07/22/2022, 04:14 AM

This following fractals appear to be the most popular piece of mathematics I've done. Nomenclature is a bit of a problem as there is no standard way to name such fractals that are Mandelbrot sets where the iterated quadradic equation is replaced by power towers. So I think of them as tetration Mandelbrot fractals.

Now I want to show what the pentation Mandelbrot set would look like, the pentation version of the above fractals. Big question is what tetration extensions should be used. I believe the method I developed for extending tetration to complex numbers would be a good candidate. My professional background is as a full stack developer, not a mathematician. So I'm in a good position to write code using my Taylor's series method. I expect I will try to write code for both Julia and Pari.

My method has been seen as being based on Schroeder's function and for almost all complex values my method should produce the same results. Well, wouldn't that be appropriate to create pentation fractals? Of course what would be really cool would be software implementations of all the major methods of extending tetration. It might be a quick way of identifying which methods produce the same results. What do folks think, is complex based tetration appropriate for creating pentation fractals in the complex plane?

JmsNxn was correct when I asked how to directly computer pentation fixed points and he pointed out what I meant to ask for was computing the tetration fixed points. What I currently need is a software approach to computing tetration fixed points because my Taylor's series method is based on fixed points. I figure I can either calculate fixed points by brute force if needed or use the fixed point associated with a pixel to begin a close approximation of the fixed point of the adjacent pixels.

Thoughts?

Now I want to show what the pentation Mandelbrot set would look like, the pentation version of the above fractals. Big question is what tetration extensions should be used. I believe the method I developed for extending tetration to complex numbers would be a good candidate. My professional background is as a full stack developer, not a mathematician. So I'm in a good position to write code using my Taylor's series method. I expect I will try to write code for both Julia and Pari.

My method has been seen as being based on Schroeder's function and for almost all complex values my method should produce the same results. Well, wouldn't that be appropriate to create pentation fractals? Of course what would be really cool would be software implementations of all the major methods of extending tetration. It might be a quick way of identifying which methods produce the same results. What do folks think, is complex based tetration appropriate for creating pentation fractals in the complex plane?

JmsNxn was correct when I asked how to directly computer pentation fixed points and he pointed out what I meant to ask for was computing the tetration fixed points. What I currently need is a software approach to computing tetration fixed points because my Taylor's series method is based on fixed points. I figure I can either calculate fixed points by brute force if needed or use the fixed point associated with a pixel to begin a close approximation of the fixed point of the adjacent pixels.

Thoughts?

Daniel