Editing Projective plane (section)

==Vector space construction==

Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other. In this construction, each "point" of the real projective plane is the one-dimensional subspace (a ''geometric'' line) through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a (''geometric'') plane through the origin in the 3-space. This idea can be generalized and made more precise as follows.{{sfnp|Baez|2002|p=165}}

Let ''K'' be any [[division ring]] (skewfield). Let ''K''<sup>3</sup> denote the set of all triples ''x'' = {{nowrap|(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>)}} of elements of ''K'' (a [[Cartesian product]] viewed as a [[vector space]]). For any nonzero ''x'' in ''K''<sup>3</sup>, the minimal subspace of ''K''<sup>3</sup> containing ''x'' (which may be visualized as all the vectors in a line through the origin) is the subset
:<math>\{ k x : k \in K \}</math>
of ''K''<sup>3</sup>. Similarly, let ''x'' and ''y'' be linearly independent elements of ''K''<sup>3</sup>, meaning that {{nowrap|1=''kx'' + ''my'' = 0}} implies that {{nowrap|1=''k'' = ''m'' = 0}}. The minimal subspace of ''K''<sup>3</sup> containing ''x'' and ''y'' (which may be visualized as all the vectors in a plane through the origin) is the subset
:<math>\{k x + m y : k, m \in K\}</math>
of ''K''<sup>3</sup>. This 2-dimensional subspace contains various 1-dimensional subspaces through the origin that may be obtained by fixing ''k'' and ''m'' and taking the multiples of the resulting vector. Different choices of ''k'' and ''m'' that are in the same ratio will give the same line.

The '''projective plane''' over ''K'', denoted PG(2,&nbsp;''K'') or ''K'''''P'''<sup>2</sup>, has a set of ''points'' consisting of all the 1-dimensional subspaces in ''K''<sup>3</sup>. A subset ''L'' of the points of PG(2,&nbsp;''K'') is a ''line'' in PG(2,&nbsp;''K'') if there exists a 2-dimensional subspace of ''K''<sup>3</sup> whose set of 1-dimensional subspaces is exactly ''L''.

Verifying that this construction produces a projective plane is usually left as a linear algebra exercise.

An alternate (algebraic) view of this construction is as follows. The points of this projective plane are the equivalence classes of the set {{nowrap|''K''<sup>3</sup> \ {(0, 0, 0)} }} modulo the [[equivalence relation]]
:''x'' ~ ''kx'',  for all ''k'' in ''K''<sup>×</sup>.
Lines in the projective plane are defined exactly as above.

The coordinates {{nowrap|(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>)}} of a point in PG(2,&nbsp;''K'') are called '''homogeneous coordinates'''. Each triple {{nowrap|(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>)}} represents a well-defined point in PG(2,&nbsp;''K''), except for the triple {{nowrap|(0, 0, 0)}}, which represents no point. Each point in PG(2,&nbsp;''K''), however, is represented by many triples.

If ''K'' is a [[topological space]], then ''K'''''P'''<sup>2</sup>  inherits a topology via the [[product topology|product]], [[subspace topology|subspace]], and [[quotient topology|quotient]] topologies.

===Classical examples===

The [[real projective plane]] '''RP'''<sup>2</sup> arises when ''K'' is taken to be the [[real number]]s, '''R'''. As a closed, non-orientable real 2-[[manifold]], it serves as a fundamental example in topology.<ref name="Bredon1993">The real projective plane appears 37 times in the index of Bredon (1993), for example.</ref>

In this construction, consider the unit sphere centered at the origin in '''R'''<sup>3</sup>. Each of the '''R'''<sup>3</sup> lines in this construction intersects the sphere at two antipodal points. Since the '''R'''<sup>3</sup> line represents a point of '''RP'''<sup>2</sup>, we will obtain the same model of '''RP'''<sup>2</sup> by identifying the antipodal points of the sphere. The lines of '''RP'''<sup>2</sup> will be the great circles of the sphere after this identification of antipodal points. This description gives the standard model of [[elliptic geometry]].

The [[complex projective plane]] '''CP'''<sup>2</sup> arises when ''K'' is taken to be the [[complex number]]s, '''C'''. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over other [[Field (algebra)|fields]] (known as '''[[pappian plane]]s''') serve as fundamental examples in [[algebraic geometry]].<ref name="Shafarevich1994">The projective planes over fields are used throughout {{harvp|Shafarevich|1994}}, for example.</ref>

The [[quaternionic projective space|quaternionic projective plane]] '''HP'''<sup>2</sup> is also of independent interest.<ref>See, e.g., {{harvtxt|Weintraub|1978}} and {{harvtxt|Gorodkov|2019}}</ref>

===Finite field planes===

By [[Wedderburn's little theorem|Wedderburn's Theorem]], a finite division ring must be commutative and so be a field. Thus, the finite examples of this construction are known as "field planes". Taking ''K'' to be the [[finite field]] of {{nowrap|1=''q'' = ''p''<sup>''n''</sup>}} elements with prime ''p'' produces a projective plane of {{nowrap|''q''<sup>2</sup> + ''q'' + 1}} points. The field planes are usually denoted by PG(2,&nbsp;''q'') where PG stands for projective geometry, the "2" is the dimension and ''q'' is called the '''order''' of the plane (it is one less than the number of points on any line). The Fano plane, discussed below, is denoted by PG(2,&nbsp;2). The [[#A finite example|third example above]] is the projective plane PG(2,&nbsp;3).

[[File:fano_plane_with_colored_lines.svg|thumb|The Fano plane. Points are shown as dots; lines are shown as lines or circles.]]

The [[Fano plane]] is the projective plane arising from the field of two elements. It is the smallest projective plane, with only seven points and seven lines. In the figure at right, the seven points are shown as small balls, and the seven lines are shown as six line segments and a circle. However, one could equivalently consider the balls to be the "lines" and the line segments and circle to be the "points" – this is an example of [[duality (projective geometry)|duality]] in the projective plane: if the lines and points are interchanged, the result is still a projective plane (see [[#Duality|below]]). A permutation of the seven points that carries [[incidence (geometry)|collinear]] points (points on the same line) to collinear points is called a ''[[collineation]]'' or ''[[symmetry]]'' of the plane. The collineations of a geometry form a [[Group (mathematics)|group]] under composition, and for the Fano plane this group ({{nowrap|1=PΓL(3, 2) = PGL(3, 2)}}) has 168 elements.

=== {{anchor|Desarguesian}} Desargues' theorem and Desarguesian planes ===

The [[Desargues' theorem|theorem of Desargues]] is universally valid in a projective plane if and only if the plane can be constructed from a three-dimensional vector space over a skewfield as [[#Vector space construction|above]].<ref>[[David Hilbert]] proved the more difficult "only if" part of this result.</ref> These planes are called '''Desarguesian planes''', named after [[Girard Desargues]]. The real (or complex) projective plane and the projective plane of order 3 given [[Projective plane#Some examples|above]] are examples of Desarguesian projective planes. The projective planes that can not be constructed in this manner are called [[non-Desarguesian plane]]s, and the [[Moulton plane]] given [[Projective plane#Some examples|above]] is an example of one. The PG(2,&nbsp;''K'') notation is reserved for the Desarguesian planes. When ''K'' is a [[Field (mathematics)|field]], a very common case, they are also known as ''field planes'' and if the field is a [[finite field]] they can be called [[Galois geometry|''Galois planes'']].