Chapter 4Circles Drawing Waves
Chapter 1 showed one spinning circle drawing one sine wave. A Fourier series is just several circles spinning at once, each one glued to the tip of the last. Watch the chain draw the exact same square/sawtooth/triangle wave from Chapter 3.
Click and drag the circles to spin the whole chain faster or slower.
Circle chain (each radius = one harmonic's amplitude)
Traced wave (height of the final tip)
What's actually being summed right now:
What's happening
Each circle in the chain corresponds to exactly one term from the Fourier series formulas in Chapter 3:
- Its radius is that harmonic's amplitude — the big first circle is the fundamental frequency, and every circle after it is smaller, matching how quickly that shape's harmonics fade.
- Its spin speed is that harmonic's frequency — the 3rd harmonic spins 3× as fast as the fundamental, the 5th spins 5× as fast, and so on.
- Circles are glued tip-to-center: the 2nd circle's center rides on the 1st circle's tip, the 3rd rides on the 2nd's tip, etc. The final tip's height at every instant is the sum of all those individual heights — exactly the addition from Chapter 2.
With only 1 circle you get a plain sine wave. Drag the harmonics slider up and watch the traced wave sharpen into the target shape, corner by corner — this is the same convergence you saw in Chapter 3, just drawn mechanically instead of algebraically.