Editing Projective geometry (section)

=== Axioms for projective planes ===
{{main|Projective plane}}
In [[incidence geometry]], most authors<ref>{{harvnb|Bennett|1995|p=4}}, {{harvnb|Beutelspacher|Rosenbaum|1998|p=8}}, {{harvnb|Casse|2006|p=29}}, {{harvnb|Cederberg|2001|p=9}}, {{harvnb|Garner|1981|p=7}}, {{harvnb|Hughes|Piper|1973|p=77}}, {{harvnb|Mihalek|1972|p=29}}, {{harvnb|Polster|1998|p=5}} and {{harvnb|Samuel|1988|p=21}} among the references given.</ref> give a treatment that embraces the [[Fano plane]] {{nowrap|PG(2, 2)}} as the smallest finite projective plane.  An axiom system that achieves this is as follows:
* (P1) Any two distinct points lie on a line that is unique.
* (P2) Any two distinct lines meet at a point that is unique.
* (P3) There exist at least four points of which no three are collinear.

Coxeter's ''Introduction to Geometry''{{sfn|Coxeter|1969|pp=229–234}} gives a list of five axioms for a more restrictive concept of a projective plane that is attributed to Bachmann, adding [[Pappus's hexagon theorem|Pappus's theorem]] to the list of axioms above (which eliminates [[non-Desarguesian plane]]s) and excluding projective planes over fields of characteristic 2 (those that do not satisfy [[Fano's axiom]]). The restricted planes given in this manner more closely resemble the [[real projective plane]].