Hi! I recently programmed a free iOS Mandelbrot and Julia renderer: "Retinamandelbrot". It allows for arbitrary deep zooms (up to 10^20000 has been tested) and realizes some new speed-up algorithms (see discussion regarding the beta version in this forum).
It is now finally available on the App Store for all devices:

https://apps.apple.com/lu/app/retinamandelbrot/id1509664268?l=de

A MacOS Version will also be (hopefully) soon ready. There does exist a rudimentary Windows version, too, but I don't have the time to polish that at the moment (the rendering code is actually written in C, so it's just a matter of creating the interface)

Ciao,
  Fritz

PS. Some more information is available on www.retinamandelbrot.com.

Linkback: https://fractalforums.org/downloads/10/retinamandelbrot/3950/

I just dowloaded your app on my iPad, it's truly beautiful and mindblowing. I love that you included so many things: the orbit the points, the corresponding Julia set for each point in the Mandelbrot set, we can clearly see which are connected and which are Fatou dusts etc... Amazing job, congrats and thanks you infinitely!

P.S.: would be really amazing to have a version of it but using the 2D Minkowski plane instead of the Euclidean plane. Any chances for that? I'm doing research on fractals in Minkowski space (2D spacetime for special relativity basically)

Hi Hoermann, this looks great. Is there any chance of a deeper explanation of the algorithms used in this renderer - I know you stated what they were in the beta thread however I would really love to have some more detail around them.

Hi Lorentzian, thank's a lot for your feedback! I am happy that you liked the app. Best, Fritz

PS. A version in 2d hyperbolic space... Oh, actually, I never saw this. Which iteration is taken there? Actually, there woulds be infinite possibilities of including other things... higher power Mandelbrots or Julias for other polynomials, for example. The algorithms should generalize partly... It's mostly a matter of finding the time!

Hi Hoermann. The iterations are the same in the sense of looking at the parameter space of the quadratic z^2+c, except we're switching our point of view from the complex plane C to the 2D Minskovski plane or hyperbolic plane or "1+1 spacetime"(1 space coordinate and 1 time coordinate). This forms the natural setting for the so-called split-complex numbers:

x+jy where j^2 =1.

Now for complex numbers z^2 is given by

(x^2-y^2)+2ixy

but for split-complex numbers z^2 is given by

(x^2+y^2)+2jxy. Every split complex number can be represented as R(cosh(alpha)+jsinh(alpha)) in analogy with C.

It would be amazing if you could implement this small change! Please let me know if you have more questions about the Minkovski plane, thanks!

Hi Hoermann. The iterations are the same in the sense of looking at the parameter space of the quadratic z^2+c, except we're switching our point of view from the complex plane C to the 2D Minskovski plane or hyperbolic plane or "1+1 spacetime"(1 space coordinate and 1 time coordinate). This forms the natural setting for the so-called split-complex numbers:

x+jy where j^2 =1.

Now for complex numbers z^2 is given by

(x^2-y^2)+2ixy

but for split-complex numbers z^2 is given by

(x^2+y^2)+2jxy. Every split complex number can be represented as R(cosh(alpha)+jsinh(alpha)) in analogy with C.

It would be amazing if you could implement this small change! Please let me know if you have more questions about the Minkovski plane, thanks!

What is the connection to Minkovski space? I understand \( ||z \bar{z}||^2 \) is the Minkovski metric, is there more?