What is the Mandelbrot set?

Rev Tyson says: "Since its discovery a quarter of a century ago, the Mandelbrot Set has become a universal icon for the doctrine of chaos--the emerging scientific idea that simple processes can result in infinite complexity. It's the set of all complex numbers z for which sequence defined by the iteration z(0) = z, z(n+1) = z(n)*z(n) + z, n=0,1,2, ... remains bounded. This means that there is a number B such that the absolute value of all iterates z(n) never gets larger than B. A bounded sequence may or not have a limit. For example, if z=0 then z(n) = 0 for all n, so that the limit of the (1) is zero. On the other hand, if z=i ( i being the imaginary unit), then the sequence oscillates between i and i-1, so remains bounded but it does not converge to a limit. I don't believe in God but I'm very interested in her. -Arthur C. Clarke Music, Pink Floyd: A Momentary Lapse Of Reason-Signs Of Life."