§1One circle is a promise
Start with the only moving part mathematics really owns: a point going around a circle at constant speed. Track its height while time slides right, and you get the sine wave — the sound of a tuning fork, the sway of a pendulum, the gentlest possible repetition.
Fig. 1 — a rotating point, unrolled into time. This diagram drew itself when you arrived at it; every figure on this page does.
A sine wave is honest but boring. It can hum; it cannot knock. To build anything with corners — a drumbeat, a clock tick, a voltage that switches — one circle is not enough.
§2Circles riding circles
Here is the whole trick: put a smaller, faster circle on the rim of the first, and add their heights. Then a smaller, faster one on that. Each circle is called a harmonic; the recipe of sizes and speeds decides what the sum traces.
f(t) = sin(t) + ⅓sin(3t) + ⅕sin(5t) + …
Fig. 2 — partial sums approaching a square wave: one, two, then four harmonics, drawn in sequence.
§3The recipe is the object
Squint at the equation and a deeper idea surfaces. The square wave is its list of coefficients — 1, ⅓, ⅕, ⅐ … — the same way a chord is its notes. Fourier's claim is that this translation always exists: shape on one side, recipe on the other, nothing lost in either direction.
Fig. 3 — the square wave's spectrum: each bar is one circle's radius, falling as 1/n. The bars rise in reading order.
This is why your phone can strip noise from a call and JPEG can throw away what your eye won't miss: in recipe-space, "unwanted" is often just a bar you can shorten.
§4The corner that rings forever
One imperfection survives. Watch the sum near a jump: it overshoots — a little horn of about 9% that never shrinks, no matter how many circles you stack. It only gets thinner. This is the Gibbs phenomenon, and it is not a bug in the math; it is the price of asking smooth circles to imitate a cliff.
Fig. 4 — zoomed on the jump: 25 harmonics, and the 9% horn (marked) still standing.
Engineers hear Gibbs as "ringing" at sharp edges in audio and see it as ghost fringes beside hard lines in over-compressed images. When you know its name, you start seeing it everywhere — which is the correct outcome of a lesson.
What you can now say at dinner
- Every repeating shape is a chord of circles — the recipe of radii is the shape.
- Corners demand infinitely many circles, which is why sharp things are expensive in signals.
- The 9% horn at a jump never dies — smoothness can imitate a cliff, but not commit to one.
- The progress bar you've been filling is itself a square wave that gained harmonics as you read. It is now as square as this page could make it.