The Beauty of Fractals

I'm a reasonably intelligent galoot, but what I can't understand is this: That beauty such as this exists neatly tucked away in common, garden-variety complex numbers. Take a small area along the periphery of the region known as the Mandlebrot Set. For each complex number c in that area, iterate the following: 

z(0) = c
z(n+1) = z(n)2 + c

Now, the magnitude of z(n+1) will either remain finite or not. If it's going to go infinite, it will exceed 2. If that happens, assign a color to that point based on the value of n when it happens. Otherwise, continue iterating until the maximum number of iterations is reached, and assign black to that point. Then plot out all of the points. I used to do this in FORTRAN on a VAX, plotting on an old low-res paintjet printer. It's a lot easier now.

fractal 1.jpg
lower lefthand corner = 0.25932155869498 + 0.00145611369495i
width = 5.459818e-008
maximum iterations = 2048 
fractal 2.jpg
lower lefthand corner = -0.75411018025570 + 0.05536016198561i
width = 6.146660e-010
maximum iterations = 2048 
fractal 3.jpg
lower lefthand corner = -0.75411017990214 + 0.05536016227582i
width = 8.817630e-011
maximum iterations = 2048 
fractal 4.jpg
lower lefthand corner = -0.75411017990214 + 0.05536016227582i
width = 8.817630e-011
maximum iterations = 10000 

at the Sign of the Three Planes