Bifurcation diagram of the logistics map in the Mandelbrot set

November 6, 2022

This is one of those thing that absolutely blew my mind when I found about it. The first the I came across this is in a video about the logistics map by Veritasium. I suggest to check out the video if you want to know more about it. The scene below renders the Mandelbrot set and uses all the real values of $z$ that are in the set as the y-axis. You can use the mouse to move it around.

To create the rendering, the following steps are taken.

1. Generate a $(x, y)$ point that is used for $c$ when iterating $z^2 + c$.
2. Store all the real values of $z$ that we encounter during the iteration.
3. Discard everything if $z$ is outside of the Mandelbrot set, e.g. the argument is greater than 2.
4. Generate points for all the real values, using those as the z-coordinate, using the $x$ and $y$ that has been generated in step 1.
5. Create a point cloud to render all the points.