This is the traditional picture of the smallest projective plane, the socalled Fano plane. This plane has 7 points and 7 lines with 3 points on every line and 3 lines through every point. Two points are on a unique line and two lines intersect in a unique point. 
Sometimes when you glance through books on our kind of geometry you get the impression that this is the only picture we can draw. Far from it! As far as I am concerned it is not even the nicest picture of the Fano plane. The following spatial model, for example, exhibits much more symmetries than the traditional one. Cross your eyes to see it in 3d. 

This is a picture of the incidence graph of the Fano
plane. The red and yellow vertices of this graph correspond to the points
and lines of the Fano plane. Two vertices of the same color are never adjacent.
Two vertices of different colors are adjacent if they correspond to a
point and a line such that the point is contained in the line.
The graph has 7 reflection axes and the reflections through these axes correspond to polarities of the Fano plane. 