Fractional-exponent fractals

To compute a point in the Mandelbrot set, the equation z = z^2+c is repeated many times. What happens if you want to use a fractional exponent such as 2.5?

The problem is that z^2.5 doesn't have a unique value. (Just like the square root of 4 is 2 or -2.) A standard formula for fractional powers will pick one of values, and there will be a "fault line" of z values where the formula suddenly switchs from one value to another. This results in artifacts in the fractal.

I'm exploring different ways of computing the "real" structure of a fractal generate from a multi-valued function such as a fractional exponent.

The following image is a sample result:

Fractional-exponent fractal

Ken Shirriff:
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Copyright 1998 Ken Shirriff. Last updated 12/21/98.