Let $P\_1$ be a set of 4 points in the Euclidean plane. Formally, $P\_1$ determines a set $L\_1$ of 6 lines, which then determine only 3 points not already in $P\_1.$ Let $P\_2$ be the set of 7 points thus far determined. Formally, $P\_2$ determines only 3 lines not already in $L\_1.$ Let $L\_2$ be the set of 9 lines thus far determined. Continuing in this manner determines sets $P\_n$ and $L\_n.$ What can be said about the cardinalities $|P\_n|$ and $|L\_n|$?



Counting points and lines in a plane
The question is discussed in Cooper, Joshua; Walters, Mark; Iterated point-line configurations grow doubly-exponentially, Discrete Comput. Geom. 43 (2010), no. 3, 554–562, MR2587837 (2011f:51016). I quote from the review by Tamas Szonyi: 


Consider a quadrangle as the starting configuration (stage one). In each stage the process considers the lines joining pairs of points that were constructed in the previous stages and adds the (affine) intersections of these lines. The starting quadrangle has to be in such a general position that no lines in any stage are parallel. The main results of the paper give doubly exponential upper and lower bounds on the number of points at each stage. 


A preprint is also available on [arXiv](https://arxiv.org/abs/0807.1549). See also <https://oeis.org/A140468> and <https://oeis.org/A243707>. <https://arxiv.org/abs/1406.5157> is entertaining. 


