Mathematics · Study 046
Fourier Epicycles
Spinning circles stacked tip to tail trace out a square wave. Add more circles with your pointer and watch the corners sharpen.
What you're looking at
Each circle spins at an odd multiple of the base speed and has a radius that shrinks in proportion: the third harmonic is one third the size, the fifth one fifth. The height of the final tip is carried across to the right and plotted over time. One circle gives a sine wave. Twenty give a convincing square wave with small overshoots at the edges.
The idea behind it
Joseph Fourier claimed in the early 1800s that any periodic shape can be built from sines, which sounded absurd for shapes with corners. The overshoot you see never fully disappears. It is known as the Gibbs phenomenon and settles at about nine percent of the jump.
Try this
Move the pointer left to right to set the number of circles from 1 to 24.
