The Fast Fourier Transform is one of the most powerful tools in modern science, powering audio mixing, medical imaging, radio, image compression and quantum measurement. It takes a complicated signal made of many waves and pulls each wave back out.
One wave at one frequency is smooth and predictable. Add two waves and the shape grows more complex. Add three and the parts become hard to see. Move the sliders to watch the components disappear into the sum.
The Fourier idea begins with a strange move. Take the signal and wrap it around a centre point at a chosen rate, rather than drawing it on a flat line. The result looks like a flower. Raise the winding rate and watch it change.
Imagine the flower as a wire of even weight, and ask where it would balance. At most rates the wire is evenly spread and the balance point sits at the centre. At certain special rates it tilts and the balance point jumps outward. Those rates are the frequencies hiding inside.
Run the winding rate from slow to fast, recording how far the balance point jumps at each step. Mostly the answer is almost nothing. At a few rates it leaps, and the leaps appear as spikes. Each spike is one of the original waves, recovered.
That was it. Wrap a signal in a circle, find the balance point, sweep the winding rate, read off the spikes. What goes in is a tangled wave. What comes out is the list of frequencies it was made of.
A simple geometric trick, running quietly inside almost everything modern.