi denotes the Euler phi function), which consist of parameters c for which f_c has an attracting cycle of period q. More specifically, for each primitive qth root of unity r=e^ (where 0<\frac<1), there is one period-q bulb called the \frac bulb, which is tangent to the main cardioid at the parameter : c_ := c(r) = \frac2\left(1-\frac2\right), and which contains parameters with q-cycles having combinatorial rotation number \frac. More precisely, the q periodic Fatou components containing the attracting cycle all touch at a common point (commonly called the ''\alpha-fixed point''). If we label these components U_0,\dots,U_ in counterclockwise orientation, then f_c maps the component U_j to the component U_. The change of behavior occurring at c_ is known as a
bifurcation Bifurcation or bifurcated may refer to: Science and technology * Bifurcation theory, the study of sudden changes in dynamical systems ** Bifurcation, of an incompressible flow, modeled by squeeze mapping the fluid flow * River bifurcation, the ...
: the attracting fixed point "collides" with a repelling period-''q'' cycle. As we pass through the bifurcation parameter into the \tfrac-bulb, the attracting fixed point turns into a repelling fixed point (the \alpha-fixed point), and the period-''q'' cycle becomes attracting.


Hyperbolic components

All the bulbs we encountered in the previous section were interior components of the Mandelbrot set in which the maps f_c have an attracting periodic cycle. Such components are called ''hyperbolic components''. It is conjectured that these are the ''only'' interior regions of M. This problem, known as ''density of hyperbolicity'', may be the most important open problem in the field of
complex dynamics Complex dynamics is the study of dynamical systems defined by Iterated function, iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Mo ...
. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components. 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 f_c(z) has a super-attracting cycle—that is, that the attraction is infinite (see the image
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Technologies, Here Television * Here TV (form ...
). 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 f_c^n(0) = 0 for some ''n''. If we call this polynomial Q^(c) (letting it depend on ''c'' instead of ''z''), we have that Q^(c) = Q^(c)^ + c and that the degree of Q^(c) is 2^. We can therefore construct the centers of the hyperbolic components by successively solving the equations Q^(c) = 0, n = 1, 2, 3, .... The number of new centers produced in each step is given by Sloane's .


Local connectivity

It is conjectured that the Mandelbrot set is
locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness ...
. This famous conjecture is known as ''MLC'' (for ''Mandelbrot locally connected''). By the work of
Adrien Douady Adrien Douady (; 25 September 1935 – 2 November 2006) was a French mathematician. Douady was a student of Henri Cartan at the École normale supérieure, and initially worked in homological algebra. His thesis concerned deformations of complex ...
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 Jean-Christophe Yoccoz (29 May 1957 – 3 September 2016) was a French mathematician. He was awarded a Fields Medal in 1994, for his work on dynamical systems. Biography Yoccoz attended the Lycée Louis-le-Grand, during which time he was a silv ...
established local connectivity of the Mandelbrot set at all finitely
renormalizable Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies.. 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.


Self-similarity

The Mandelbrot set is self-similar under magnification in the neighborhoods of the
Misiurewicz point In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point (mathematics), critical point is strictly p ...
s. It is also conjectured to be self-similar around generalized Feigenbaum points (e.g., −1.401155 or −0.1528 + 1.0397''i''), in the sense of converging to a limit 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. These little copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set.


Further results

The
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
of the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
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
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
nature of the Mandelbrot set boundary. In fact, Shishikura's result intuitively states that the Mandelbrot set boundary is so "wiggly" that it manages to locally fill up space as efficiently as a two-dimensional planar region. This opens up the potential possibility that the Mandelbrot set boundary, despite being a curve, has a ''nonzero'' area (or, more formally, has a positive planar Lebesgue measure). Whether or not this is actually the case remains an open problem today. It has been shown that the generalized Mandelbrot set in higher-dimensional hypercomplex number spaces (i.e. when the power \alpha of the iterated variable z tends to infinity) is convergent to the unit (\alpha-1)-sphere. In the Blum–Shub–Smale model of
real computation In computability theory, the theory of real computation deals with hypothetical computing machines using infinite-precision real numbers. They are given this name because they operate on the set of real numbers. Within this theory, it is possibl ...
, the Mandelbrot set is not computable, but its complement is
computably enumerable In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ...
. 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 In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carried out in a co ...
, 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.


Relationship with Julia sets

As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
of the Mandelbrot set at a given point and the structure of the corresponding
Julia set In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
. For instance, a point is in the Mandelbrot set exactly when the corresponding Julia set is connected. Thus, the Mandelbrot set may be seen as a map of the connected Julia sets. 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 In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
two, and then transfers this information to the parameter plane.. Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters.
Adrien Douady Adrien Douady (; 25 September 1935 – 2 November 2006) was a French mathematician. Douady was a student of Henri Cartan at the École normale supérieure, and initially worked in homological algebra. His thesis concerned deformations of complex ...
eloquently summarizes this principle as: :


Geometry

For every rational number \tfrac, where ''p'' and ''q'' are
relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
, 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 In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
of \tfrac. 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 In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
of the limb tends to zero like \tfrac. The best current estimate known is the '' Yoccoz-inequality'', which states that the size tends to zero like \tfrac. 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. We can also find the numerator of the rotation number, ''p'', by numbering each antenna counterclockwise from the limb from 1 to q-1 and finding which antenna is the shortest.


Pi in the Mandelbrot set

In an attempt to demonstrate that the thickness of the ''p''/''q''-limb is zero, David Boll carried out a
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
experiment in 1991, where he computed the number of iterations required for the
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...
to diverge for z = -\tfrac+ i\varepsilon (-\tfrac being the location thereof). As the series does not diverge for the exact value of z = -\tfrac, the number of iterations required increases with a small \varepsilon. It turns out that multiplying the value of \varepsilon with the number of iterations required yields an approximation of \pi that becomes better for smaller ''\varepsilon''. For example, for ''\varepsilon'' = 0.0000001 the number of iterations is 31415928 and the product is 3.1415928. In 2001, Aaron Klebanoff proved Boll's discovery.


Fibonacci sequence in the Mandelbrot set

It can be shown that the
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
is located within the Mandelbrot set and that a relation exists between the main cardioid and the
Farey Diagram In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ''n'', arranged in order ...
. Upon mapping the main cardioid to a disk, one can notice that the amount of antennae that extends from the next largest Hyperbolic component, and that is located between the two previously selected components, follows suit with the Fibonacci sequence. The amount of antennae also correlates with the Farey Diagram and the denominator amounts within the corresponding fractional values, of which relate to the distance around the disk. Both portions of these fractional values themselves can be summed together after \frac to produce the location of the next Hyperbolic component within the sequence. Thus, the Fibonacci sequence of 1, 2, 3, 5, 8, 13, and 21 can be found within the Mandelbrot set.


Image gallery of a zoom sequence

The boundary of the Mandelbrot set shows more intricate detail the closer one looks or 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 present in the Mandelbrot set boundary 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
computer monitor A computer monitor is an output device that displays information in pictorial or textual form. A discrete monitor comprises a visual display, support electronics, power supply, housing, electrical connectors, and external user controls. The di ...
, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers. Its border shows an astronomical number of different fractal structures. Mandel zoom 00 mandelbrot set.jpg, Start. Mandelbrot set with continuously colored environment. Mandel zoom 01 head and shoulder.jpg, Gap between the "head" and the "body", also called the "seahorse valley" Mandel zoom 02 seehorse valley.jpg, Double-spirals on the left, "seahorses" on the right Mandel zoom 03 seehorse.jpg, "Seahorse" upside down 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 In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point (mathematics), critical point is strictly p ...
. Between the "upper part of the body" and the "tail" a distorted small copy of the Mandelbrot set called satellite may be recognized. Mandel zoom 04 seehorse tail.jpg, The central endpoint of the "seahorse tail" is also a
Misiurewicz point In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point (mathematics), critical point is strictly p ...
. 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. Mandel zoom 06 double hook.jpg, Satellite. The two "seahorse tails" are the beginning of a series of concentric crowns with the satellite in the center. 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". Mandel zoom 08 satellite antenna.jpg, "Antenna" of the satellite. Several satellites of second order may be recognized. Mandel zoom 09 satellite head and shoulder.jpg, The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear. 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. 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". Mandel zoom 12 satellite spirally wheel with julia islands.jpg, In the outer part of the appendices, islands of structures may be recognized; they have a shape like
Julia set In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
s ''Jc''; the largest of them may be found in the center of the "double-hook" on the right side. Mandel zoom 13 satellite seehorse tail with julia island.jpg, Part of the "double-hook" Mandel zoom 14 satellite julia island.jpg, Islands Mandel zoom 15 one island.jpg, Detail of one island Mandel zoom 16 spiral island.jpg, Detail of the spiral
Open this location in an interactive viewer.
The islands in the third-to-last step seem to consist of infinitely many parts like
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Thr ...
s, as is actually the case for the corresponding Julia set J_c. However, they are connected by tiny structures, so that the whole represents a simply connected set. The tiny structures meet each other at a satellite in the center that is too small to be recognized at this magnification. The value of ''c '' for the corresponding ''J_c'' 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.


Inner structure

While the Mandelbrot set is typically rendered showing the outside boundary detail, structure within the bounded set can also be revealed. For example, while calculating whether or not a given c value is bound or unbound, while it remains bound, the maximum value that this number reaches can be compared to the c value at that location. If the sum of squares method is used, the calculated number would be max:(real^2 + imaginary^2) - c:(real^2 + imaginary^2). The magnitude of this calculation can be rendered as a value on a gradient. This produces results like the following, gradients with distinct edges and contours as the boundaries are approached. The animations serve to highlight the gradient boundaries. 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


Generalizations


Multibrot sets

Multibrot set In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a port ...
s are bounded sets found in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
for members of the general monic univariate
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
family of recursions : z \mapsto z^d + c.\ For an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''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 \mapsto z^3 + 3kz + c , whose two critical points are the
complex square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
s of the parameter ''k''. A parameter is in the cubic connectedness locus if both critical points are stable. For general families of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s, the ''boundary'' of the Mandelbrot set generalizes to the
bifurcation locus In complex dynamics, the bifurcation locus of a parameterized family of one-variable holomorphic functions informally is a locus of those parameterized points for which the dynamical behavior changes drastically under a small perturbation of the pa ...
, which is a natural object to study even when the connectedness locus is not useful. The
Multibrot set In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a port ...
is obtained by varying the value of the exponent ''d''. The
article Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article may also refer to: G ...
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 perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pr ...
. In general, when ''d'' is a positive integer, the central region in each of these sets is always an
epicycloid In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an ''epicycle''—which rolls without slipping around a fixed circle. It is a particular kind of roulette. Equati ...
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 In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid cre ...
of (1-d) cusps.


Higher dimensions

There is no perfect extension of the Mandelbrot set into 3D. This is because there is no 3D analogue of the complex numbers for it to iterate on. However, there is an extension of the complex numbers into 4 dimensions, called the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s, 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 Projected is an American rock supergroup consisting of Sevendust members John Connolly and Vinnie Hornsby, Alter Bridge and Creed drummer Scott Phillips, and former Submersed and current Tremonti guitarist Eric Friedman. The band released t ...
into a 3D structure. However, 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 largely uninteresting to look at, as it does not have any of the properties that would be expected from a 4-dimensional fractal. Taking a 3-dimensional cross section at d = 0\ (q = a + bi +cj + dk) results in a solid of revolution of the 2-dimensional Mandelbrot set around the real axis.


Other, non-analytic, mappings

Of particular interest is the
tricorn The tricorne or tricorn is a style of hat that was popular during the 18th century, falling out of style by 1800, though actually not called a "tricorne" until the mid-19th century. During the 18th century, hats of this general style were referr ...
fractal, the connectedness locus of the anti-holomorphic family : z \mapsto \bar^2 + c. The tricorn (also sometimes called the ''Mandelbar'') was encountered by
Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
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 The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function: :z_ = (, \operatorname \left(z_n\right), +i, \operatorname \left(z_n\right), )^2 + c, \quad z_0= ...
, which is obtained by iterating the following: : z \mapsto (, \Re \left(z\right), +i, \Im \left(z\right), )^2 + c.


Computer drawings

There exist a multitude of various algorithms for plotting the Mandelbrot set via a computing device. Here, the most widely used and simplest algorithm will be demonstrated, namely, the naïve "escape time algorithm". In the escape time algorithm, a repeating calculation is performed for each ''x'', ''y'' point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel. The ''x'' and ''y'' locations of each point are used as starting values in a repeating, or iterating calculation (described in detail below). The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout". If that condition is reached, the calculation is stopped, the pixel is drawn, and the next ''x'', ''y'' point is examined. The color of each point represents how quickly the values reached the escape point. Often black is used to show values that fail to escape before the iteration limit, and gradually brighter colors are used for points that escape. This gives a visual representation of how many cycles were required before reaching the escape condition. To render such an image, the region of the complex plane we are considering is subdivided into a certain number of
pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the smal ...
s. To color any such pixel, let c be the midpoint of that pixel. We now 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, we know that 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 Some programming languages provide a complex data type for complex number storage and arithmetic as a built-in (primitive) data type. In some programming environments the term ''complex data type'' (in contrast to primitive data types) is a synony ...
. 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*x + y*y ≤ 2*2 AND iteration < max_iteration) do xtemp := x*x - y*y + x0 y := 2*x*y + y0 x := xtemp iteration := iteration + 1 color := palette teration 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 = \mathop(z^2+c) = x^2-y^2 + x_0 and y = \mathop(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.).


References in popular culture

The Mandelbrot set is widely considered the most popular
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
, and has been referenced several times in
popular culture Popular culture (also called mass culture or pop culture) is generally recognized by members of a society as a set of practices, beliefs, artistic output (also known as, popular art or mass art) and objects that are dominant or prevalent in a ...
. * The
Jonathan Coulton Jonathan William Coulton (born December 1, 1970), often called "JoCo" by fans, is an American folk/comedy singer-songwriter, known for his songs about geek culture and his use of the Internet to draw fans. Among his most popular songs are " Co ...
song "Mandelbrot Set" is a tribute to both the fractal itself and to the man it is named after, Benoit Mandelbrot. * The second book of the ''
Mode series The ''Mode'' series is a quartet of novels by Piers Anthony. Like many of Anthony's other fictional works, it explores many themes and ideas. This series has themes of violence, the abuse of power, sexism and male dominance, gender roles, the envi ...
'' by
Piers Anthony Piers Anthony Dillingham Jacob (born 6 August 1934) is an American author in the science fiction and Fantasy (genre), fantasy genres, publishing under the name Piers Anthony. He is best known for his :Xanth books, long-running novel series set in ...
, ''Fractal Mode'', describes a world that is a perfect 3D model of the set. * The Arthur C. Clarke novel '' The Ghost from the Grand Banks'' features an artificial lake made to replicate the shape of the Mandelbrot set. * Benoit Mandelbrot and the eponymous set were the subjects of the Google Doodle on 20 November 2020 (the late Benoit Mandelbrot's 96th birthday). * The American rock band
Heart The heart is a muscular organ in most animals. This organ pumps blood through the blood vessels of the circulatory system. The pumped blood carries oxygen and nutrients to the body, while carrying metabolic waste such as carbon dioxide t ...
has an image of a Mandelbrot set on the cover of their 2004 album, ''
Jupiters Darling ''Jupiters Darling'' is the thirteenth studio album by American Rock music, rock band Heart (band), Heart, released on June 22, 2004, by Sovereign Artists. Sovereign Artist's marketing director, Paul Angles, simultaneously released their album ...
''. * The British black metal band
Anaal Nathrakh Anaal Nathrakh are a British extreme metal band formed in 1999 in Birmingham by multi-instrumentalist Mick Kenney and vocalist Dave Hunt. They are currently signed to Metal Blade Records. The band's name is Irish for ''snake's breath'' ( anál ...
uses an image resembling the Mandelbrot set on their Eschaton album cover art. * The television series ''
Dirk Gently's Holistic Detective Agency ''Dirk Gently's Holistic Detective Agency'' is a humorous detective novel by English writer Douglas Adams, published in 1987. It is described by the author on its cover as a "thumping good detective-ghost-horror-who dunnit-time travel-romantic- ...
'' (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. * In Ian Stewart's 2001 book ''
Flatterland ''Flatterland'' is a 2001 book written by mathematician and science popularizer Ian Stewart about non-Euclidean geometry. It was written as a sequel to ''Flatland'', an 1884 novel that discussed different dimensions. Plot summary Almost 100 ...
'', there is a character called the Mandelblot, who helps explain fractals to the characters and reader.


See also

*
Buddhabrot The Buddhabrot is the probability distribution over the trajectories of points that escape the Mandelbrot fractal. Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha Siddhartha Gautama, most commonly ...
* Collatz fractal *
Fractint Fractint is a freeware computer program to render and display many kinds of fractals. The program originated on MS-DOS, then ported to the Atari ST, Linux, and Macintosh. During the early 1990s, Fractint was the definitive fractal generating pro ...
*
Gilbreath permutation A Gilbreath shuffle is a way to shuffle a deck of cards, named after mathematician Norman Gilbreath (also known for Gilbreath's conjecture). Gilbreath's principle describes the properties of a deck that are preserved by this type of shuffle, and ...
*
Mandelbox In mathematics, the mandelbox is a fractal with a boxlike shape found by Tom Lowe in 2010. It is defined in a similar way to the famous Mandelbrot set as the values of a parameter such that the origin does not escape to infinity under iteration ...
*
Mandelbulb The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and in 2009 further developed by Daniel White and Paul Nylander using spherical coordinates. A canonical 3-dimensional Mandelbrot set does not e ...
*
Menger Sponge In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Si ...
*
Newton fractal The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial or transcendental function. It is the Julia set of the meromorphic function which is given by Newton's method. ...
*
Orbit portrait In mathematics, an orbit portrait is a combinatorial tool used in complex dynamics for understanding the behavior of one-complex dimensional quadratic maps. In simple words one can say that it is : * a list of external angles for which rays lan ...
*
Orbit trap In mathematics, an orbit trap is a method of colouring fractal images based upon how close an iterative function, used to create the fractal, approaches a geometric shape, called a "trap". Typical traps are points, lines, circles, flower shapes ...
*
Pickover stalk Pickover stalks are certain kinds of details to be found empirically in the Mandelbrot set, in the study of fractal geometry. They are so named after the researcher Clifford Pickover, whose "epsilon cross" method was instrumental in their discove ...
* Plotting algorithms for the Mandelbrot set


References


Further reading

*
(First appeared in 1990 as
Stony Brook IMS Preprint
available a
arXiV:math.DS/9201272
) *
(includes a DVD featuring Arthur C. Clarke and
David Gilmour David Jon Gilmour ( ; born 6 March 1946) is an English guitarist, singer, songwriter, and member of the rock band Pink Floyd. He joined as guitarist and co-lead vocalist in 1967, shortly before the departure of founding member Syd Barrett. P ...
) *


External links

*
Video: Mandelbrot fractal zoom to 6.066 e228

Relatively simple explanation of the mathematical process
by Dr Holly Krieger, MIT
Mandelbrot Viewer
Browser based Mandelbrot set renderer including
gallery with examples

Various algorithms for calculating the Mandelbrot set
(on
Rosetta Code Rosetta Code is a wiki-based programming website with implementations of common algorithms and solutions to various programming problems in many different programming languages. It is named for the Rosetta Stone, which has the same text inscribe ...
)
Fractal calculator written in Lua by Deyan Dobromiroiv, Sofia, Bulgaria
{{DEFAULTSORT:Mandelbrot Set Fractals Articles containing video clips Articles with example pseudocode Complex dynamics