In 1822 Joseph Fourier proved something that initially seemed absurd:
any periodic signal — any shape, any sound, any waveform —
can be perfectly reconstructed as a sum of pure sine waves.
The more sine waves you add, the closer the approximation.
With infinite terms, it becomes exact.
The epicycle view (left panel) shows what this looks like geometrically.
Each frequency component is a circle rotating at a different speed.
The tip of the outermost circle traces the waveform. This is not a metaphor —
it is the literal geometric interpretation of complex exponentials.
Ptolemy used epicycles to model planetary motion in 150 AD.
Fourier showed the same circles describe everything from heat flow to music.
The wave panel (top right) shows the reconstructed signal f(t).
The equation f(t) = Σ aₙcos(nt) + bₙsin(nt) accumulates
term by term. A pure sine is one circle. A square wave needs infinitely many —
odd harmonics only, amplitudes 1/n. Notice the Gibbs phenomenon: the slight overshoot
near sharp edges that never fully disappears no matter how many terms you add.
The spectrum (bottom right) is the Fourier transform itself — the
signal viewed from the frequency domain rather than the time domain. Each vertical bar
is one frequency component. Its height is amplitude; its color is its phase.
This is how your phone recognizes speech, how MRI machines reconstruct images,
how JPEG compression works, how telescopes filter noise from starlight.
Sound and light can both be understood through frequencies.
Your ear performs a real-time Fourier transform — the cochlea physically separates
frequencies along its length, each hair cell tuned to one. Your visual cortex does
something similar with spatial frequencies in images.
Perception is Fourier analysis happening in wet tissue.
adjust harmonic amplitudes with the sliders · draw a custom shape · watch the spectrum update live