Base -1

[Daniel]

See fractals at tetration of -1 

Mandelbrot of exponential map  at -1


Julia set of exponential map at -1

[JmsNxn]

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.

[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
  }

[JmsNxn]

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.

[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
  }

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.

Tetration (exponential map) Mandelbrot set - the Mandelbrot set with the quadratic equation replaced by a=pixel and \( a^z \) 

Code:

TetrationM (XAXIS) {; \\ x axis symmetry
  z = pixel:          \\ initialize setting z to the value of the current pixel converted to a complex number
   z = pixel ^ z      \\ iterate once the original complex value of the pixel
    |z| <= 100000     \\ iterate until |z| becomes larger than 100,000. The number of iterations is the escape value
                      \\ represented by the color of the pixel.
  }

 
Tetration (exponential map) Julia set

Code:

TetraJ (XAXIS) {;   \\ x axis symmetry
  z = pixel:        \\ initialize setting z to the value of the current pixel converted to a complex number
   z = P1 ^ z       \\ P1 is a user defined complex variable set at runtime.
                    \\ iterate once P1 to the power of z
    |z| <= 100000   \\ iterate until |z| becomes larger than 100,000. The number of iterations is the escape value
                    \\ represented by the color of the pixel.
  }

[Catullus]

.  is a fixed point of . A fixed point is also a 1-cycle, so the issue with iterated exponentials at n-cycles happens with .
What if ?