|| Experimental mathematics perhaps may never be accepted by mathematicians, but for many it has or it will become a passion. The stimulus it will provide enhancing our intuition in the future is beyond our imagination. Also, without question it contains enough potential to develop a sophisticated art form.
When we were struck by our first findings in the spring of 1983 playing with Julia sets in a computer graphics lab we were quite ignorant with regad to the mathematical beauty and depth of the subject. Since then our addiction exploded and the questions became numerous.
The goal of this exposition is to embed our computer graphical findings into the more classical theory and give some flavour of the subject. Most of the more recent progress which seems to be due to Bowen, Douady, Hubbard, Ruelle and Sullivan is out of the scope of this note. We refer the interested reader to [Bo,D,R,S]. We think that it is fair to say that much of the present interest in the subject of iterations of rational functions in the complex plane is due to the influence of Mandelbrot's work [Ma1, Ma2].