[math-fun] Planar density question
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
On Thursday 02 February 2006 09:10, dasimov@earthlink.net wrote:
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.)
0. Instead of using rectangles (-oo,a) x (-oo,b) you can use various other families of sets. I'm not sure what the exact criterion is, but I bet you can work with, say, "polar sectors" of the form S(a,b1,b2) := { r exp(it) : 0 <= r < a, b1 < t < b2 }. 1. Consider any two fixed S(-) having the same values of a and b2-b1. For large n, everything is fixed under a set of rotations about 0 that comes arbitrarily close to taking one S(-) onto the other. Therefore (mumble, wave hands) the limiting distribution or non-distribution has circular symmetry, and it suffices to look at the S(a) := S(a,0,2pi). 2. So, when will an intersection occur within S(a)? Working that out in detail is a bit fiddly, but we don't need to do it. Line segments exp(it1)..exp(it2) and exp(it3)..exp(it4) intersect iff ... *some* condition on the signs and sizes of the various tj-tk holds, which means that there's some region of (t1 t2 t3 t4)-space where it holds, and then the number of such intersections for large n is just (n/2pi)^4 times the measure of that region plus an error term of smaller order; and the fraction of all intersections for which it holds is just the measure of that region plus some error term that -> 0 as n -> oo. So, modulo a whole lot of handwaving: yes, there is a limiting distribution, it's rotation-invariant, and (though you'd need to fill in more details than I have done to be sure) it seems pretty likely that it's well enough behaved to have a pdf that's at worst a continuous function plus some delta functions. Or have I waved my hands *too* much? :-) -- g
dasimov@earthlink.net wrote:
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.)
There's a paper on this by Bjorn Poonen and Mike Rubinstein: http://math.berkeley.edu/~poonen/papers/ngon.pdf Gary McGuire
On Thu, 2 Feb 2006 dasimov@earthlink.net wrote:
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.)
Consider the set C(n) of points in the plane obtained by intersecting all line segments joining the n-th roots of unity. Then plot as a single picture the set D(n) = union of C(k) for k from 3 to n. Why should the face of a monkey appear in D(25)? If you can read postscript (or eps) files you can see the Maple generate picture of this set here: http://www.math.usf.edu/~eclark/D25.eps --Edwin
participants (4)
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dasimov＠earthlink.net -
Edwin Clark -
Gareth McCaughan -
Gary McGuire