  Consider a plane graph G, drawn with straight lines. For every pair a,b of
vertices of G, we compare the shortest-path distance between a and b in G (with
Euclidean edge lengths) to their actual distance in the plane. The worst-case
ratio of these two values, for all pairs of points, is called the dilation of
G. All finite plane graphs of dilation 1 have been classified. They are closely
related to the following iterative procedure. For a given point set P in R^2,
we connect every pair of points in P by a line segment and then add to P all
those points where two such line segments cross. Repeating this process
infinitely often, yields a limit point set P*.
  The main result of this paper is the following gap theorem: For any finite
point set P in the plane for which the above P* is infinite, there exists a
threshold t > 1 such that P is not contained in the vertex set of any finite
plane graph of dilation at most t. We construct a concrete point set Q such
that any planar graph that contains this set amongst its vertices must have a
dilation larger than 1.0000047.
