The Mandelbrot set

This is a set of elements that belongs to the plane of complex numbers (in the figure, the abscissa corresponds to the real part and the ordinate corresponds to the imaginary part of a particular complex number c) that, even after repeated substitution into the (recurrent) equation

zn+1 = zn2 + c (where z0 = 0),

does not exceed a value of 2. Some points in the plane of complex numbers exceed a value of 2 in the very first substitution into the equation, while this occurs for others only when the given procedure, i.e. addition to its square and substitution of the result into the right-hand side of the equation, is repeated many times. The number of these repetitions (iterations) required to exclude that a particular point belongs to the Mandelbrot set is depicted by the degree of grey in the figure. (A much nicer picture is created when the numbers of repetitions are depicted in various colours.)

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