A model of PG(2,3)
See also the Wikipedia page on projective planes.
A very short introduction to PG(2,3)
The points in PG(2,3) are the sets {A1,B1}, {A2,B2}, ..., {A13,B13}. There are 13 points in PG(2,3).
The lines can also be identified with sets {A1',B1'}, {A2',B2'}, ..., {A13',B13'}, they can be represented with the same 3D points (but the objects "in reality" differ).
The vector Bn always equals to 2An where the coordinates are computed modulo 3.
A point P represented with coordinates (x,y,z) and a line l represented with coordinates (a,b,c) are incident if x·a+y·b+z·c=0 (modulo 3).
These assumptions lead to a matrix M that contains all lines, in each row the points of a certain line are given, in PG(2,3). For example, {A2,B2}, {A5,B5}, {A8,B8}, {A11,B11} are collinear points of PG(2,3) on a line in PG(2,3).