Here's a hypnotic version with one sum:

https://youtu.be/-qgreAUpPwM

Perhaps so the animations are outside of the drawing area?

And I would guess you get fewer artifacts - the version linked in a sibling comment with just one end has a lot of factors and yet is not very accurate.

Yeah, this is pretty annoying. Starting with a function from the circle to the complex plane, the Fourier series decomposes it into a sum of exponentials, which are circular motions about 0. This version with x and y coordinates separated overcomplicates and confuses. And with a four-dimensional output, there are 2 unused dimensions whose values are arbitrary.

I wonder if this means the problem is under-determined (too many degrees of freedom) and that leads to numerical instability. Certainly one set of equations is enough, as 3Blue1Brown’s Fourier transform video shows.

I implemented such a version in Scratch back in 2016! https://scratch.mit.edu/projects/117304786/

Here’s a gif of that if the project player doesn’t work for you: https://github.com/kach/gifblocks/blob/master/gallery/fourie...