# Negative-exponent fractals

To compute a point in the Mandelbrot set, the equation z = z^2+c is repeated many times. If z exceeds 4, it will grow to infinity, and the point is outside the Mandelbrot set. Similar sets can be generated for exponents of 3, 4, etc.

What happens if you want to use a negative exponent such as -2?

You can use a similar mechanism, and declare that a point has escaped if it exceeds a bound. However, unlike the positive exponent case, large values don't really escape, since z^-2 is very small if z is very large, so the z values come back. Thus, the results depend on the bound selected and show artifacts, which appear as concentric circles.

A better solution is to see if the sequence of z values forms a cycle. For some values of c, z will converge to a fixed value. For others, z will oscillate between two or more values. In other cases, z will chaotically wander. By coloring each point according to the cyclic behavior, we can see the true structure of the fractal without artifacts that depend on the escape bound, as shown on the left. (The result is reminiscent of Mobius fractals.)

With c fixed and z varying, a Julia set is created. Negative-exponent fractals show a variety of Julia sets, depending on which c value is selected. To compute the Julia set, again the cycles must be detected. However, for the Julia set, every point will converge to the same cycle. The figure can be colored according to the "phase" of the cycle, though. This yields the following images.
For more details, see my paper, which was published as ``An Investigation of z -> 1/z^n+c,'' Computers & Graphics, 17(5), Sep. 1993, pp 603-607.

Ken Shirriff: shirriff@eng.sun.com