Editing Projective plane (section)

==Plane duality==
{{Main|Duality (projective geometry)}}

{{further|Incidence structure#Dual structure}}
A projective plane is defined axiomatically as an [[incidence structure]], in terms of a set ''P'' of points, a set ''L'' of lines, and an [[incidence relation]] ''I'' that determines which points lie on which lines. As ''P'' and ''L'' are only sets one can interchange their roles and define a '''plane dual structure'''.

By interchanging the role of "points" and "lines" in
: ''C'' = (''P'', ''L'', ''I'')
we obtain the dual structure
: ''C''* = (''L'', ''P'', ''I''*),
where ''I''* is the [[converse relation]] of ''I''.

In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words "point" and "line" and making whatever grammatical adjustments that are necessary, is called the '''plane dual statement''' of the first. The plane dual statement of "Two points are on a unique line." is "Two lines meet at a unique point." Forming the plane dual of a statement is known as ''dualizing'' the statement.

If a statement is true in a projective plane ''C'', then the plane dual of that statement must be true in the dual plane ''C''*. This follows since dualizing each statement in the proof "in ''C''" gives a statement of the proof "in ''C''*."

In the projective plane ''C'', it can be shown that there exist four lines, no three of which are concurrent. Dualizing this theorem and the first two axioms in the definition of a projective plane shows that the plane dual structure ''C''* is also a projective plane, called the '''dual plane''' of ''C''.

If ''C'' and ''C''* are isomorphic, then ''C'' is called '''''self-dual'''''. The projective planes PG(2,&nbsp;''K'') for any division ring ''K'' are self-dual. However, there are [[non-Desarguesian plane]]s which are not self-dual, such as the Hall planes and some that are, such as the [[Hughes plane]]s.

The '''''Principle of plane duality''''' says that dualizing any theorem in a self-dual projective plane ''C'' produces another theorem valid in ''C''.