An artist's question about the Fourier series

So, the Fourier theorem states that any curve you could want, can be approximated by combining lots of sine waves (the same waves that you get when you swing a weight on a string)

The video is not combining waves together to generate arbitrary functions, so it’s not really a Fourier series. But since the technique is so ubiquitous it’s mentioned in all sorts of contexts when you see waves

As for the math in the video below, check out “harmonic oscillators” as a starting point
Fourier analysis is about combining sinusoidal waves to get more complex shapes

Each string is a slightly different length and there fore the balls swing at different rates and go in and out of phase with each other

It’s definitely related to complex analysis, albeit mostly only tangentially
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That video title is BS. Pendulums of different periods simply swinging in no way represents the important part of Fourier analysis.  The only connection you could say is, those pendulums have different frequencies of oscillation and Fourier analysis involves representing a function as a linear combination of complex exponentials which correspond to specific frequencies. But this is just saying they both involve frequencies. But it's in different contexts.
This is just ratios. Say you just have two of these, one that does 65 swings a minute, the other 70 swings a minute. Here a swing is just HALF of a period (one extreme to the other). When will they create some kind of synced pattern?

They'll make a noticeable pattern when both are at some outstanding position at the same time, probably when both are at an end. This happens when both have made an integer number of swings. After T seconds, the first will have T\*65/60 swings and the second will have T\*70/60 swings. Or, simplified, 13T/12 swings and 7T/6. When will both of these be integers? Whenever T is a multiple of 12, so every 12 seconds, they'll both line up and it'll look cool.

If the numbers were different, say 65 swings and 67 swings, then the times would be different. If you had more, then there are more combinations that can line up creating different patterns and, eventually, they would all line up and reset. This is what is in the clip.

Fourier Series are a more complicated combining of things like this, rather than just a comparison like this.
Think it this way: You can make any image with the sum of semi-infinite simple polygons (usually triangles or pyramids), right? That's what the computer does, that's why there's anti-aliasing and stuff, to make it look like a single image instead of the sum of many pixels.

Fourier series is almost the same stuff, but for waves: There's the most fundamental wave of them all, the sine, and any wave can be made by the sum of infinite (or semi-infinite) different sine waves. If you're into music, Fouriers Principle is the magic of synths!

Obs: Some mathematicians will get furious with my explanation and they will probably give me an intellectual whipping for it (fair enough), that's because I'm an engineer and my approach is more... practical, and less rigorous with the theory lol

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