The Mandelbrot Set

The most famous fractal is called the Mandelbrot Set, after the Polish-born mathematician Benoit Mandelbrot. To construct the Mandelbrot Set, consider the following sequence of numbers:

z, z^2 + z, (z^2 + z)^2 + z, ((z^2 + z)^2 + z)^2 + z, . . .

The behavior of this sequence depends on the value of the complex number z. For some values of z this sequence is bounded, and for other values it is unbounded. If the sequence is bounded, the complex number z is in the Mandelbrot Set. If the sequence is unbounded, the complex number z is not in the Mandelbrot Set.

Members of the Mandelbrot Set

1. The complex number -2 is in the Mandelbrot Set because for z = -2, the corresponding Mandelbrot sequence is -2, 2, 2, 2, 2, 2, . . . which is bounded. (No number in the sequence has an absolute value greater than 2.)
2. The complex number i is also in the Mandelbrot Set because for z = i, the corresponding Mandelbrot sequence is i, -1 + i, -i, -1 + i, -i, -1 + i, . . . which is bounded. (No number in the sequence has an absolute value greater than .)
3. The complex number 1 + i is not in the Mandelbrot Set because for z = 1 + i, the corresponding Mandelbrot sequence is 1 + i, 1 + 3i, -7 + 7i, 1 - 97i, -9407 - 193i, . . . which is unbounded. (The absolute values of the numbers in the sequence become arbitrarily large.)