Add enough sine waves and you can draw a closed heart. Each harmonic is a gear turning at a steady rate, its radius the amplitude and its speed an integer multiple of the fundamental frequency; chain the gears together and the pen on the last link traces out the target waveform. A square wave needs only the odd harmonics with amplitude 4/(πk), while a triangle wave has coefficients falling as 1/k², so a handful of gears already fits. At a jump discontinuity, though, an overshoot of about 9% refuses to fade no matter how many terms are added: the Gibbs phenomenon.
Add enough sine waves and you can draw a closed heart. Each harmonic is a gear turning at a steady rate, its radius the amplitude and its speed an integer multiple of the fundamental frequency; chain the gears together and the pen on the last link traces out the target waveform. A square wave needs only the odd harmonics with amplitude 4/(πk), while a triangle wave has coefficients falling as 1/k², so a handful of gears already fits. At a jump discontinuity, though, an overshoot of about 9% refuses to fade no matter how many terms are added: the Gibbs phenomenon.
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