Editing Projective plane (section)

==Projective planes in higher-dimensional projective spaces==
Projective planes may be thought of as [[Projective geometry|projective geometries]] of dimension two.<ref>There are competing notions of ''dimension'' in geometry and algebra (vector spaces). In geometry, lines are 1 dimensional, planes are 2 dimensional, solids are 3 dimensional, etc. In a vector space however, the dimension is the number of vectors in a basis. When geometries are constructed from vector spaces, these two notions of dimension can lead to confusion, so it is often the case that the geometric concept is called ''geometric'' or ''projective'' dimension and the other is ''algebraic'' or ''vector space'' dimension. The two concepts are numerically related by: algebraic dimension = geometric dimension + 1.</ref>
Higher-dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane.

The smallest projective space of dimension 3 is [[PG(3,2)]].

These turn out to be "tamer" than the projective planes since the extra degrees of freedom permit [[Desargues' theorem]] to be proved geometrically in the higher-dimensional geometry. This means that the coordinate "ring" associated to the geometry must be a [[division ring]] (skewfield) ''K'',  and the projective geometry is isomorphic to the one constructed from the vector space  ''K''<sup>''d''+1</sup>, i.e. PG(''d'',&nbsp;''K''). As in the construction given earlier, the points of the ''d''-dimensional [[projective space]]  PG(''d'',&nbsp;''K'') are the lines through the origin in ''K''<sup>''d''+1</sup> and a line in PG(''d'',&nbsp;''K'') corresponds to a plane through the origin in ''K''<sup>''d''+1</sup>.  In fact, each ''i''-dimensional object in PG(''d'',&nbsp;''K''), with {{nowrap|''i'' &lt; ''d''}},  is an {{nowrap|(''i'' + 1)}}-dimensional (algebraic) vector subspace of ''K''<sup>''d''+1</sup> ("goes through the origin"). The projective spaces in turn generalize to the [[Grassmannian|Grassmannian spaces]].


It can be shown that if Desargues' theorem holds in a projective space of dimension greater than two, then it must also hold in all planes that are contained in that space. Since there are projective planes in which Desargues' theorem fails ([[non-Desarguesian plane]]s), these planes can not be embedded in a higher-dimensional projective space. Only the planes from the vector space construction PG(2,&nbsp;''K'') can appear in projective spaces of higher dimension. Some disciplines in mathematics restrict the meaning of projective plane to only this type of projective plane since otherwise general statements about projective spaces would always have to mention the exceptions when the geometric dimension is two.<ref>{{harvp|Bruck|Bose|1964|loc=Introduction}}. "One might say, with some justice, that projective geometry, in so far as present day research is concerned, has split into two quite separate fields. On the one hand, the researcher into the foundations of geometry tends to regard Desarguesian spaces as completely known. Since the only possible non-Desarguesian spaces are planes, his attention is restricted to the theory of projective planes, especially the non-Desarguesian planes. On the other hand stand all those researchers – and especially, the algebraic geometers – who are unwilling to be bound to two-dimensional space and uninterested in permitting non-Desarguesian planes to assume an exceptional role in their theorems. For the latter group of researchers, there are no projective spaces except the Desarguesian spaces."</ref>