In linear algebra, lines are sets of points, but in modern geometry, points and lines remain undefined terms, implicitly defined by their incidence relations (which points are "on" which lines).

You may find the Fano plane (a three-dimensional finite projective space) interesting:

  * http://en.wikipedia.org/wiki/Fano_plane (brief description)
  * http://math.ucr.edu/home/baez/octonions/node4.html (connections with higher math)