# Plane configurations from pairs of lines

The \( (9_5) \)-configuration of points and planes that I've been discussing is an instance of a more general family of \( (ab_{a+b-1}) \)-configurations, formed by the intersection points of two sets of \( a \) and \( b \) lines from the two families of lines on a doubly ruled surface. The two sets of lines in this case are the diagonals of the prism sides; they belong to a common hyperboloid. This also shows that the Levi graph of the configuration is vertex-transitive though I think not edge-transitive.

More generally, if we have a family of points and lines in space, with two lines per point and an equal number of points per line, such that the plane through any two coincident lines doesn't contain any others of the points, we get a configuration from the planes through pairs of lines. It looks like it may be possible to form a \( (24_5) \) in this way, different from the prism one, by taking a torus tiled by a \( 4\times 4 \) grid of quadrilaterals: use as lines the diagonals of an alternating subset of the quadrilaterals, and as points the vertices of the quadrilaterals and the points where the diagonals cross.