A dense planar point set from iterated line intersections

Given S1, a starting set of points in the plane, not all on a line, we define a sequence of planar point sets {Si}∞i=1 as follows. With Si already determined, let Li be the set of all the lines determined by pairs of points from Si , and let Si+1 be the set of all the intersection points of lines in Li . We show that with the exception of some very particular starting configurations, the limiting point set ⋃∞ i=1 Si is everywhere dense in the plane.  2003 Elsevier B.V. All rights reserved.