This is great as it doesn't just explain the basics, it also shows how they are used in MP3 and JPEG files.
I always had a very vague "throws away data that's not perceived" idea of how MP3 works. With the background from this I was able to really understand the wikipedia page and found this great Ars article from 2007 to cement my understanding.
At the time of writing the Gibbs phenomenon -- the overshoot occurring at discontinuities of a square wave, which does not go away no matter how many harmonics are included (see https://en.wikipedia.org/wiki/Gibbs_phenomenon) -- is not correctly represented in these visualisations, implying that important aspects of the underlying maths are obscured. In my opinion this limits the didactic value of an otherwise impressive presentation.
Yes, apologies I made some approximates in the article!
Interestingly if you're talking about discrete time fourier transforms (which is what we usually deal with in computers with jpegs, mp3s, etc), then you can perfectly represent a signal with say, 1024 samples using exactly 1024 sine waves without worrying about that effect. It's only in the continuous time variant that you have to worry about things like the Gibbs phenomenon (which you do run into once you start translating to real world output)
Right! I did assume a continuous space because that's what it looked like to me. Hence my pedantic comment which was sort of comically at odds with the light-hearted spirit of your article :) Very nice work. Thanks for sharing, and for the clarifying remarks!