ΣAk·sin(k t+φk)
A_k = 4 / (π k), odd k only
Stack rotating circles until a smooth sine grows corners and becomes a square wave, then read amplitude, phase, error, and captured energy at once across a three-axis space of time, height, and frequency.
A single circle turns quietly in front of you. Take only the height of one point on it and let that height stream to the right — and a smooth wave appears. A wave is not a shape drawn by hand; it is the shadow of a rotation. One turn of the circle is one period of the wave, and the radius of the circle is the height of the wave. Time runs right, height runs up and down. These two axes carry everything that follows.
Follow one full turn of the circle and watch exactly one period of the wave get drawn.
Seen from the side a rotation is a sine; seen head-on it is a circle. Two faces of the same motion.
