Julia Sets

This page includes an interactive visualization of representing sets of points in the complex plane. We depict a complex number $z$ by the screen-relative coordinates $(x, y)$ (i.e., rightward, upward) such that $z = x + iy.$

The Julia set(s) that we visualize on this page are the set of starting points that will stay close to the origin when repeatedly mapped according to

\begin{equation} \label{eqn-1}\tag{1} z \rightarrow z^{2} + c \quad ; \quad \begin{bmatrix} x \\ y \end{bmatrix} \mapsto \begin{bmatrix} x^{2} - y^{2} + c_{x} \\ 2xy + c_{y} \end{bmatrix}, \end{equation}

where $c$ is a point determined by the coordinates of your mouse. Each value of $c$ thus defines a unique set, which we visualize by mapping how long an initial point (a pixel in this image) stays close to the origin using color. The longer an initial point stays close to the origin, the more color (i.e., the brighter) it gets. Go ahead, move your mouse over the image below!

View fullscreen

You can click on the image as well! Doing so animates a single iteration of the map Eq. (1). Note that, in the complex plane, multiplication looks like a combination of rotation and a scaling, while addition looks like a translation.

Notice that, because the points in the set must approximately stay in this set (lest they soon fly off to infinity), the Julia set describes fixed points and cycles of the mapping (Eq. (1)).

Also note that, if you could record all of the mouse positions for which the origin of the image is in the Julia set, you would map the Mandelbrot set! In fact, the Julia set(s) and the Mandelbrot set are different two-dimensional slices of the same four-dimensional object, where the Julia sets fix values of $c$ and the Mandelbrot set fixes $x = 0$ . We can visualize the Mandelbrot set in the background of the set above.

Show Mandelbrot set above

Pay attention to the origin, noting that the origin is in the Julia set if and only if your mouse is in the Mandelbrot set.