M-Set on a Partial/Total Mobius Cylinder looks neat

[Levi]

Here are some escape time pics of a power-2 Mandelbulb with the Mobius cylinder rules applied. Same configuration as in OP. I also tried inverting the 'supplementary' component (I don't know if the Mandelbulb's 3rd dimension has a name, I just call it supplementary) in addition to inverting the imaginary component, but it didn't change much.

Top view


Side view

[PeaBrain]

Hello, I'm new in the forum. Can you please explain me how I can redo the formula in UltraFractal5?

[claude]

Nice!  Wrapping in the other direction looks good too, just change the formula line

Code:

p=mobius(p);

to

Code:

p.yx=mobius(p.yx);

(I like this version with the width slider around 1.3).  Also, using the regular quadratic Mandelbrot set distance estimator seems to work well enough.  I tried some experiments by rotating instead of reflecting, basing the angle on the continuous X value, got a sort of sinusoidal strip with cusps opposite needles alternating up and down but it wasn't as interesting.

Not very mathematically satisfying, as Levi mentioned, but the pictures look cool.  A Möbius strip isn't orientable, and squaring complex numbers has to do with doubling orientation relative to a distinguished origin (which also isn't really possible to pick on a loop).  But, it might be worth trying some iterated function systems (using the chaos game method) on a Moebius cylinder: http://math.stackexchange.com/questions/1369588/is-this-a-valid-example-of-a-non-euclidean-sierpinski-attractor