The Mandelbrot set, named after Benoît Mandelbrot, is a famous example of a fractal. It begins with this equation: zn+1 = zn2 + c, where c and z are complex numbers and n is zero or a positive integer (natural number). Starting with z0=0, c is in the Mandelbrot set if the absolute value of zn never exceeds a certain number (that number depends on c) however large n gets. An equivalent definition of the M-set is the set of all complex numbers c such that the Julia Set (2) of c is connected (i.e., 0 is a member of the J-set of c (where the exponent is 2)).
For example, if c = 1 then the sequence is 0, 1, 2, 5, 26,…, which goes to infinity. Therefore, 1 is not an element of the Mandelbrot set.
On the other hand, if c is equal to the square root of -1, also known as i, then the sequence is 0, i, (−1 + i), −i, (−1 + i), −i…, which does not go to infinity and so it belongs to the Mandelbrot set.
When graphed, the Mandelbrot set is very pretty and recognizable.
A generalization of Mandelbrot sets allows any exponent: zn+1 = znd + c. These sets are called Multibrot sets. The Multibrot set for d = 2 is the Mandelbrot set.