(You may click on each of the pictures on this page to get a larger version)
In these factals, we use iterative equations z = c + f(z) for various functions f. The iterations may result in converging or diverging values of z. Scanning the screen as if it were the complex plane, we use the iterative equation a fixed number of times to compute the values of z. We then color the "points" (pixels) according to various convergence criteria. For example, we color a point one color if the values of z converge, another color if they oscillate between two values, another color if they oscillate among three values, etc..., and still another color if they become unbounded. This method produces "interior" (non-black) colors in Mandelbrot sets and other fractals.
Here are three fractals of the form z =c + kz^4 + (1-k)/z , with k = 1, .95 and 1.05. The k=1 case results in a "three-lobed" Mandelbrot type fractal but the addition of a pole at the origin,(1-k)/z,interrupts the three-fold symmetry.
z= c + z^4 z = c + z^4 + .95/z z = z^4 + 1.05/zmathfile home page