# Mandelbrot set

**The Mandelbrot set is the set of complex numbers for which the function fc(z)=z2+c does not diverge when iterated from z=0. Its definition is credited to Adrien Douady who named it in tribute to the mathematician Benoit MandelBrot, a pioneer of fractal geometry. It is one of the best-known examples of mathematical visualization and mathematical beauty and motif.**

### About Mandelbrot set in brief

The Mandelbrot set is the set of complex numbers c{displaystyle c} for which the function fc(z)=z2+c does not diverge when iterated from z=0. Its definition is credited to Adrien Douady who named it in tribute to the mathematician Benoit MandelBrot, a pioneer of fractal geometry. Images of the Mandel Brot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The Mandel brot 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 and mathematical beauty and motif. It was first defined and 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 Yorktown Heights, New York, BenoitMandelbroT first saw a visualization of the set. 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 a wide audience to the algorithm for computing the MandlebrotSet. The 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.

It 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 Mikhail Lyubich, Curt McMullen, John Milnor, Mitsuhiro Shishikura and Jean-Christophe Yocco. The study of theMandel Brott 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. For example, for c = 1, the sequence is 1, 0, 1, which tends to infinity, so 1 is not an element of the Set. On the other hand, if c = 0, c is a quadratic map, and the sequence does not belong to 0, so 0, 0.1, 0 does belong to the set, which is not bounded, so c is not a member of theSet. If c is in orbit around a point in the complex plane for which z =0, the critical point under iteration of the critical map is a member. If z = 0 and zn is 0, the set is a set if, when applying the iteration repeatedly repeatedly, the absolute value of zn remains bounded in absolute value. For instance, for the complex number c=1, c does not belongs to the Set if and only if the sequence f c(0),fc(fc(0)),…,…, etc., remains bounded.

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