Given integer n > 0, draw a line segment in C connecting each pair of nth roots of unity. Then as n -> oo, does the set of intersection points in C (assume each is given equal weight and the weights sum to 1) approach a continuous density on the unit disk? (Note: we care only about the intersection points, not the rest of the line segments.) Since for n even and > 4, the line segments must have at least one triple intersection (and I think usually more), it may be simpler to just consider n odd (in which case I think no triple intersections can occur, at least not in the *interior* of the unit circle). ------------------------------------------------------------------------ (((It's kind of clunky how you'd define a limiting distribution (if any), but it's done all the time in probability theory: Suppose for k = 1,2,3,... you're given a set of points S_k in R^2 -- and a corresponding discrete distribution obtained by giving each point q in S_k the probability Pr(q). Then there's a cumulative distribution F_k(a,b), defined by F_k(a,b) = sum of Pr(q) for all q = (qx, qy) in S_k such that qx <=a and qy <= b. Then the discrete distributions on the S_k's approach a "limit" as k -> oo if for almost all (a,b) in R^2, the limit as k -> oo of F_k(a,b) exists. In this case call the limiting cumulative distribution F(a,b). Then the limiting density will be given by d(q) = d^2 (F(a,b)) / (da db) assuming it exists and equals d^2 (F(a,b)) / (db da) almost evereywhere.))) There ought to be a cleaner way to define the limiting density, but I don't know what it is. So: Do these string figures' intersection points approach a limiting density, at least through n odd? --Dan