The Mandelbrot set is a set of points in the complex plane that forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex c-values for which the orbit of 0 under iteration of the complex quadratic polynomial x2 + c remains bounded.
Eg. 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 gives the sequence 0, i, (-1 + i), –i, (-1 + i), -i… which is bounded, and so it belongs to the Mandelbrot set.
When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies it as a fractal.
The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition.