Orbits and Mandelbrot set explorer with Numerical Elixir and Zig

Introduction

Source:

• Mandelbrot set
• Plotting algorithm

This repo is about visualising the orbits of points in the 2D-plane under the map z -> z*z + c known as Julia or Mandelbrot sets.

We are looking whether the iterates stay bounded are not. When we associate a colour that reflects this stability, this gives rise to Mandelbrot images.

These images are therefor a colourful representation of where the sequence is stable and how fast does these sequences diverge.

This repo contains:

• a pur Zig computation
• two Livebook to explore the orbits of points and to zoom into the Mandelbrot set.

The two Livebooks are:

• one proposing a pur Elixir orbit explorer. You click and it displays the orbit of any point.
• the other proposing a Mandelbrot set explorer with a clickable image to zoom in (and out):
  • we have a pur Elixir implementation using Numerical Elixir with EXLA backend of the Mandelbrot set.
  • we have an enhanced version where the heavy computations are made with running embedded Zig code thanks to the library Zigler.

The Zig code is compiled to WebAssembly to demonstrate quickly how this renders:

https://ndrean.github.io/zig-assembly-test/

Orbit explorer

Given a complex number c and the polynomial p_c(x)=x^2+c, we compute the sequence of iterates:

p(0)=c
p(p(0))=p(c)=c^2+2
p(p(c)) = c^4+2c^3+c^2+c
...

The set of this sequence is the orbit of the number c under the map p.

For example, for c=1, we have the orbit O_1 = { 0, 1, 2, 5, 26,...} but for c=-1, we have a cyclic orbit, O_{-1} = {−1, 0, −1, 0,...}.

The code below computes "orbits". You select a point and a little animation displays the orbit.

Mandelbrot set explorer

The algorithm

Given an image of size W x H in pixels,

• loop over 1..H rows, i, and over 1..W columns, j
  • compute the coordinate c in the 2D-plane projection corresponding to the pixel (i,j)
  • compute the "escape" iterations n,
  • compute the RGB colours for this n,
  • append to your final array.

A Livebook

You can explore the fractal by clicking into the 2D-plane.

This happens thanks to KinoJS.Live.

We have a pur Elixir version that uses Nx with the EXLA backend, and another one that uses embedded Zig code for the heavy computations. The later is 3-4 magnitude faster mostly because we are running compiled Zig code versus interpreted Elixir code in a Livebook running uncompiled on a BEAM.

Details of the mandelbrot set

[Needs latex]

In fact, this set is contained in a disk $ D_2$ of radius 2. This does not mean that $ 0_c$ is bounded whenever $|c|\leq 2$ as seen above. Merely $0_c$ is certainly unbounded - the sequence of $z_n$ is divergent whenever $|c| > 2$.

We have a more precise criteria: whenever the absolute value $|z_n|$ is greater that 2, then the absolute values of the following iterates grow to infinity.

The Mandelbrot set $M$ is the set of numbers $c$ such that its sequence $O_c$ remains bounded (in absolute value). This means that $| z_n (c) | < 2$ for any $n$.

When the sequence $O_c$ is unbounded, we associate to $c$ the first integer $N_c$ such that $|z_N (c)| > 2$.

Since we have to stop the computations at one point, we set a limit $m$ to the number of iteration. Whenever we have $|z_{n}|\leq 2$ when $n=m$, then point $c$ is declared "mandelbrot stable".

When the sequence $O_c$ remains bounded, we associate to $c$ a value $max$.

So to each $c$ in the plane, we can associate an integer $n$, whether null or a value between 1 and $m$. Furthermore, we decide to associate each integer $n$ a certain RGB colour.

With this map:

$$c \mapsto n(c) \Leftrightarrow \mathrm{colour} = f\big(R(n),G(n),B(n)\big)$$

we are able to plot something.

By convention it is black when the orbit $O_c$ remains bounded.

When you represente the full Mandelbrot set, you can take advantage of the symmetry; indeed, if $c$ is bounded, so is its conjugate.

---

Math details:

Firstly consider some $|c| \leq 2$ and suppose that for some $N$, we have $|z_N|= 2+a$ with $a > 0$. Then:

$$|z_{N+1}| = |z_N^2+c|\geq |z_N|^2 -|c| > 2+2a = |z_N|+a$$

so $|z_{N+k}| \geq |z_N| +ka \to \infty$ as $k\to \infty$.

Lastly, consider $|c| > 2$. Then for every $n$, we have $|z_n| > |c|$. So:

$$|z_{n+1}| \geq |z_n|^2 -|c| \geq |z_n|^2-|z_n| = |z_n|(|z_n|-1) \geq |z_n|(|c|-1) > |z_n|$$

so the term grows to infinity and "escapes".

---

The mandelbrot set $M$ is compact, as closed and bounded (contained in the disk of radius 2). It is also surprisingly connected.

Fix an integer $n\geq 1$ and consider the set $M_n$ of complex numbers $c$ such that there absolute value at the rank $n$ is less than 2. In other words, $ M_n={c\in\mathbb{C}, , |z_n(c)|\leq 2} $. Then the complex numbers Mandelbrot-stable are precisely the numbers in all these $ M_n$, thus $M = \bigcap_n M_n$. We conclude by remarking that each $M_n$ is closed as a preimage of the closed set $ [0,2]$ by a continous function, and since $M$ is an intersection of closed sets (not necesserally countable), it is closed.