Indexing the archive…
Your Universe of Digital Possibilities
Stack one spinning circle per harmonic, tip to tip, and the pen they trace draws a wave. Add more circles and any shape emerges — a square, a ramp, a peak — from nothing but sines. Flip to the spectrum to read the recipe back out as bars, and click one to mute that harmonic and watch its job vanish. At every sharp edge a ripple refuses to die: that is Gibbs — pile on harmonics and the overshoot only narrows, closing on ~9% of the jump, never flattening away.
Fourier claimed something outrageous in 1822: any periodic signal — even one with corners and jumps — is a sum of smooth sine and cosine waves at integer multiples of one fundamental frequency. The harmonics slider adds these terms one by one.
Euler’s formula turns each harmonic into a rotating vector of radius |cₙ| spinning at speed n. Stacked tip-to-tip they are the epicycle chain on the left; the pen’s height is the signal.
Run it backwards: the discrete Fourier transform reads the amplitude of each harmonic out of the samples — the spectrum bars. The same operation lives inside MP3, JPEG and every spectrum analyser.
At a jump, no finite sum of smooth waves ever closes it — a ripple overshoots the edge and, as you add harmonics, only narrows, never flattening. Its height homes in on a fixed ≈ 8.95% of the jump: the square hits it almost at once, the sawtooth climbs toward it from below.
The old Signal Lab claimed five worlds — a market, a sound, a heartbeat, a climate, a city — share one grammar. This is the grammar: the theorem that any signal in time is a sum of pure circular motions, a chord of frequencies you can read off like notes. It is the same decomposition The String plays as a particle spectrum, and the very thing The Strobe destroys when it samples a wave too slowly to tell its harmonics apart. Sound, light, tides, the wobble of a planet’s orbit — all of it is built from circles going round. This instrument shows the circles.