So what is this thing?

A wave!

The wave can be split up into sine waves.

The Fourier transform is a way to take the combined wave, and get each of the sine waves back out.

Sound wave! When we hear a sound, we hear the different frequencies of the sine waves that make up the sound.

Play Full Wave

Play High Frequency

Play Low Frequency

A square wave!

It can be split up into sine waves.

We need a lot of them this time.

Play Wave

Drag the slider above to play with how many sine waves there are.

Visually, the first few sine waves make the biggest impact, but the smaller waves smooth it out.

When you listen to the wave, you'll hear the sound get lower, because we're removing the higher frequencies.

This works for any repeating line—try drawing your own!

Play Wave

Move the slider to see how as we add more sine waves, it gets closer and closer to your drawing

By using a Fourier transform, we can get the important parts of a sound, and only store those to end up with something that's pretty close to the original sound.

On a computer we store a wave as a series of points.

We can represent it as sine waves and compress by dropping smaller frequencies. The result isn’t identical, but it sounds similar to us.

This is essentially what MP3s do!

We use Fourier transforms to understand the fundamental properties of a wave, and then we can use that for compression.

Ok, now let's dig more into the Fourier transform.

Epicycles

At the start, everything is splitted into sine waves. Interestingly, the sine waves are not just regular sine waves, but they’re '3D'. It is "complex sinusoids" or "spirals".

If we take a look from the side, they look like sine waves. From front on, though, these look like circles.

So far, we’ve only required the regular '2D' sine waves. When we do a Fourier transform on '2D' waves, the complex parts cancel out so we just end up with sine waves.

But we can use the '3D' sine waves to make something fun looking like this:

What’s going on here?

Well, we can think of the drawing as a '3D' shape because of the way it moves around in time. The x and y dimensions tell us the position, and then the time dimension is the time at that moment.

Now that we have a '3D' pattern, we can't use the regular 2D sine waves to represent it. So we need something else.

What we can use is the 3D spiral sine waves from before.

Remember, these waves look like circles when we look at them from front on. The name for the pattern of a circle moving around another circle is an epicycle.

Use the slider above to control how many circles there are.

Like before, we get a pretty good approximation of our pattern with just a few circles.

Use the slider to control how many circles are used for your drawing

We can approximate them fairly well with just a small number of circles, instead of saving all the points.