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really nice, claude.

from your code
https://mathr.co.uk/blog/2016-04-10_collatz_fractal.html
i was able to generalize the formula to 3d with good distance estimation, this is the formula in mandelbulber:

void CollatzIteration(CVector3 &z, sExtendedAux &aux)
{
   CVector3 xV(1.0, 1.0, 1.0);
   z = xV + 4.0 * z
      - CVector3(xV + 2.0 * z)
            * z.RotateAroundVectorByAngle(xV, M_PI);
   z /= 4.0;
   aux.DE = aux.DE * 4.0 + 1.0;
}

attached images are all the same object from different points of view.

with Add Cpixel.  Step multiplier a slow .01

(I have to put something here so the forum doesn't reject this as a duplicate post... )

This is really cool! I've seen this fractal before, but the use of DE coloring almost made me not recognize it at first - it really brings out all the details.

Anyway, I realized that you could change the constants in the formula to make a whole family of Julia sets based on this formula. There are 4 complex constants in the formula, so that makes an 8-dimensional parameter space (the "/4" at the end of the formula is technically another constant, but it's redundant with the others).

Next up would be to make a Mandelbrot set for this formula, however to render it properly I'd need to derive the critical points ( I think the sin/cos based fractals have an infinite no. of critical points and are therefore hard to render properly, but I think there's a trick to it. I read some pdf's on it a long time ago, I'll see what I can dig up).

Two more!

Here's the .frag with the Julia parameters:

awesome collatz, it would be interesting to see collatz in ultrafractal or some kind of deep zoom program, and see it in color.

The menger sponge is awesome! i want to print it in 3d!!! perhaps just pring one of them, but in distorted timespace!:)

I've been fascinated on and off for some time by the Collatz postulate, with how every positive integer always reverts back to a 4-2-1 loop, no matter what. And if you allow negatives, there are a finite number of additional known loops. Some particularly hard numbers grow to quite impressively high values before eventually regressing to the aformentioned 4-2-1 loop.

I'm more interested in this unique fractal and any hidden details it may hold at higher values. Is it Mandelbrot-like increasing complexity or relative sameness? I've seen a few renders but I don't understand how it is calculated. If it uses trig, then an arbirary precision library for it would be bog-slow compared to deep-zoom Mandelbrots.

Also is there a plugin for UF5? I own a basic license for Ultra Fractal but haven't done much with it in the past several years since I got obsessed with deep zoom fractal exploration, and my second and third order abs() fractal sets got incorporated into Fractal Extreme plugins and Kalles Fraktaler. UF5 is far too slow for doing any arbitrary deep zoom stuff.
 

Edit: No trig at all! I just read the following writeup which is quite fascinating:
http://jlpe.tripod.com/collatz/cfractal.pdf

It does require a complex abs() necessitating the square root function which would complicate arbitrary precision zooming. The Collatz fractal has two different versions based upon whether it uses the ceiling or floor of the complex absolute value to determine evenness or oddity, and computes the result based on whether the fractal eventually converges on 4-2-1 or escapes without bound. And I suppose one could develop an exterior coloring technique based on how quickly it escapes to some arbitrarily large bailout, or internally based upon how long slowly it converges on 4-2-1. This might make the boundary of the set highly ornate, even in monochrome, with fractal details on both edges of the curve if both inner and outer coloring methods are employed.