Animation Sequence with Fractals from Variation on the Mandelbrot Set

The increasing interest in fractal functions over the past few years can be attributed in part to computer graphics for two reasons: Computer-generated images can help mathematics form intuitions to study the properties of fractals and, at the same time, they have popularized the subject, drawing more people in this field. A typical fractal function is the Mandelbrot set, which is the locus of all points in the complex c plane for which the magnitude of the iterants zk+1=zmk+c remains bounded for m=2. We have experimented with variations in the iteration formula and we report on the most interesting case, in which m is varied over a wide range of values. This choice results in a sequence of fractal surfaces exhibiting self-similarity and suggesting smooth evolution under animation. One such sequence suggests a mathematical conjecture, which further illustrates the interaction between computer graphics and fractal geometry. Finally, we offer an extension for adapting fractal graphics algorithm for massively parallel computers.