(06/20/2022, 12:40 AM)JmsNxn Wrote: Hey, Daniel--could you elaborate further on how you are constructing these graphs/the mathematical theory behind this?

I know you are using the fixed point formula \((-1)^{-1} = -1\) but could you elaborate further? Which branch of the exponential are you using particularly. I assume this is the Schroder iteration (your Bell matrix approach). But which branch of \((-1)^z\) are you choosing. Which is to mean: \((-1)^z = f_k(z) = e^{\pi i(2k+1) z}\) for some \(k \in \mathbb{Z}\). And each has a repelling fixed point at \(z=-1\) with multiplier \((2k+1)\pi i\). I assume that you are doing the entire iteration about these fixed points (every entire function about a repelling fixed point admits an entire iteration).

Just curious because this looks really interesting. I'm just interested to know more about the backstory of how these graphs are made!  

Please, elaborate!

Regards, James.

(06/20/2022, 01:01 AM)Daniel Wrote:

(06/20/2022, 12:40 AM)JmsNxn Wrote: Hey, Daniel--could you elaborate further on how you are constructing these graphs/the mathematical theory behind this?

I know you are using the fixed point formula \((-1)^{-1} = -1\) but could you elaborate further? Which branch of the exponential are you using particularly. I assume this is the Schroder iteration (your Bell matrix approach). But which branch of \((-1)^z\) are you choosing. Which is to mean: \((-1)^z = f_k(z) = e^{\pi i(2k+1) z}\) for some \(k \in \mathbb{Z}\). And each has a repelling fixed point at \(z=-1\) with multiplier \((2k+1)\pi i\). I assume that you are doing the entire iteration about these fixed points (every entire function about a repelling fixed point admits an entire iteration).

Just curious because this looks really interesting. I'm just interested to know more about the backstory of how these graphs are made!  

Please, elaborate!

Regards, James.

These fractals were made thirty years ago with FractInt, a versatile fractal generator with a programming language. As you can see in the code, the algorithms are simple that generated the fractals.  

Tetration (exponential map) Mandelbrot set

Code:

TetrationM (XAXIS) {;
  z = pixel:
   z = pixel ^ z
    |z| <= 100000
  }

 
Tetration (exponential map) Julia set

Code:

TetraJ (XAXIS) {;
  z = pixel:
   z = P1 ^ z
    |z| <= 100000
  }

(06/20/2022, 02:37 AM)JmsNxn Wrote: Daniel, please explain better. I get that that makes sense to you. Please elaborate further. At the risk of sounding stupid. Explain more. Elaborate.

Can you elaborate further from the Fractint reference? I didn't get much from this that I could use to answer my original question. Any help would be greatly appreciated. Are these just \(f(z) = e^{\pi i z}\) and \(F(s)\) is the iterate? Please, elaborate futher.