Some years ago we discussed the optimal not-too-symmetrical egg shape. I sometimes enjoy a hobby of trying to represent a positive integer by an arrangement of dots in the plane (or sometimes 3-space), as symmetrical as possible. This can be fun, but there are still plenty of degrees of freedom. Say N = 4. Should it be the 4 corners of a square in R^2, or maybe the corners of a regular tetrahedron in R^3 ??? (We're not concerned here with how the dot pattern is depicted, just with the conceptual arrangement. Or technically, generalized to any given integer N > 0: Question: ----- What metric should we put on N points to make it best represent N-ness??? ----- NOTE: This is *more general* than the original question. I'm most interested in this more general question. So, a challenge: Come up with the most symmetrical N-dot arrangements, in your opinion, for N equal to each of the first 100 positive integers. No, ignore that. Here's the *real question*: The *real question*: -------------------- If you were the judge of such a contest, what is the optimal mathematical criterion to use to judge it as fairly as possible? ----- Note: ----- * If morally speaking, ties for first place are inevitable (as they may be for at least some N), then the optimal judging criterion should recognize such ties. * Even better than merely picking best or tied for best would be a mathematically defined: *partial ordering on all entries*. ----- —Dan