Editing Projective geometry (section)

== Axioms of projective geometry ==
Any given geometry may be deduced from an appropriate set of [[axiom]]s. Projective geometries are characterised by the "elliptic parallel" axiom, that ''any two planes always meet in just one line'', or in the plane, ''any two lines always meet in just one point''. In other words, there are no such things as parallel lines or planes in projective geometry.

Many alternative sets of axioms for projective geometry have been proposed (see for example Coxeter 2003, Hilbert & Cohn-Vossen 1999, Greenberg 1980).

=== Whitehead's axioms ===
These axioms are based on [[Alfred North Whitehead|Whitehead]], "The Axioms of Projective Geometry". There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are:
* G1: Every line contains at least 3 points
* G2: Every two distinct points, A and B, lie on a unique line, AB.
* G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).

The reason each line is assumed to contain at least 3 points is to eliminate some degenerate cases. The spaces satisfying these
three axioms either have at most one line, or are projective spaces of some dimension over a [[division ring]], or are [[non-Desarguesian plane]]s.

=== Additional axioms ===
One can add further axioms restricting the dimension or the coordinate ring. For example, Coxeter's ''Projective Geometry'',{{sfn|Coxeter|2003|pp=14–15}} references Veblen{{sfn|Veblen|Young|1938|pp=16,18,24,45}} in the three axioms above, together with a further 5 axioms that make the dimension 3 and the coordinate ring a commutative field of characteristic not&nbsp;2.

=== Axioms using a ternary relation ===
One can pursue axiomatization by postulating a ternary relation, [ABC] to denote when three points (not all necessarily distinct) are collinear. An axiomatization may be written down in terms of this relation as well:
* C0: [ABA]
* C1: If A and B are distinct points such that [ABC] and [ABD] then [BDC]
* C2: If A and B are distinct points then there exists a third distinct point C such that [ABC]
* C3: If A and C are distinct points, and B and D are distinct points, with [BCE] and [ADE] but not [ABE], then there is a point F such that [ACF] and [BDF].
For two distinct points, A and B, the line AB is defined as consisting of all points C for which [ABC]. The axioms C0 and C1 then provide a formalization of G2; C2 for G1 and C3 for G3.

The concept of line generalizes to planes and higher-dimensional subspaces. A subspace, AB...XY may thus be recursively defined in terms of the subspace AB...X as that containing all the points of all lines YZ, as Z ranges over AB...X. Collinearity then generalizes to the relation of "independence". A set {{nowrap|{{mset|A, B, ..., Z}}}} of points is independent, [AB...Z] if {{nowrap|{{mset|A, B, ..., Z}}}} is a minimal generating subset for the subspace AB...Z.

The projective axioms may be supplemented by further axioms postulating limits on the dimension of the space. The minimum dimension is determined by the existence of an independent set of the required size. For the lowest dimensions, the relevant conditions may be stated in equivalent
form as follows. A projective space is of:
* (L1) at least dimension 0 if it has at least 1 point,
* (L2) at least dimension 1 if it has at least 2 distinct points (and therefore a line),
* (L3) at least dimension 2 if it has at least 3 non-collinear points (or two lines, or a line and a point not on the line),
* (L4) at least dimension 3 if it has at least 4 non-coplanar points.

The maximum dimension may also be determined in a similar fashion. For the lowest dimensions, they take on the following forms. A projective space is of:
* (M1) at most dimension 0 if it has no more than 1 point,
* (M2) at most dimension 1 if it has no more than 1 line,
* (M3) at most dimension 2 if it has no more than 1 plane,
and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar lines intersect—the very principle that projective geometry was originally intended to embody. Therefore, property (M3) may be equivalently stated that all lines intersect one another.

It is generally assumed that projective spaces are of at least dimension 2. In some cases, if the focus is on projective planes, a variant of M3 may be postulated. The axioms of (Eves 1997: 111), for instance, include (1), (2), (L3) and (M3). Axiom (3) becomes vacuously true under (M3) and is therefore not needed in this context.

=== 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]].