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The Mandelbrot set is a particular mathematical set of points whose boundary is a distinctive and easily recognisable two-dimensional fractal shape. The set is closely related to Julia sets (which include similarly complex shapes), and is named after the mathematician Benoît Mandelbrot , who studied and popularised it.
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 z n +1 = z n 2 + c remains bounded . That is, a complex number c is part of the Mandelbrot set if, when starting with z 0 = 0 and applying the iteration repeatedly, the absolute value of z n 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 i 2  = −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 its 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 visualisation .

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