The shape an oscilloscope draws when two tones play at a 3:2 ratio. About 20 circles bring it within 1%.
Measuring the shape…
x = sin(3t + π/2), y = sin 2t
Jules Antoine Lissajous studied these in 1857 by bouncing light off two vibrating tuning forks. The ratio of the two frequencies decides the knot: 3:2 gives this one. Each sine wave on its own is a pair of opposite circles, which is why the biggest bars come in matching pairs.
Each path of the lissajous figure is traced by a chain of circles, each one spinning on the tip of the one before. A Fourier transform measures the path and works out every circle's size, speed and starting angle. The circles are sorted biggest first, so the first few give the rough shape and the rest add detail.
"Within 1%" means the drawing is, on average, less than 1% of its width away from the real outline. The bars show the circles of the main outline: height is size, position is how many times it turns per loop and in which direction. Click a bar to remove that circle and see what it was doing. Learn how it works.