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..