Imagine you have a complex drawing—a butterfly, a heart, or even your own signature. It looks like a messy, complicated shape. But what if I told you that this drawing is actually just a combination of simple, spinning circles?

This is the magic of the Fourier Transform. Discovered by a French mathematician named Joseph Fourier over 200 years ago, this is a mathematical trick that lets us break down any complicated shape or wave into a bunch of very simple, perfectly smooth waves. When we talk about drawing shapes, these simple waves look like spinning circles, which we call epicycles.

Adding Up Circles

Think about how you might draw a picture. You start with the big outlines, then you add medium details, and finally, you draw the tiny, fine lines. The Fourier Transform works in a very similar way!

Instead of a pencil, imagine you have a giant clock hand spinning around in a circle. Now, attach a slightly smaller clock hand to the tip of the first one, spinning at a different speed. Then attach an even smaller one to that, and so on. If you attach a pen to the very last clock hand and let them all spin, it will draw a shape.

Fourier's mind-blowing discovery was that any closed shape you can possibly draw can be perfectly recreated by chaining together enough spinning circles.

The Math Behind the Magic

To a computer, a drawing is just a path traveling through space over time. We can use complex math (specifically, something called the complex plane) to figure out exactly what circles we need. The formula to find these circles looks like this:

$$ z(t) = \sum \, _{k \in \mathbb{Z}} ~ c_{k}\, e^{ i\, k\, t} $$

Don't let the symbols scare you! Here is what it means in plain English:

• The big sideways "M" symbol ($\sum$) just means "add all these things together".
• The $c_k$ part tells us how big a specific circle should be, and what angle it should start at.
• The $e^{ikt}$ part is the mathematical engine that makes the circle spin at a specific speed (the frequency, $k$).

Details vs. The Big Picture

To perfectly recreate a drawing with razor-sharp corners, the math says we need an infinite number of circles. But in reality, we don't need that many.

The biggest circles in the chain sketch out the general, blocky shape of your drawing. The tiny, super-fast spinning circles at the end of the chain are responsible for drawing the sharp corners and fine details. If we just throw away the smallest circles, we still get a shape that looks almost exactly like the original, just a little smoother and wobblier.

Why Should I Care?

The Fourier Transform isn't just for making cool animations of shapes. It is arguably one of the most important mathematical tools ever discovered!

Because it lets us break down complicated signals into simple waves, it is the math that allows your phone to compress photos into small JPEG files, and audio into MP3s. It's used by noise-canceling headphones to figure out what sound waves to block, by scientists analyzing earthquakes, and even by astronomers looking for new planets!

You can see this math in action by drawing your own shapes and watching the spinning circles recreate them in our interactive Fourier Drawing Machine experiment.