I am trying to write program, that builds fractal, like a mandelbrot, but somewhat special... I could explain my "thought flow":
A mandelbrot fractal (I call it "operation order 1") is done by iteratively calculating complex number $Z(n+1) = Z(n)^2 + C$, $Z(0)=0$, $C$ is zoomed and offset screen coordinate. After an infinite number of steps (supertask) we do determine, wherether series for a pixel - does series converge or diverge? and "how fast" does it diverges? Faster are colored by blue, slower - by white. For non-divergent $C$ pixels, belonging to Mandelbrot Set - are black:
Then, we may proceed to next order, in hyperoperation sequence, and produce "operation order 2" fractal, by iterated exponentiation $Z(n+1)=c\exp(Zn)*C$, basically resulting in this:
You may increase to the next, order 3 fractal, and get tetration fractal by doing Z(n+1)=cpow(ctet(some_const, Z(n)),C). But I virtually ran into a wall.. NO traces of how can I write ctet(base,height) function, for integer base, but complex height..
It's EASY to write for complex base and positive integer height: it's just equal to base^(base^(base^... and do it height times. But I need that not only for height being positive integers, but for height being complex argument. And base being.. $2.0$, or $e$, or whatever constant, that would just be convenient (that's of minor importance for the fractal structure, rather than bringing height variable to complex space)
That is called analytic continuation, or somehow similar. I am not interested in getting big numbers, but rather in iterating and detecting "Does it converge or diverge?", and, secondary "How many iterations does it take to diverge?"
Can you offer it in simple words, without going really heavy math?
Ive found this but I really fail to understand what is he doing, with insane symbols that aren't easily understandable by programmer on "how to convert that to code". and his taylor series has only X variable, but ctet(base,height) hyperoperation should take 2 arguments indeed..
This is not an answer, just a comment, but having an image in it
Here is an example of complex heights. I've the base $b=4$, initial value $z_0=4$ . Then -in steps of 128- computations for heights $h=1, \cdots, h=I , \cdots, h=-1 , \cdots, h=-I , \cdots, h=1$ with $h$ on the complex unit circle. (The computations are done with a version of the Kneser-ansatz for bases $b$ larger than $b=e=\exp(1)$ (the choice in Kneser's original text) programmed by the tetration-forum member S. Levenstein). You might not yet have an idea how iteration with complex heights looks in a trajectory. Here is one picture.
You can see, that from $z_k= \;^{\exp(2\pi I \cdot k/128)}4$ with $k=0$ and $z_0=4$ the trajectory does roughly an egg-formed path via $z_{64}=0$ back to $z_{128}=4=z_0$.
I don't know whether such a trajectory would really be useful for your project-intentions... Is it anyway?
remark: I'd like more to use the notation $$ \exp_{\log 4}^{\circ h_k} (z) \text{ with } h_k=\exp(2 \pi I \cdot k/128) \text{ and } z=1 $$ but for whatever reason the public prefers the less expressive notation $^h 4$ .
remark 2: the legend of the image does not perfectly fit.
In the title it should be "$\small \exp(2*Pi*î*k/128)$ for $\small k=0..128$"
and the x/y-legends should contain "z" instead of "x".
It's an older picture, just reused.