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Author Topic: z^fickle + c  (Read 699 times)
Description: What happens when the power gets deeper as you zoom?
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kjknohw
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Posts: 45


« on: April 09, 2017, 09:01:35 PM »

We all know about c -> z^n + c for some n. Lets add a wrinkle: n is max(2,2+round(-k*log(abs(z)))) for some k>0. This will produce discontinuities in the fractal when n jumps but it also introduces a lot of interesting phenomena. Furthermore, we can multiply by a constant: z -> f(n)*z^n + c. It's natural to set f(n) = exp(0.5*x*x/k), so that abs(f(n)*z^n) is continuous. There is a transition around k = 0.32, lets start with k = 0.3, which superficially looks like the m-set.

The jumps in n quickly produces strange effects such as scrambling the "period n-folding" rule. Note the 3 fold symmetry and 12-fold symmetry deeper in for a power 4 minibrot:



This one can't seem to make up it's mind:



Zooming in, the mandelbrot set becomes a collage of various features. Most of these cannot be unseen (these features will repeat themselves when we find deeper baby mandelbrot sets):



Changing k upward causes the different powers to interact and the mini-mandelbrots don't even resemble z^n mandelbrots (image for k=0.4):



Perhaps the most interesting parameters are at k back to 0.3 and f(n) = exp(1.1*x*x/k) instead of f(n) = exp(0.5*x*x/k). This produces a more open fractal with islands of activity buried inside mini-msets (values much above 1.1 destroy the existence of deeper mini m-sets in the first place):



Attached is the ultra fractal formula file.

* fickle.ufm (2.01 KB - downloaded 153 times.)
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