Julia Set: Math Art and its Workings

One of the most familiar examples of math used in algorithmic art is the Julia Set. It is a simple mathematical process with surprisingly complex and beautiful results.

At the heart of the Julia Set is this seemingly simple equation:

z = z² + c

Where:

• z is a complex number (a number with a real and imaginary part).

• c is another complex number, but a constant.

• We start with an initial value of z (which corresponds to a pixel on the screen) and then keep plugging it into the formula again and again.

Mathematically this is called iteration. We calculate a new z, plug it back in as the new vaue of Z and repeat.

As we iterate, the number z either:

• Escapes to infinity (its value gets bigger and bigger) or

• Stays bounded (it keeps bouncing around without getting too large)

We define the pixel as escaping as when the magnitude (or absolute size) of z grows larger than 2 (threshold). Once that happens, it’s guaranteed to grow forever. So we say that point has escaped. If it never escapes, even after hundreds or thousands of iterations, we consider it part of the Julia Set.

Here’s where the art happens:

1. Each pixel on the screen represents a different starting point (z) in the complex plane.

2. For each pixel, we run the formula over and over.

3. We color the pixel based on how many iterations it took to escape- for example:

   • Escapes fast → Bright color (e.g. blue)

   • Escapes slowly → Medium tone (e.g. green)

   • Never escapes → Black or deep color (part of the set)

This process paints the algo art picture. And by changing the constant c, the entire image can morph into something completely new say- sharp spikes, spirals or even blobs. Just a tiny change in c produces radically different visuals.

The boundary between escaping and non-escaping points is infinitely complex. It never smooths out, no matter how far we zoom in. That’s what makes the Julia Set a fractal which is a shape that repeats its structure at every scale.

And to think that all of this comes from a single equation of math!