Editing Finite geometry (section)

{{Short description|Geometric system with a finite number of points}}
[[Image:Order 2 affine plane.svg|thumb|200px|right|Finite affine plane of order 2, containing 4 "points" and 6 "lines". Lines of the same color are "parallel". The centre of the figure is not a "point" of this affine plane, hence the two green "lines" don't "intersect".]]
{{General geometry}}

A '''finite geometry''' is any [[geometry|geometric]] system that has only a [[finite set|finite]] number of [[point (geometry)|points]].
The familiar [[Euclidean geometry]] is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the [[pixel]]s are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite [[projective space|projective]] and [[affine space]]s because of their regularity and simplicity.  Other significant types of finite geometry are finite [[Möbius plane|Möbius or inversive plane]]s and [[Laguerre plane]]s, which are examples of a general type called [[Benz plane]]s, and their higher-dimensional analogs such as higher finite [[inversive geometry|inversive geometries]].

Finite geometries may be constructed via [[linear algebra]], starting from [[vector space]]s over a [[finite field]]; the affine and [[projective plane]]s so constructed are called [[Galois geometry|Galois geometries]].  Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite [[projective space]] of dimension three or greater is [[isomorphism|isomorphic]] to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the [[non-Desarguesian plane]]s.  Similar results hold for other kinds of finite geometries.