Here is a paper I wrote that discusses the symmetry that found for a given order of a Mandelbrot or Julia fractal. Fractal Symmetry For those interested the methods section of the paper gives a very brief idea of how to generate a Mandelbrot or Julia fractal. Below is a graphic that shows the basic changes in symmetry as the order of the function f(z)=zⁿ + c, where n is a integer and is the order of the function.