| Experiments Home |
|
| Mathematics |
|
|
| Benoît Mandelbrot & Mandelbrot Set |
|
|
|
|
  |
|
|
|
|
|
Benoît Mandelbrot & Mandelbrot Set See also Fractals
Benoît B. Mandelbrot (born November 20, 1924) is a French-Jewish-American mathematician, best known as the "father of fractal geometry (Mandelbrot set). He was born in Poland, but his family moved to France when he was a child; he is a dual French and American citizen and was educated in France. Mandelbrot now lives and works in the United States. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Center; and Battelle Fellow at the Pacific Northwest National Laboratory. He is most known for his Mandelbrot set fractal (see below). Early yearsMandelbrot was born in Warsaw in a Jewish family from Lithuania. Anticipating the threat posed by Nazi Germany, the family fled from Poland to France in 1936 when he was 11. He remained in France through the war to near the end of his college studies. He was born into a family with a strong academic tradition — his mother was a medical doctor and he was introduced to mathematics by two uncles. His uncle, Szolem Mandelbrojt, was a famous Parisian mathematician. His father, however, made his living trading clothing.[1] Mandelbrot attended the Lycée Rolin in Paris until the start of World War II, when his family moved to Tulle. In 1944 he returned to Paris. He studied at the Lycée du Parc in Lyon and in 1945-47 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 studied aeronautics. Back in France, he obtained a Ph.D. in Mathematical Sciences at the University of Paris in 1952.[1] From 1949 to 1957 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 then Lille, France.[2] 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.[2] He remained at IBM for thirty-two years, becoming an IBM Fellow, and later Fellow Emeritus.[1] Later yearsFrom 1951 onwards Mandelbrot worked on problems and published papers not only in mathematics but also in real-world fields including information theory, economics and fluid dynamics. He became convinced that two key themes, fat tails and self-similar structure, ran through a multitude of these problems. Mandelbrot found that price changes in financial markets did not follow a Gaussian distribution, but rather other 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.[3] Mandelbrot also put his ideas to work in cosmology and offered in 1974 a different resolution to 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 (e.g. like a Cantor dust), it would not be necessary to rely on the Big Bang theory to explain Olbers' Paradox (aka. the "dark night sky 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. 
In 1975 Mandelbrot coined the term fractal to describe these structures, and published his ideas in Les objets fractals, forme, hasard et dimension (1975; an English translation Fractals: Form, chance and dimension was published in 1977).[4] In 1979, while on secondment as Visiting Professor of Mathematics 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 of the formula z² - μ. While investigating how the topology of these Julia sets depended on the complex parameter μ he studied the Mandelbrot set fractal that is now named after him (note that the Mandelbrot set is now usually defined in terms of the formula z² + c, so Mandelbrot's early plots in terms of the earlier parameter μ are left-right mirror images of more recent plots in terms of the parameter c) . 
In 1982 Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature.[5] This influential work brought fractals into the mainstream of both professional and popular mathematics. On his retirement from IBM in 1987, Mandelbrot joined the Yale Department of Mathematics. At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences. His 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, and the Einstein Lectureship of the American Mathematical Society in 2006. The small planet 27500 Mandelbrot was named in his honour. On November 23, 1990, he was made a knight in the French Legion of honour. In 2004, Mandlebrot was the subject of a pop song written by Jonathan Coulton. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory.[6] Mandelbrot was promoted to officer of the French Legion of honour on January 1, 2006.[7] Mandelbrot, fractals, and the new theme of regular roughnessAlthough Mandelbrot coined the term fractal, some objects featured in The Fractal Geometry of Nature had been previously described by other mathematicians. However; they had been regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them around into essential tools for the long-stalled effort of extending the scope of science to non-smooth parts of the real world. He highlighted their common properties, such as self-similarity (linear, non-linear, or statistical), scale invariance and (usually) non-integer Hausdorff dimension. He also emphasized the use of fractals as realistic and useful models of many phenomena in the real world that can be viewed as rough. Natural fractals include the shapes of mountains, coastlines and river basins; the structure of plants, blood vessels and lungs; the clustering of galaxies; Brownian motion. Man-made fractals include stock market prices but also music, painting and architecture. Far from being unnatural, Mandelbrot held the view that fractals were, in many ways, more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry.
Images portraying various views of the Mandelbrot set.
Mandelbrot has been called a visionary.[8] 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. It sparked a widespread popular interest in fractals as well as contributing to chaos theory and other fields of science and mathematics. PronunciationBenoît is read as "ben-wa" [bənwa]. The pronunciation of the name "Mandelbrot", which is a Yiddish and German word meaning "almond bread", is given variously in dictionaries. The Merriam-Webster Collegiate Dictionary and the Longman Pronouncing Dictionary give [ˈmæn.dəlˌbɹoʊt] (first syllable sounds like "man"; last syllable rhymes with "boat"); the Bollard Pronouncing Dictionary of Proper Names gives the quasi-French pronunciation [ˈmæn.dəlˌbɹɔː] (last syllable rhymes with "draw"); the American Heritage Dictionary gives [ˈmɑːn.dəlˌbɹɑt] (first syllable has the vowel sound of the 'a' in "father"; last syllable rhymes with "pot"). Mandelbrot himself, as most Frenchmen do, pronounces his name as [mɑ̃dɛlbʁot] (roughly maw-dell-brote) when speaking in French.[9] Honours and awardsPartial list of awards[10]
See also
References
Further reading
External links
The Mandelbrot set is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex c-values for which the orbit of 0 under iteration of the complex quadratic polynomial xn+1=xn2 + c remains bounded.[1] Eg. c = 1 gives the sequence 0, 1, 2, 5, 26… which leads to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = i gives the sequence 0, i, (-1 + i), –i, (-1 + i), -i… which is bounded, and so it belongs to the Mandelbrot set. When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies the boundary as a fractal. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition. Benoît Mandelbrot and others worked hard to communicate this area of mathematics to the public.
HistoryThe 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. The first pictures of it were drawn in 1978 by Robert Brooks and Peter Matelski as part of a study of Kleinian Groups.[2] Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980.[3] The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard[4], who established many of its fundamental properties and named the set in honour of Mandelbrot. The mathematicians Heinz-Otto Peitgen and Peter Richter became well-known for promoting the set with glossy photographs, books, and a touring gallery.[5] The cover article of the August 1985 Scientific American featured an image created by Mandelbrot, Peitgen, and Hubbard. [6] 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,[7] [8], Curt McMullen, John Milnor, Mitsuhiro Shishikura, and Jean-Christophe Yoccoz. Formal definitionThe Mandelbrot set M is defined by a family of complex quadratic polynomials given by
where c is a complex parameter. For each c, one considers the behavior of the sequence ![A mathematician's depiction of the Mandelbrot set M, a point c is coloured black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of c.](http://en.wikipedia.org/wiki/Image:Mandelset_hires.png)
More formally, if 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 colouring all the points c which belong to M
black, and all other points white. The more colourful pictures usually
seen are generated by colouring points not in the set according to how
quickly or slowly the sequence The Mandelbrot set can also be defined as the connectedness locus of the family of polynomials Pc(z). That is, it is the subset of the complex plane consisting of those parameters c for which the Julia set of Pc is connected. Basic propertiesThe Mandelbrot set is a compact set, contained in the closed disk of radius 2 around the origin. In fact, a point c belongs to the Mandelbrot set if and only if The intersection of M with the real axis is precisely the interval The correspondence is given by In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set. The area of the Mandelbrot set is estimated to be 1.506 591 77 ± 0.000 000 08. 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 which 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. The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of M, gives rise to external rays of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle. The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of parameters c for which the dynamics changes abruptly under small changes of c. It can be constructed as the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates. The Mandelbrot curves are defined by setting p0=z, pn=pn-12+z, and then interpreting the set of points |pn(z)|=1 in the complex plane as a curve in the real Cartesian plane of degree 2n+1 in x and y. Other propertiesThe main cardioid and period bulbs
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 for which Pc has an attracting fixed point. It consists of all parameters of the form for some To the left of the main cardioid, attached to it at the point c = − 3 / 4, a circular-shaped bulb is visible. This bulb consists of those parameters There are infinitely many other bulbs tangent to the main cardioid: for every rational number 
This bulb is called the 
The change of behavior occurring at Hyperbolic componentsAll the bulbs we encountered in the previous section were interior components of the Mandelbrot set in which the maps 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. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" 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). Local connectivityIt 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 celebrated work of Jean-Christophe Yoccoz established local connectivity of the Mandelbrot set at all finitely-renormalizable parameters; that is, roughly speaking those which are contained only in finitely many small Mandelbrot copies. 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 points. It is also conjectured to be self-similar around generalized Feigenbaum points (e.g. -1.401155 or -.1528 + 1.0397i), in the sense of converging to a limit set.[9][10] 
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 resultsThe Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro Shishikura.[11] 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. Relationship with Julia sets
As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set. 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. Similarly, Yoccoz first proves the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters. Adrien Douady phrases this principle as
Geometry
Recall that, for every rational number 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. Image gallery of a zoom sequenceThe mandlebrot 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 60,000,000,000 to 1. Relating to an ordinary monitor, it represents a section of a Mandelbrot set with a diameter of 20 million kilometres. Its border would show an inconceivable number of different fractal structures.
Start: Mandelbrot set with continuously coloured environment.
Generalizations  Sometimes the connectedness loci of families other than the quadratic family are also referred to as the Mandelbrot sets of these families. The connectedness loci of the unicritical polynomial families 
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. It is also possible to consider similar constructions in the study of non-analytic mappings. Of particular interest is the tricorn, the connectedness locus of the anti-holomorphic family 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. Computer drawings
Algorithms :
Every algorithm can be implemented in sequential or parallel version. Mirror symmetry can be used to speed-up calculations. Escape time algorithmThe 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 behaviour of that calculation, a colour is chosen for that pixel. The x and y location 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 if they have reached a critical 'escape' condition. 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 other starting values, 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 maximum number of iterations, the more detail and subtlety emerge in the final image, but the longer time it will take to calculate the picture. The colour 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 colours are used for points that escape. This gives a visual representation of how many cycles were required before reaching the escape condition. For programmersThe definition of the Mandelbrot set, together with its basic
properties, suggests a simple algorithm for drawing a picture of the
Mandelbrot set. The region of the complex plane we are considering is
subdivided into a certain number of pixels. To colour any such pixel, let If this is the case, we know that the midpoint does not belong to the Mandelbrot set, and we colour our pixel. (Either we colour it white to get the simple mathematical image or colour it according to the number of iterations used to get the well-known colourful images). Otherwise, we keep iterating for a certain (large, but 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 colour the pixel black. In pseudocode, this algorithm would look as follows. For each pixel on the screen do:
{
x = x0 = x co-ordinate of pixel
y = y0 = y co-ordinate of pixel
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
}
if ( iteration == max_iteration )
then
colour = black
else
colour = iteration
plot(x0,y0,colour)
}
where, relating the pseudocode to
and so, as can be seen in the pseudocode in the computation of x and y:
To get colourful images of the set, the assignment of a colour 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 to do it, without slowing down the calculations, is to use the number of executed iterations as an entry to a look-up colour palette table initialized at startup. If the colour table has, for instance, 500 entries, then you can use n mod 500, where n is the number of iterations, to select the colour to use. You can initialize the colour palette matrix in various different ways, depending on what special feature of the escape behavior you want to emphasize graphically. Continuous (smooth) coloring
The Escape Time Algorithm is popular for its simplicity. However, it creates bands of color, which can detract from an image's value. This can be fixed using the Normalized Iteration Count Algorithm[13], which provides a smooth transition of colors between iterations. The equation is
where n is the number of iterations for z, B is the bailout radius (it is normally 2 for a Mandelbrot set, but it can be changed), and P is the power for which z is raised to in the Mandelbrot set equation (zn+1=znP+c, P is generally 2). Another equation for this is
Note that this new equation is simpler than the first, but it only
works for Mandelbrot sets with a bailout radius of 2 and a power of 2. Distance estimatesOne can compute distance from point c ( in exterior or interior ) to nearest point on the boundary of Mandelbrot set. Exterior distance estimationThe pro | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
,
obtained by 
,
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 
denotes the nth iterate of 
diverges to infinity. See the section on 
for all 
. In other words, if the 
ever becomes larger than 2, the sequence will escape to infinity.![[-2 , 0.25]\,](mandelbrot_set_files/be6c2c2f723244f8201cc824c532b217.png)
. The parameters along this interval can be put in one-to-one correspondence with those of the real ![z\mapsto \lambda z(1-z),\quad \lambda\in[1,4].\,](mandelbrot_set_files/14ea2bcd4986ea856f321c077a157def.png)
in the 
for which 
, with p and q 
-fixed point). If we label these components 
in counterclockwise orientation, then 
.
is known as a 
have an attracting periodic cycle. Such components are called hyperbolic components.
. The best current estimate known is the famous 
.
for 
and 
