While scouring reddit for a programming challenge I stumbled across the programming daily subreddit that provides daily challenges to readers. One looked at generating Julia fractals so sensing an opportunity for some maths and gnarly images I thought I’d have a go.

The idea behind Julia fractals is taking a complex function and a point in the complex plane and repeatedly applying the function until the value of the function exceeds some threshold. Given a pixel whose location corresponds to the point in the plane, we color it according to the number of iterations required and repeat for lots of points in the plane.

We used the function f with \omega = - (0.221 + 0.731i)

\begin{matrix} f_\omega: & \mathbb{C} & \rightarrow & \mathbb{C} \\ & z & \rightarrow & z^{2} - \omega \end{matrix}

Iterating over the intervals [-1,1] for both the real and imaginary parts in steps of 0.001, gives the following which I will definitely be using as the new header for the blog.

julia set