why on earth do such configurations happen to have the capacity to approximate any function you prescribe?? Well they don't. Only certain classes of functions

why on earth do such configurations happen to have the capacity to approximate any function you prescribe?? Well they don't. Only certain classes of functions with bounded varitation will have a convergent Fourier series approximation. I think the best way for these demos to introduce this stuff would be to focus on the one with the steps, point out the Gibbs phenomena around the jumps (ringing). Then show for smooth blobby things you are all good though. I think historically a lot of mathematicians had a lot of problems with Fourier's methods applied to the heat equation for these types of reasons (which initial/boundary conditions are ok, etc) hence we have the whole field of harmonic analysis now...