# Fourier Series

Canonical: https://manyspace.aporia.systems/tools/fourier-series
Updated: 2026-10-13

Every repeating wave is a stack of pure sines. Add harmonics one at a time and watch a square wave assemble itself — bumps, ringing, and all.

## Try this

- Take the square wave from 1 harmonic to 25 and watch the corners sharpen.
- Toggle parts on and see which sine does the heavy lifting.
- Drag the probe into a jump and compare target vs sum.

## How Fourier series work here

Any periodic wave can be written as a sum of sines and cosines — one per harmonic. This lab uses the exact textbook formulas for three classic waves, so each harmonic you add is the mathematically correct next term.

Near a jump, the sum always overshoots by about 9% no matter how many harmonics you add. That is the Gibbs phenomenon — a theorem about partial sums, not a rendering bug. The marker shows it; the lab never hides it.

## Related tools

- [Derivative Visualizer](https://manyspace.aporia.systems/tools/derivative)
- [Integral Explorer](https://manyspace.aporia.systems/tools/integral)
- [Unit Circle Explorer](https://manyspace.aporia.systems/tools/unit-circle)
- [Complex Plane Explorer](https://manyspace.aporia.systems/tools/complex-plane)
- [Probability Lab](https://manyspace.aporia.systems/tools/probability)
- [Bayes Lab](https://manyspace.aporia.systems/tools/bayes)
- [Eigen Explorer](https://manyspace.aporia.systems/tools/eigenvectors)