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 fo ...
s
for which the function
does not
diverge to infinity when
iterated from
, i.e., for which the sequence
,
, 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 groups.
Afterwards, in 1980,
Benoit Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of p ...
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.
Images of the Mandelbrot set exhibit an elaborate and infinitely complicated
boundary that reveals progressively ever-finer
recursive 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 ill ...
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
whether the sequence
goes to infinity. Treating the
real and
imaginary parts of
as
image coordinates 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
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
is held constant and the initial value of
is varied instead, one obtains the corresponding
Julia set for the point
.
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 beauty
Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as ...
, and
motif.
History
The Mandelbrot set has its origin in
complex dynamics, 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
...
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 cl ...
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.
[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,
New York
New York most commonly refers to:
* New York City, the most populous city in the United States, located in the state of New York
* New York (state), a state in the northeastern United States
New York may also refer to:
Film and television
* '' ...
,
Benoit Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of p ...
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 ...
of
quadratic polynomials in an article that appeared in 1980. The mathematical study of the Mandelbrot set really began with work by the mathematicians
Adrien Douady and
John H. Hubbard (1985),
[Adrien Douady and John H. Hubbard, ''Etude dynamique des polynômes complexes'', Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)] who established many of its fundamental properties and named the set in honor of Mandelbrot for his influential work in
fractal geometry.
The mathematicians
Heinz-Otto Peitgen
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, ...
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 ...
(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 ...
'' 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 problems or to perform a computation. Algorithms are used as specifications for performing ...
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 te ...
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
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...
, 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,
Mitsuhiro Shishikura,
Curt McMullen,
John Milnor 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 ...
of the
critical point under
iteration of the
quadratic map
:
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 fo ...
''c'' is a member of the Mandelbrot set if, when starting with
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), ...
of
remains bounded for all
.
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 called ...
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 am ...
, so 1 is not an element of the Mandelbrot set. On the other hand, for
, the sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set.
The Mandelbrot set can also be defined as the
connectedness locus of the family of quadratic
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 exampl ...
s, while its
boundary can be defined as the
bifurcation locus of this quadratic family.
Basic properties
The Mandelbrot set is a
compact set, since it is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
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
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
. More specifically, a point
belongs to the Mandelbrot set if and only if
for all
. 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), ...
of
must remain at or below 2 for
to be in the Mandelbrot set,
, and if that absolute value exceeds 2, the sequence will escape to infinity. Since
, it follows that
, establishing that
will always be in the closed disk of radius 2 around the origin.
The
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of
with the real axis is precisely the interval