This picture https://dl.dropboxusercontent.com/u/3507527/Chebyshev175nrpi.png shows the Chebyshev 175-point set -- whose (unweighted) first 175 moments agree with those of the (uniform distribution on the) real interval [-1,1]. Caveats: Unfortunately the polynomial root solver has numerical difficulties so it is dubious these are very accurate. (If I try for 201 points it totally dies.) Only points with non-positive real part and non-negative imaginary part are shown, i.e. this is one quarter of the full picture. So it appears Chebyshev ultimately is approximating the real interval with an ellipse (or something close to it) plus a few stragglers, perhaps on another curve. Pretty interesting, whatever the hell it is. A natural guess is that the ellipse has foci at +-1. It would be good to produce a more-accurate plot of, say, the 1001-point set. Here is how to tell "maxima" to compute the 51 point set: f(N) := taylor( exp( (N/2)*(log(1-z^2) + (2/z)*atanh(z) - 2) ), z, 0, N); g(N) := subst(1/x, z, trunc(f(N))) * x^N; expand(g(51)); allroots(%); In words: find the Maclaurin series of exp( (N/2)*(log(1-z^2) + (2/z)*atanh(z) - 2) ) in powers of z from z^0 up to and including z^N, this is a polynomial in z. Now replace z by 1/x, then multiply the whole thing by x^N, to get the "reversed" degree-N polynomial, which is monic. Now find its roots. The result is the Chebyshev N-point set. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)