In a configuration, each point is incident to the same number of lines and each line is incident to the same number of points.
1. Any three points not on a line define a plane with a configuration.
2. On any plane, the points and lines make a configuration.
My question -- are there 3D analogs of projective spaces involving configurations that are not projective spaces? For example is there a 3D set of points featuring lots of Desargues configurations?