Mandelbrot set

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Initial image of a Mandelbrot set zoom sequence with continuously colored environment.

The Mandelbrot set is a fractal that has become popular outside of mathematics both for its aesthetic appeal and its complicated structure, arising from a simple definition. This is largely due to the efforts of Benoît Mandelbrot and others, who worked hard to communicate this area of mathematics to the public.

[edit] History

The Mandelbrot set has its place in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. The first pictures of it were drawn in 1978 by Brooks and Matelski as part of a study of Kleinian Groups.^[1]

Mandelbrot studied the parameter space of quadratic polynomials in an article which appeared in 1980.^[2] The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard,^[3] who established many fundamental properties of $M$ , and named the set in honor of Mandelbrot.

The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. It would be futile to attempt to make a list of all the mathematicians who have contributed to our understanding of this set since then, but such a list would certainly include Mikhail Lyubich, Curt McMullen, John Milnor, Mitsuhiro Shishikura and Jean-Christophe Yoccoz.

[edit] Formal definition

The Mandelbrot set $M$ is defined by a family of complex quadratic polynomials

$f_c:\mathbb C\to\mathbb C$

given by

$f_c(z) = z^2 + c.\,$

where $c$ is a complex parameter. For each $c$ , one considers the behaviour of the sequence $(0, f_c(0), f_c(f_c(0)), f_c(f_c(f_c(0))), \ldots)$ obtained by iterating $f c (z)$ starting at $z = 0$ , which either escapes to infinity or stays within a disk of some finite radius. The Mandelbrot set is defined as the set of all points $c$ such that the above sequence does not escape to infinity.

A mathematician's depiction of the Mandelbrot set M. A point c is colored black if it belongs to the set, and white otherwise.

More formally, if $f^n_c(z)$ denotes the nth iterate of $f c (z)$ (i.e. $f c (z)$ composed with itself n times) the Mandelbrot set is the subset of the complex plane given by

$M = \left\{c\in \mathbb C : \sup_{n\in \mathbb N}|f^n_c(0)| < \infin\right\}.$

Mathematically, the Mandelbrot set is just a set of complex numbers. A given complex number $c$ either belongs to $M$ or it does not. A picture of the Mandelbrot set can be made by coloring all the points $c$ which belong to $M$ black, and all other points white. The more colorful pictures usually seen are generated by coloring points not in the set according to how quickly or slowly the sequence $|f^n_c(0)|$ diverges to infinity. See the section on computer drawings below for more details.

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

[edit] Basic properties

The 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 $|f^n_c(0)|\leq 2$ for all $n\geq 0$ . In other words, if the absolute value of $f^n_c(0)$ ever becomes larger than 2, the sequence will escape to infinity.

The intersection of $M$ with the real axis is precisely the interval $[ - 2,0.25]$ . The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family,

$z\mapsto \lambda z(z-1),\quad \lambda\in[1,4].\,$

The correspondence is given by

$c = \frac{1-(\lambda-1)^2}{4}.$

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. This is conjectured to be $\sqrt{6\pi -1} - e$ = 1.506591651… exactly. [1]

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, p_n=p_n-1²+z, and then interpreting the set of points |p_n(z)|=1 in the complex plane as a curve in the real Cartesian plane of degree 2ⁿ⁺¹ in x and y.

[edit] Image gallery of a zoom sequence

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 amount of different fractal structures.

Start	Step 1	Step 2	Step 3	Step 4
Step 5	Step 6	Step 7	Step 8	Step 9
Step 10	Step 11	Step 12	Step 13	Step 14

Start: Mandelbrot set with continuously colored environment.
Step 1: Gap between the "head" and the "body" also called the "seahorse valley".
Step 2: On the left double-spirals, on the right "seahorses".
Step 3: "Seahorse" upside down. Its "body" is composed by 25 "spokes" consisting of 2 groups of 12 "spokes" each and one "spoke" connecting to the main cardioid. These 2 groups can be attributed by some kind of metamorphosis to the 2 "fingers" of the "upper hand" of the Mandelbrot set. Therefore the number of "spokes" increases from one "seahorse" to the next by 2. The "hub" is a so called Misiurewicz point. Between the "upper part of the body" and the "tail" a distorted small copy of the Mandelbrot set called satellite can be recognized.
Step 4: The central endpoint of the "seahorse tail" is also a Misiurewicz point.
Step 5: Part of the "tail". There is only one path consisting of the thin structures which leads 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". It makes sure, that the Mandelbrot set is a so called simply connected set. That means there are no islands and no loop roads around a hole.
Step 6: Satellite. The two "seahorse tails" are the beginning of a series of concentric crowns with the satellite in the center.
Step 7: Each of this 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 mentioned in zoom step 5 passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head".
Step 8: "Antenna" of the satellite. Several satellites of second order can be recognized.
Step 9: The "seahorse valley" of the satellite. All the structures from the image of zoom step 1 reappear.
Step 10: Double-spirals and "seahorses". Unlike the image of zoom step 2 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.
Step 11: Double-spirals with satellites of second order. Analog to the "seahorses" the double-spirals can be interpreted as a metamorphosis of the "antenna".
Step 12: In the outer part of the appendices islands of structures can be recognized. They have a shape like Julia sets J_c. The largest of them can be found in the center of the "double-hook" on the right side.
Step 13: Part of the "double-hook".
Step 14: On the first sight these islands seem to consist of infinitely many parts like Cantor sets, as it is actually the case for the corresponding Julia set J_c. Here they are connected by tiny structures so that the whole represents a simply connected set. These tiny structures meet each other at a satellite in the center which 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 has relative to the main body of the Mandelbrot set the same position as the center of this image relative to the satellite shown in zoom step 7.

[edit] The main cardioid and period bulbs

Periods of hyperbolic components

Upon looking at a picture of the Mandelbrot set, one immediately notices the large cardioid-shaped region in the center. This main cardioid is the region of parameters $c$ for which $f c$ has an attracting fixed point. It consists of all parameters of the form

$c = \frac{1-(\mu-1)^2}{4}$

for some $\mu\,$ in the open unit disk.

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 $c\,$ for which $f c$ has an attracting cycle of period 2. This set of parameters is an actual circle, namely that of radius 1/4 around -1.

There are infinitely many other bulbs attached to the main cardioid: for every rational number $\frac{p}{q}$ , with p and q coprime, there is such a bulb attached at the parameter : $c_{\frac{p}{q}} = \frac{1 - \left(e^{2\pi i \frac{p}{q}}-1\right)^2}{4}.$

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

This bulb is called the $\frac{p}{q}$ -bulb of the Mandelbrot set. It consists of parameters which have an attracting cycle of period $q$ and combinatorial rotation number $\frac{p}{q}$ . 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 $U_0,\dots,U_{q-1}$ in counterclockwise orientation, then $f c$ maps the component $U j$ to the component $U_{j+p\,(\operatorname{mod} q)}$ .

Attracting cycles and Julia sets for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4 and 1/5 bulbs

The change of behavior occurring at $c_{\frac{p}{q}}$ is known as a bifurcation: the attracting fixed point "collides" with a repelling period $q$ -cycle. As we pass through the bifurcation parameter into the $\frac{p}{q}$ -bulb, the attracting fixed point turns into a repelling fixed point (the $α$ -fixed point), and the period $q$ -cycle becomes attracting.

[edit] 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. 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).

[edit] Little Mandelbrot copies

Little Mandelbrot copies

The Mandelbrot set is self-similar in the sense that small distorted versions of itself can be found at arbitrarily small scales near any point of the boundary of the Mandelbrot set. This phenomenon is explained by Douady and Hubbard's theory of renormalization.

[edit] Local connectivity of the Mandelbrot set

It is conjectured that the Mandelbrot set is locally connected. This famous conjecture is known as MLC (for Mandelbrot Locally Connected). By the work of Douady and 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 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.

[edit] Further results

The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 by a result of Mitsuhiro Shishikura.^[4] 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 BBS 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.

[edit] Relationship with Julia sets

An "embedded Julia set"

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

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

[edit] Geometry of the Mandelbrot set

Recall that, for every rational number $\frac{p}{q}$ , where $p$ and $q$ are relatively prime, there is a hyperbolic component of period $q$ bifurcating from the main cardioid. The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the $\frac{p}{q}$ -limb. Computer experiments suggest that the diameter of the limb tends to zero like $\frac{1}{q^2}$ . The best current estimate known is the famous Yoccoz-inequality, which states that the size tends to zero like $\frac{1}{q}$ .

A period $q$ -limb will have $q - 1$ "antennas" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas.

cycle periods and antennas

[edit] 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 $f_c = z^d + c\,$ for $d > 2$ are often called Multibrot sets.

Multibrot sets of degrees 3 and 4

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

$z \mapsto \bar{z}^2 + c.\,$

The tricorn (also sometimes called the Mandelbar set) was encountered by Milnor in his study of parameter slices of real cubic polynomials. It is not locally connected. This property is inherited by the connectedness locus of real cubic polynomials.

[edit] Computer drawings of the Mandelbrot set

Buddhabrot method

Still image of a movie of increasing magnification on 0.001643721971153 + 0.822467633298876i

Algorithms :

parallel
sequential
- Escape time algorithm
  - boolean version ( draws M-st and its exterior using 2 colors ) = Mandelbrot algorithm
  - discrete (integer) version = level set method ( LSM/M ); draws Mandelbrot set and color bands in its exterior
  - continous version
  - level curves version = draws lemniscates of Mandelbrot set
  - decomposition of exterior of andelbrot set
- Hubbard-Douady potential of Mandelbrot set (CPM/M)
- Distance Estimation Method for Mandelbrot set = Milnor algorithm (DEM/M)
- abstract M-set

[edit] Escape time algorithm

The simplest algorithm for generating a representation of the Mandelbrot set is known as the "escape time" algorithm. A repeating calculation is performed for each x, y point in the plot area and based on the 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 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 colours are used for points that escape. This gives a visual representation of how many cycles were required before reaching the escape condition.

[edit] For Programmers

The 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 color any such pixel, let $c\,$ be the midpoint of that pixel. We now iterate the critical value $c\,$ under $f_c\,$ , checking at each step whether the orbit point has modulus larger than 2.

If this is the case, we know that the midpoint does not belong to the Mandelbrot set, and we color our pixel. (Either we color it white to get the simple mathematical image or color it according to the number of iterations used to get the well-known colorful 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 color 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

  x2 = x*x
  y2 = y*y
 
  iteration = 0
  maxiteration = 1000
 
  while ( x2 + y2 < (2*2)  AND  iteration < maxiteration ) 
  {
 
    y = 2*x*y + y0
    x = x2 - y2 + x0

    x2 = x*x    
    y2 = y*y

    iteration = iteration + 1
  }
 
  if ( iteration == maxiteration )
    colour = black
  else
    colour = iteration
}

where, relating the pseudocode to $c\, , z\,$ and $f_c\,$ :

$z = x + i y$
$z 2 = x 2 + i 2 x y - y 2$
$c = x 0 + i y 0$

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

$x = R e (z 2 + c) = x 2 - y 2 + x 0$ and y = $I m (z 2 + c) = 2 x y + y 0$

The code above has some speed optimizations. The use of the variables x2 and y2 means we save several operations. It saves us two multiplications since then we don't need to do "x*x" and "y*y" in the while-expression. Also note that we put "y = 2*x*y + y0" before "x = x2 - y2 + x0", otherwise we would need to store away "x" in a temporary variable before calculating "y = 2*x*y + y0". The reason we then don't need to store "y" in a temporary variable is that "x=" now doesn't do "y*y" but instead uses the variable "y2".

[edit] Continuous (smooth) colouring

Instead of the separate bands of colour produced by the basic algorithm, a more aesthetically-pleasing smoothly-coloured image can be achieved by using a suitable formula to convert the discrete iteration count into a fractional (continuous) number whenever the iteration process 'escapes'. One such formula is the "renormalized" fraction iteration count, an overview of which is provided here, with a more detailed explanation and sample images provided here. The fractional iteration count can then be converted to a suitable colour; one method is to interpolate between the colours defined for the basic discrete algorithm.

[edit] Problems

A portion of the Mandelbrot set centered at (0.282, -0.01)

A portion of the Mandelbrot set.

The Mandelbrot set has some very thin filaments, so even if a given pixel does intersect the Mandelbrot set, it is quite likely that its midpoint will nevertheless escape.

As a result, the algorithm does not behave stably under small perturbations, and will generally not detect small features of the Mandelbrot set. It is precisely this property which led Mandelbrot to conjecture that $M$ is disconnected.

A common way around this problem, which also results in more aesthetically pleasing pictures, is to color any pixel whose midpoint is determined to be an escaping parameter according to the number of iterations it requires to escape. Since parameters closer to the Mandelbrot set will take longer to escape, this method will make the connections between different parts of the Mandelbrot set visible.

[edit] Distance estimates

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

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

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

[edit] Optimizations

One way to improve calculations is to find out beforehand whether the given point lies within the cardioid or in the period 2 bulb.

To prevent having to do huge numbers of iterations for other points in the set, one can do "periodicity checking"—which means check if a point reached in iterating a pixel has been reached before. If so, the pixel cannot diverge, and must be in the set. This is most relevant for fixed-point calculations, where there is a relatively high chance of such periodicity—a full floating-point (or higher accuracy) implementation would rarely go into such a period.

[edit] Art and the Mandelbrot set

"Budding Turbines"

Searching the Mandelbrot set for interesting pictures and transforming them into artistic images often produces remarkably beautiful works. The fractal art article has some examples.

Here is an animation showing increasing levels of detail (warning, large file). Although it does not start from a view of the whole set, towards the end it clearly shows the recursive nature of the fractal at all scales when a shape similar to the whole comes into view.

[edit] The Mandelbrot set in popular culture

St. Louis filmmaker Bill Boll has produced a DVD called 'The Amazing Mandelbrot Set' containing several high-quality deep zooms of the Mandelbrot set, along with a tutorial.

The Australian band GangGajang has a song Time (and the Mandelbrot set) where the term Mandelbrot set is used liberally in the lyrics.

The American singer Jonathan Coulton has a song titled Mandelbrot Set on his EP Where Tradition Meets Tomorrow about the history of the Mandelbrot set, and of Benoît Mandelbrot himself.

Blue Man Group's first album Audio features tracks titled "Mandelgroove", "Opening Mandelbrot", and "Klein Mandelbrot". The album was nominated for a Grammy in 2000. Also, a hidden track on their second album, The Complex, is entitled Mandelbrot No. 4.

The artwork for Canadian band Rush's 16th album, Test for Echo, includes the Mandlebrot Set.

The British post-rock band Mandelbrot Set released their debut album All Our Actions Are Constantly Repeated on October 2006 on Highpoint Lowlife.

Detroit-area melodian Bonedaddy, of Scrummage Records, gives a shout-out to Mandlebrot and his set in the song "Epstein's Face", wherein he states clearly: "Mandlebrot set; 2 legit 2 quit; Epstein's Face is floating through time and space!" In the music video for this song, an image of the Mandlebrot fractal enlarges in front of the viewer's eyes.

A Mandelbrot poster is visible in Eric's room in the That '70s Show sitcom (3rd season), which is an anachronism since those posters only appeared in the 80s.

In the opening credits of Casino Royale (2006) a plume of gunsmoke coming from a pistol is stylistically represented as clubs (the playing cards suit) repeated as a Mandlebrot pattern.

The Australian movie "The Bank" (2001) directed by Robert Connolly is based on the idea that the Mandelbrot set can predict financial markets.

[edit] References

^ Robert Brooks and Peter Matelski, The dynamics of 2-generator subgroups of PSL(2,C), in "Riemann Surfaces and Related Topics", ed. Kra and Maskit, Ann. Math. Stud. 97, 65–71, ISBN 0-691-08264-2
^ Benoît Mandelbrot, Fractal aspects of the iteration of $z\mapsto\lambda z(1-z)\,$ for complex $\lambda,z\,$ , Annals NY Acad. Sci. 357, 249/259
^ Adrien Douady and John H. Hubbard, Etude dynamique des polynômes complexes, Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)
^ Mitsuhiro Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math. 147 (1998) p. 225-267. (First appeared in 1991 as a Stony Brook IMS Preprint, available as arXiv:math.DS/9201282.)

[edit] Further reading

Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993, ISBN 0-387-97942-5
John W. Milnor, Dynamics in One Complex Variable (Third Edition), Annals of Mathematics Studies 160, Princeton University Press 2006, ISBN 0-691-12488-4 (First appeared in 1990 as a Stony Brook IMS Preprint, available as arXiV:math.DS/9201272.)
Nigel Lesmoir-Gordon. "The Colours of Infinity: The Beauty, The Power and the Sense of Fractals." ISBN 1-904555-05-5 (The book comes with a related DVD of the Arthur C. Clarke documentary introduction to the fractal concept and the Mandelbrot set.

[edit] See also

Udo of Aachen, a fictional monk, who supposedly discovered the Mandelbrot set in the thirteenth century in an April Fool's hoax perpetrated by British technical writer Ray Girvan
External ray
Fractint - an open source fractal generator
Julia Set

[edit] External links

Wikimedia Commons has media related to:

Mandelbrot set

The Mandelbrot Set and Julia Sets by Michael Frame, Benoit Mandelbrot, and Nial Neger
The Encyclopedia of the Mandelbrot Set by Robert P. Munafo
The Mandelbrot and Julia sets Anatomy by Evgeny Demidov
papers of G. Pastor and M. Romera
FAQ on the Mandelbrot set
PI and the Mandelbrot set, A very interesting connection between PI and the Mandelbrot set, by Dave Boll
Chaos and Fractals at the Open Directory Project (suggest site)

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