Editing Incidence geometry (section)

===Projective planes===
{{main|projective plane}}
A ''projective plane'' is a linear space in which:
* Every pair of distinct lines meet in exactly one point,
and that satisfies the non-degeneracy condition:
* There exist four points, no three of which are [[collinear]].

There is a [[bijection]] between {{math|''P''}} and {{math|''L''}} in a projective plane. If {{math|''P''}} is a finite set, the projective plane is referred to as a ''finite'' projective plane. The '''order''' of a finite projective plane is {{math|1=''n'' = ''k'' – 1}}, that is, one less than the number of points on a line. All known projective planes have orders that are [[prime power]]s. A projective plane of order {{math|''n''}} is an {{math|((''n''<sup>2</sup> + ''n'' + 1)<sub>''n'' + 1</sub>)}} configuration.

The smallest projective plane has order two and is known as the ''Fano plane''.
[[File:Fano plane.svg|thumb|{{center|Projective plane of order 2 <br> the Fano plane}}]]

==== Fano plane ====
{{main|Fano plane}}

This famous incidence geometry was developed by the Italian mathematician [[Gino Fano]]. In his work<ref>{{citation|first=G.|last=Fano|title=Sui postulati fondamentali della geometria proiettiva|year=1892|journal=Giornale di Matematiche|volume= 30|pages=106–132}}</ref> on proving the independence of the set of axioms for [[Projective space|projective ''n''-space]] that he developed,<ref>{{harvnb|Collino|Conte|Verra|2013|loc=p. 6}}</ref> he produced a finite three-dimensional space with 15 points, 35 lines and 15 planes, in which each line had only three points on it.<ref>{{harvnb|Malkevitch}} Finite Geometries? an AMS Featured Column</ref> The planes in this space consisted of seven points and seven lines and are now known as [[Fano plane]]s.

The Fano plane cannot be represented in the [[Euclidean plane]] using only points and straight line segments (i.e., it is not realizable). This is a consequence of the [[Sylvester–Gallai theorem]], according to which every realizable incidence geometry must include an ''ordinary line'', a line containing only two points. The Fano plane has no such line (that is, it is a [[Sylvester–Gallai configuration]]), so it is not realizable.{{sfnp|Aigner|Ziegler|2010}}

A [[complete quadrangle]] consists of four points, no three of which are collinear. In the Fano plane, the three points not on a complete quadrangle are the diagonal points of that quadrangle and are collinear. This contradicts the ''Fano axiom'', often used as an axiom for the Euclidean plane, which states that the three diagonal points of a complete quadrangle are never collinear.