You know how the mandelbrot is z = z^2 + c? What happens when you define it as z = z^z + c taking the principal value for the exponent. Expanded out it's:
 exp(i(a arg z + b ln |z|))<br />)
where
and using the principal value for arg z.
You get some really interesting properties and a fractal that is especially sensitive to slight changes in the escape time algorithm distance function. One thing I've noticed that only that purple section i'm zooming in on the attached image is actually growing in complexity with magnification. All other sections are just recursive "more of the same", like the spirals on the mandelbrot. One other thing that's noteworthy is the tree like structure that has uncanny resemblance to the simple pythagoras tree fractal, even the ingrowing smaller trees and voids you see inside!
It seems like something that someone must've plotted and studied before, I'd love to hear what you fractal experts have to say about it.
August 04, 2011, 10:50:27 PM
A new 3d mandelbrot variant - Mandelcup
The 3D Mandelbulb
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