Let's look at the Julia set shown above in "Cloisonné Lion IV." First off, it is a totally disconnected Cantor set of points. Is it easy to tell from the picture? Not really.
The nongoldish colors are used to paint the complement of the filledin Julia set, hence, the purplish color at the center of the image indicates that the orbit of z_{0} = (0, 0) diverges to ∞. Since the orbit coincides with the critical orbit of p, it follows that p does not belong to the Mandelbrot set, or equivalently, the Julia set of p is a Cantor set by the alternative definition of the Mandelbrot set. In short, we can tell if the Julia set is connected or not by looking at the color of the image at z_{0} = (0, 0) on some occasions.
So, math helps sometimes. Another fun fact in fractal plotting is that changing the value of a parameter p very slightly may result in a drastically different image. Compare the parameters of the images shown in "Cloisonné Lion IV" and "Cloisonné Lion V." An ugly fractal may turn to beauty by a tiny modification, so we should always be patient!
