...because it confuses presenting the basis [groan] of the Fourier transform via already-understood concepts with showing a gee-whiz incidental property of the

...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.