Equal ripples

What you see is the error itself: f(x) minus a polynomial, magnified until it shows, with the true size of the error marked on the scale at the left. The coral dots are n + 2 reference points. Each move first finds the polynomial of degree at most n whose error at those points alternates in sign with equal size (the two gold dashed lines are that height, |h|), then moves the points to the peaks of the error. Set the move count back to 0 and step it up, or press play (after 12 it starts again from 0): at first (for eˣ at degree 4, say) the error is low on one side and runs off the picture on the other; for eˣ, after three to five moves the n + 2 peaks and troughs are pressed to one height, just touching the dashed lines, while the curve still crosses 0 between them. Now the largest error over the whole stretch equals |h| (the two numbers in the corner agree to the digits shown), and all n + 2 peaks and troughs reach it: of every polynomial of degree at most n, this one has the smallest maximum error. Change the degree and the number of peaks and troughs follows; switch to |x − 0.3|, with its corner, and the ripples still level out, only much higher, and it takes a few more moves.