Base 1  Printable Version + Tetration Forum (https://tetrationforum.org) + Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1) + Forum: Mathematical and General Discussion (https://tetrationforum.org/forumdisplay.php?fid=3) + Thread: Base 1 (/showthread.php?tid=1004) Pages:
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RE: Base 1  Daniel  06/19/2022 See fractals at tetration of 1 Mandelbrot of exponential map at 1 Julia set of exponential map at 1 RE: Base 1  JmsNxn  06/20/2022 Hey, Danielcould 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. RE: Base 1  Daniel  06/20/2022 (06/20/2022, 12:40 AM)JmsNxn Wrote: Hey, Danielcould you elaborate further on how you are constructing these graphs/the mathematical theory behind this? 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) {; Tetration (exponential map) Julia set Code: TetraJ (XAXIS) {; RE: Base 1  JmsNxn  06/20/2022 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. RE: Base 1  Daniel  06/20/2022 (06/20/2022, 01:01 AM)Daniel Wrote:(06/20/2022, 12:40 AM)JmsNxn Wrote: Hey, Danielcould you elaborate further on how you are constructing these graphs/the mathematical theory behind this? (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. 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 Tetration (exponential map) Julia set Code: TetraJ (XAXIS) {; \\ x axis symmetry RE: Base 1  Catullus  07/06/2022 . is a fixed point of . A fixed point is also a 1cycle, so the issue with iterated exponentials at ncycles happens with . What if ? 