interesting *brot variant

[vinecius]

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:



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.

[rethread]

z^z is alot higher magnitude.   blasting it to total hell there.

[ottomagus]

Hi. Its known as tetration. A while back I wrote a couple of formulas for Ultra Fractal exploring the idea. Had intended to return to it, and your post has reminded me to do so.

A Google search for 'tetration fractal' turns up a fair amount of info https://www.google.co.uk/search?q=tetration+fractal&ie=&oe=

I'm sure there are people here who know a lot more about it than I do 

[vinecius]

Yup, tetration fractal it is.  I looked it up on google images and it's the same sort of structures that I've been getting.

The next logical thing to try was z = (z ^ z) ^ c which is shown below, bottom zoomed in.  I also tried z = z ^ (z ^ c) and there appears to be content there but I get overflow errors, might need bigger numbers if i want to go further.

[ottomagus]

Some Julias:

z=z^z^z-1.275


z=(z-3.2)^z


z=(z-2)^(z+2)


z=(z-0.703)^(z-0.703)


z=z^z+z-0.5277


Mandelbrots to follow...

[ottomagus]

Mandelbrots:

z=z^z+c


z=z^z^z+c


z=z^c^z


z=(z+c)^z


z=(z+c)^z


z=(z+c)^z


z=(z+c)^z


z=(z+c)^z


z=(z+c)^(z-c)


z=(z+c)^(z-c)


z=z^z+z+c


z=z^z+z+c


z=z^z+z+c


z=z^z+z+c

[ottomagus]

Mandelbrots again:

z=(z^z)^2+c


z=(z^z)^2+c


z=c^z^z


z=(z/c)^(z/c)


z=(z/c)^(z/c)


z=c/z^z


z=(c/z)^(c/z)


z=(z^z+c)^(z^z+c)


z=(z^z+c)^(z^z+c)