In a configuration, each point is incident to the same number of lines and each line is incident to the same number of points.
The Fano plane is a configuration, with 3 points on each line, and 3 lines on each point. It's possible to make the smallest 3D projective space with 15 points, 15 planes, and 35 lines.
Features:
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?