§ 04 · Sandbox · Signals & geometry Filed 2026.05

Draw any shape, see the circles that draw it for you.

The Fourier transform is usually introduced as an integral and met with apprehension. Here it is a chain of rotating arrows. Sketch a closed shape with your finger or mouse, and watch a tower of nested circles retrace it — perfectly when you allow enough of them, cartoonishly when you don’t. Or hum into the microphone and see your voice as the same kind of stack, only made of audible frequencies instead of geometric ones.

Mode Draw Samples 0 Harmonics 0 / 0

Phase 0.00 τ State ▶ idle

Drag to draw a closed shape · release to decompose

Harmonics · N

Speed

1.0×

Show

Presets

Reading the picture

§ 04 · Notes

Every closed curve, no matter how knotty, can be written as a sum of circles spinning at integer rates. Draw a square; the diagonals tell you which rates matter.

A circle’s radius is how much of that frequency the shape contains. Its starting angle is where in time that ingredient peaks. Magnitude and phase, made visible.

Slide harmonics from 1 upward. The first ring gives the rough wobble; later rings carve the corners. A square needs roughly 30 to look square; a star needs more.

Audio mode runs the same trick, only the input is air pressure over time instead of (x, y) over arc length. Whistling pure tones produces single tall bars. A vowel blooms.

Show the math

Sample a closed curve at N equally-spaced points along its arc length. Treat each sample as a complex number z_k = x_k + i·y_k. The discrete Fourier transform decomposes that sequence into coefficients that label rotating arrows.

c_n = (1/N) Σ_k=0^N−1 z_k · e^{−2πi·nk/N} analysis

Each coefficient c_n describes one arrow. Its length |c_n| is the radius of that arrow’s circle; its angle arg(c_n) is where the arrow points at time zero; the integer n is how many full revolutions per period it makes. Positive n turn one way; negative n turn the other.

z(t) = Σ_n c_n · e^2πi·nt, t ∈ [0, 1) synthesis

Sort the coefficients by magnitude and keep only the top K. Stack them tip-to-tail and let t advance. The final tip retraces the original curve — exactly when K = N, and progressively more cartoonishly as K shrinks.

For the audio mode, the sampled signal is real (just amplitude over time) and the visualisation shows |c_n| against frequency, which is exactly what your eardrum already does: the cochlea is a biological spectrum analyser.

Fourier’s 1822 theorem was so unbelievable to his contemporaries that the prize committee refused to print it for fifteen years.