JuliaBrots

Consider the equation

zn+1 = z2n + c

Each value of the complex constantcwill generate a different sequence. With a graph with the X axis (real axis) covering the range -2.5 to 1.5 and the Y axis (imaginary axis) covering the range -1.5 to 1.5 the Mandelbrot Set image is obtained. (See Mathematics of Divergent Fractals). Now assign to ca point in the Mandelbrot set or its immediate surrounding region. This will be called a Julia set index. On a new graph ranging from -2 to 2 on the real X axis and ranging from -1.5 to 1.5 on the imaginary Y axis, choose a point on the graph and assign it toz0. Now iterate the equation. Do this for every point on the graph, using the same value ofcfor all iterations. This will generate a Julia set for the pointc.

Clearly, ccan also be treated as a variable, and so the above equation has 4 dimensions, two for the real and imaginary parts ofz, and two for the real and imaginary parts ofc. This 4 dimensional object was named a JuliaBrot by Mark Peterson, one of the original developers of the Fractint fractal generating software. By holding one of the dimensions constant a 3DJuliaBrot can be generated. The three images below illustrate a JuliaBrot generated with

Fractint and UltraFractal. For these images crealis held constant and cimag is the z axis. The UltraFractalJuliaBrot code was written by this author and is in the public UltraFractal library.