Base -1

[Daniel]

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
  }