...because it confuses presenting the basis [groan] of the Fourier transform via already-understood concepts with showing a gee-whiz incidental property of the Fourier transform.
The thing that bothers me about the epicycle example in particular is that it's not clear (from the example) what the domain of the function we're trying to approximate is, and what, if any, restrictions need to be placed on the function. For example, is it OK for the function cross itself? Can I represent an arbitrary shape in 2D with this method? What happens if the line has to jump? Is that OK or not? Is it accidental, or on purpose, that the line comes back to meet itself at the end? Why is the function we're representing (in the epicycle example) in 2D but we usually take the Fourier transform of 1D functions? Where did complex numbers come in? What is complex exponentiation? What's an integral?
The square wave example does better. You can just use pure sines or cosines, so you don't have to introduce complex numbers. You don't have the 2D vs. 1D issue. You don't need an integral. So with less folderol, you get to about the same level of understanding as in the given example.
On the other hand, I really like the epicycle example as a machine learning or sparse basis example. That is, explaining that you can fit anything if you have enough degrees of freedom. After the Fourier transform is already understood by other means.
Because it introduces and relies upon far too many other mathematical concepts. Angular frequencies, complex numbers, integrations and a lots of formulas. It was a '...for dummies' question and you can explain FTs with much simpler means.
The example that worked best for me is with audio sampling. A demonstration of how you can build up a 'wiggly line' to match an audio sample by adding together multiple waves of different frequencies. This then leads into talking about a real world example - mp3 encoding.
In my experience, math.stackexchange likes mathematical rigour and presupposes some mathematical maturity. There are a fair number of professional mathematicians using the site.
Sure, if you just want a "for dummies" understanding, a hand-wavy example like the second highest voted answer suffices. But if you want to learn it properly (prove theorems, etc), you need to know concepts like complex numbers and integration.
If you wanted something more deep than the square wave example, you could explain why the complex exponential is used. This would provide important insight, because a smart reader might say, OK fine, square wave approximation, but why don't we use step functions or gaussian-shaped bumps or anything else as basis functions.
The story I tell myself about that is, "complex exponentials are the (unique) eigenfunctions of linear time-invariant systems".
Simply put, this means that if S(.) is a system that has derivatives and integrals, and scaling and shifting in time, then
S(exp(i omega t)) = const * exp(i omega t)
I.e., you put in a complex exponential, you get out a (scaled) complex exponential. That's not going to happen, in general, for other basis functions.
This property is (under some not too restrictive conditions) basically a defining property of the complex exponential. (Basically, notice that d/dt [exp (a t)] = a exp(a t) -- which turns out to have a converse as well.)
Then, because S is linear, a linear combo of complex exponentials (on the input of S) transforms into a weighted combo of the same exponentials on the output.
To me, that's the deeper reason that Fourier analysis is important. Those sines and cosines are mathematically inevitable, right from the moment in high school when you learned d/dx e^x = e^x.
Noted on that SE page (the OP) is a related, but subsidiary property, that the Fourier bases are also the eigenfunctions of the wave equation. But the wave equation is linear, so this follows from the above reasoning.
I imagine, but I don't know, that the usefulness of the Fourier basis for quantum-mechanical analysis also follows from linearity.
I agree; the top answer sounded simple, but it wasn't simple at all to understand. I think there's more to the FT than you get from a working knowledge of how it's applied to audio, though. In other words, there's understanding the FT, and then there's understanding the FT. I admit that I'm not in the second group.
The top-rated answer is actually very interesting (I had never considered Fourier space in terms of circles before) but it's less visually intuitive to me, and less obviously related to the most common applications of FT, than the way I was taught (and the way I presume it is most commonly taught), which can be seen at the top of this Wikipedia page: http://en.wikipedia.org/wiki/Fourier_series
Understanding that the square wave (and any other wave) can be represented as a superposition of sine waves at various frequencies seems like a better place to start because it illustrates the most common use case of FTs (alternating between time and frequency domain for a signal).
