alexfikl / netbrot

Mandelbrot set on networks

Netbrot

This repository contains some experiments for vector Mandelbrot sets. We look at the map

$$f(z) = (A z)^2 + c,$$$$

where $A \in \mathbb{C}^{n \times n}$ and $z \in \mathbb{C}^n$ with just $c \in \mathbb{C}$. This gives some interesting results that are not directly analogous to the standard scalar case:

• The escape radius is no longer just $2$.
• The periodicity of the various points is weirder.
• There are (possibly) multiple attractive or repelling fixed points, not just $z = 0 + 0\imath$.

Additional math needed! Most of these ideas have no proofs at the moment, but seem fun to investigate!

Install

This is a Rust app and uses all the standard build infrastructure. To build it, just run

cargo build --release

Usage

This is currently very experimental and just meant for playing around. Even so, it's nicely parallelized with rayon and colored.

It can do the usual orbit iteration

netbrot --color orbit -- out.png

and it can also look at periodicity of various points

netbrot --color period -- out.png

Example

As a simple example, we take the matrix

$$\begin{bmatrix} 1 & 0.8 \\\ 1 & -0.5 \end{bmatrix}$$

License

The code is MIT licensed (see LICENSES/MIT.txt). It was originally copied from the Rust Programming example here and has since evolved a bit.