A Projective Plane is an incidence structure denoted by \( \pi=(\mathscr{P,L,I}) \) that satisfies following axioms.
- [P 1] If p and q are two points, there is exactly one line on both.
- [P 2] If L and M are two lines, there is at least one point on both.
- [P 3] If L is a line, there are at least three points on L.
- [P 4] If L is a line, there is at least one point not on L.
- [P 5] There is at least one line.
Exanple
The incidence structure with \(\mathscr{P}=\{a_0,a_1,a_2,a_3,d_1,d_2,d_3\}\) and \(\mathscr{L}=\{L_1, L_2,L_3,L_1',L_2',L_3',D\}\)
and incidence relation as in the figure is a projective plane.
This structure has seven points each on three lines
and seven lines each on three points
This structure is called “fano-configuration” and it is the smallest projective plane that exists.
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