Mandelbrot set

Progressive infinite iterations of the "Nautilus" section of the Mandelbrot Set rendered using webGL

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

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

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

Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980.[4] The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard (1985),[1] 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 and Peter Richter became well known for promoting the set with photographs, books (1986),[5] and an internationally touring exhibit of the German Goethe-Institut (1985).[6][7]

The cover article of the August 1985 Scientific American introduced a wide audience to the algorithm for computing the Mandelbrot set. The cover was created by Peitgen, Richter and Saupe at the University of Bremen.[8] The Mandelbrot set became prominent in the mid-1980s as a computer graphics demo, when personal computers became powerful enough to plot and display the set in high resolution.[9]

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, Mitsuhiro Shishikura, Curt McMullen, John Milnor and Mikhail Lyubich.[10][11]

The Mandelbrot set can also be defined as the connectedness locus of a family of polynomials.

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

External rays of wakes near the period 1 continent in the Mandelbrot set

has an attracting fixed point. It consists of all parameters of the form

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

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).

It is conjectured that the Mandelbrot set is locally connected. This famous conjecture is known as MLC (for Mandelbrot locally connected). By the work of Adrien Douady and John H. Hubbard, this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important hyperbolicity conjecture mentioned above.

The Mandelbrot set is self-similar under magnification in the neighborhoods of the Misiurewicz points. It is also conjectured to be self-similar around generalized Feigenbaum points (e.g., −1.401155 or −0.1528 + 1.0397i), in the sense of converging to a limit set.[18][19] 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.

The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro Shishikura.[20] 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.

As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set. For instance, a point is in the Mandelbrot set exactly when the corresponding Julia set is connected.

This principle is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proved 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.[20] Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters.[17] Adrien Douady phrases this principle as:

The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually called "zooming in". The following example of an image sequence zooming to a selected c value gives an impression of the infinite richness of different geometrical structures and explains some of their typical rules.

The magnification of the last image relative to the first one is about 1010 to 1. Relating to an ordinary monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers. Its border would show an astronomical number of different fractal structures.

The seahorse "body" is composed by 25 "spokes" consisting of two groups of 12 "spokes" each and one "spoke" connecting to the main cardioid. These two groups can be attributed by some kind of metamorphosis to the two "fingers" of the "upper hand" of the Mandelbrot set; therefore, the number of "spokes" increases from one "seahorse" to the next by 2; the "hub" is a so-called Misiurewicz point. Between the "upper part of the body" and the "tail" a distorted small copy of the Mandelbrot set called satellite may be recognized.

Animations of the Multibrot set for d from 0 to 5 (left) and from 0.05 to 2 (right).
A 4D Julia set may be projected or cross-sectioned into 3D, and because of this a 4D Mandelbrot is also possible.

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

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

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

Another non-analytic generalization is the Burning Ship fractal, which is obtained by iterating the following:

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.

In pseudocode, this algorithm would look as follows. The algorithm does not use complex numbers and manually simulates complex-number operations using two real numbers, for those who do not have a complex data type. The program may be simplified if the programming language includes complex-data-type operations.

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

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.).

The Mandelbrot set is considered by many the most popular fractal,[27][28] and has been referenced several times in popular culture.