A result about the density of iterated line intersections in the plane

Abstract

Let S be a finite set of points in the plane and let T ( S ) be the set of intersection points between pairs of lines passing through any two points in S. We characterize all configurations of points S such that iteration of the above operation produces a dense set. We also discuss partial results on the characterization of those finite point-sets with rational coordinates that generate all of Q 2 through iteration of T .