I have an idea of a method for recursively dividing a plane, and as I'd like to do more research about this algorithm and the set of points that it produces, I'd like to know what it's formally known as.
In a plane $X$, consider the set of four points $P$ = {$P_1..P_4$} that form an irregular quadrilateral.
Consider the set $L$ of lines that can be drawn through these points, {$\overleftrightarrow{P_1P_2}$, $\overleftrightarrow{P_3P_4}$, $\overleftrightarrow{P_1P_4}$, $\overleftrightarrow{P_2P_3}$, $\overleftrightarrow{P_1P_3}$, $\overleftrightarrow{P_2P_4}$}.
Consider any points defined by intersection of the lines in $L$, such that these points that do not already exist in $P$. Because any two of the quadrilateral's sides are nonparallel, the lines passing through the sides will intersect at the two "new" points $P_5$ and $P_6$. The diagonals of the quadrilateral will also intersect at a new point $P_7$. Add these points to $P$.
Consider the line $\overleftrightarrow{P_5P_6}$, as well as any lines that can be drawn through ($P_5$, $P_6$, or $P_7$) and any of the other points in $P$, such that these new lines are not collinear with the existing lines in $L$. Add these lines to $L$.
Repeat steps 3 and 4 until no more new lines can be added to $L$, that is, no new intersections are formed by constructing lines from the existing intersections. (This may be infinite, but I haven't proven that yet.)
What is this method of space division called?