This Math Formula Can Build ANYTHING!

Added: 2026-05-24

[00:00:00]Ever wondered how complex signals are made? Welcome to the world of Fourier series. This massive formula can replicate almost any periodic function you can imagine. But if our wave is symmetric to the origin, an odd function, things get beautifully simple. The constants and cosines drop to zero. All we need are signs. Let's try to build a perfect square wave using nothing but smooth curves. We start with the first harmonic, a basic fundamental sine wave.

[00:00:28]Not quite a square yet, right? Now we add up to the fifth harmonic, combining just three non-zero terms.

[00:00:34]Look at those corners forming. Let's push it to the 19th harmonic. 10 terms combined and it tightly hugs the square shape. Finally, the 99th harmonic with 50 terms.

[00:00:45]Notice those sharp ripples at the edges?

[00:00:47]That's the famous Gibbs phenomenon. Want to see the full visual journey? Check out the link below.