On Circles and Waves

Joseph Fourier’s central claim, published in 1822^[1], was that any periodic signal — no matter how jagged or irregular — can be expressed as a sum of simple sine waves. Fourier made this claim in the context of heat diffusion. His contemporaries, including Lagrange, were deeply skeptical. It took decades for the result to be fully accepted and properly formalised.

This seems implausible at first. A square wave has sharp corners. A sine wave is smooth. How can you sum smooth things and get something with corners?

The answer is that you can get arbitrarily close with enough terms. Each sine wave you add fills in the corners a little more. The square wave is the limit of an infinite sum — something you approach but never quite reach with finitely many terms. The overshoot you see near the corners when using a finite number of terms is called the Gibbs phenomenon. It was first noticed by Henry Wilbraham in 1848, and then independently rediscovered by J. Willard Gibbs in 1899. The overshoot is always about 9% of the jump height, no matter how many terms you add.

The widget below makes this concrete. Each rotating arm corresponds to one term in the series. The first arm rotates once per cycle and accounts for the fundamental frequency. Each subsequent arm rotates at an odd multiple of that speed and is proportionally shorter.

The formula for the square wave approximation is worth looking at directly:

Notice that only odd harmonics appear. This follows from the symmetry of the square wave. Because it is an odd function — symmetric under both time-reversal and sign-flip — its Fourier series contains only sine terms. And because it has half-wave symmetry, only odd harmonics survive. Even harmonics cancel out. The coefficient on each term shrinks as $\frac{1}{n}$, which is why higher harmonics matter less and less.

What Fourier gave us was not just a mathematical trick but a new way of seeing signals. Instead of thinking about a waveform as a shape in time, you can think of it as a recipe: a list of frequencies and their amplitudes. This is exactly what your phone’s microphone does when it listens to music, The Fast Fourier Transform algorithm, developed by Cooley and Tukey in 1965, makes this computation practical. It runs in $O(n \log n)$ rather than $O(n^2)$, which matters enormously when n is large. and what an MRI scanner does when it reconstructs an image from radio signals.