# How is this fractal produced? It is stated here:

Iterating the above optimized map $$f(z)=\frac{1}{4}(1 + 4z - (1 + 2z)\cos(\pi z))$$in the complex plane produces the Collatz fractal.

The point of view of iteration on the real line was investigated by Chamberland (1996), and on the complex plane by Letherman, Schleicher, and Wood (1999).

However, in the 2 mentioned publications I did not find this image. I would like to know which start value $x_0$ created this image.

Am I correct, that this image is simply a visualization of the sequence $(f^n(x_o))_{n \in \mathbb{N}}$ where the black parts show where the sequence remained for a long time? Does this sequence also end in a finite orbit?

• why the collatz tag? – Arjang Mar 22 '16 at 14:57
• @Arjang if you restrict the function $f$ on the natural numbers you get the collatz function – Adam Mar 22 '16 at 15:04
• Usually in these images, black points corresponds to convergent starting values while pixels are colored according to some measurement of how quickly divergence occurs. I think Wikipedia on Mandelbrot talks about ways of doing this. – D. Wagner Mar 22 '16 at 15:42

## 1 Answer

The start value $x_0$ is determined by the coordinates of each pixel. Pixels are coloured acccording to how quickly the orbit for that pixel diverges (escape time colouring). Black pixels remained bounded within the iteration limit. I wrote a small GLSL implementation as demonstration: https://www.shadertoy.com/view/Ms3XDn (it could be improved with smooth colouring, and user interface for moving around/zooming)

Here is a screenshot of the shadertoy with center (2.66, 0) and size 0.5 units: And here's one with center (0, 0) and size 4 units: • Nice! THis is actually reproducing the image :D. Where is the origin (x,y)=(0,0) in your image? – Adam Mar 22 '16 at 18:00
• @Adam the center of the image is around (2.5, 0) (but not exactly, would have to take into account the aspect ratio of the image more appropriately..) – Claude Mar 22 '16 at 18:27
• Thank you. Could you also state the width of the picture? Is the length like 10000 or more like 10 units? – Adam Mar 23 '16 at 16:50
• I second @adam 's last comment: I'd like to see the places,where the known cycles in the positive and the negative numbers are. (Unfortunately the link doesn't work for me, something like WebGL is missing at my system) – Gottfried Helms Mar 25 '16 at 3:03
• I added a couple of screenshots with location information. – Claude Mar 25 '16 at 11:35