Mandelbrot set with Numerical Elixir and Zigler

We build a Mandelbrot set explorer with a Livebook. It is an Elixir code runner that lets you write Elixir in the browser.

Algorithm

An explanation can be found here

Given an image with dimensions (eg w x h of 1500 x 1000), max iteration (eg 100), we want to assign to each pixel a colour which represents the "stability" of the orbit of the underlying point.

Iterate over each pixel (i,j):

• map it into the 2D plane: compute its "complex coordinates"
• compute the iteration number
• compute a colour
• Sum-up and draw.

Explorer

We have to parts:

• one Livebook to visualise the orbits of a point, the iterates
• one Livebook to visualise the Mandelbrot set and zoom into any point.

This is done with the Numerical Elixir library Nx. A solution is also proposed using embedded Zig within the Livebook.

The Kino.JS.Live library makes it possible to interact with the browser.

Exploring orbits

In this Livebook, you click on a point c of the 2D-plane and it computes the succession of points, the orbit, under the iterative map z(n+1) = z(n) * z(n) + c and z(0)=c.

Code source: https://github.com/ndrean/mandelbrot/blob/main/livebook/orbits.livemd

Explore the Mandelbrot set

In this Livebook, you can click into the 2D-plane to zoom into the area of interest.

We propose two solutions:

• one using only Elixir for the computations,
• one using Zig for the heavy computations (you need to have Zig installed).

Although Zig gives you a huge boost, the performance of the Elixir code is good. This is because we use the numerical library Nx with the EXLA backend.

Code source: https://github.com/ndrean/mandelbrot/blob/main/livebook/mandelbrot.livemd