The Fast Fourier Transform is one of the most powerful tools in modern science, powering audio mixing, medical imaging, image compression and quantum measurement. It takes a complex multi-wave signal and isolates each wave.
One wave at one frequency is smooth and predictable. Add two waves and the shape grows more complex. Add three and it becomes hard to see. Move the sliders to watch how the wave evolves in complexity.
The Fast Fourier Transform works by wrapping the complex signal around the centre point of a circle at a given rate, and creates a pattern like a flower. Changing the winding rate makes the pattern change.
Imagine the pattern as being made of metal wire and see 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 moves outward. This shows us where the individual wave frequencies are hiding.
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 everything.