The Mandelbrot set is a mathematical set of points whose boundary is a distinctive and easily recognizable two-dimensional fractal shape. The set is closely related to Julia sets (which include similarly complex shapes), and is named after the mathematician Benoit Mandelbrot, who studied and popularized it.
images of the Mandelbrot set are made by sampling complex numbers and determining for each whether the result tends towards infinity when a particular mathematical operation is iterated on it. Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all.
More precisely, the Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.
For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = i (where i is defined as i2 = ?1) gives the sequence 0, i, (?1 + i), ?i, (?1 + i), ?i, ..., which is bounded, and so i belongs to the Mandelbrot set.
images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.
The Mandelbrot set has become popular outside mathematics both for
ts aesthetic appeal and as an example of a complex structure arising from the application of simple rules, and is one of the best-known examples of mathematical visualization.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 this fractal were drawn in 1978 by Robert W. Brooks and Peter Matelski as part of a study of Kleinian groups. On 1 March 1980, at IBM's Thomas J. Watson Research Center in upstate New York, Benoit Mandelbrot first saw a visualization of the set.
Mandelbrot studied the parameter space 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, 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 photographs, books, and an internationally touring exhibit of the German Goethe-Institut.
The cover article of the August 1985 Scientific American introduced the algorithm for computing the Mandelbrot set to a wide audience. The cover featured an image created by Peitgen, et al.
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, Curt McMullen, John Milnor, Mitsuhiro Shishikura, and Jean-Christophe Yoccoz.