Projective planes may be thought of as projective geometries of dimension two.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. Higher-dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane. 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 Kd+1, i.e. PG(d, K). As in the construction given earlier, the points of the d-dimensional projective space PG(d, K) are the lines through the origin in Kd+1 and a line in PG(d, K) corresponds to a plane through the origin in Kd+1. In fact, each i-dimensional object in PG(d, K), with , is an -dimensional (algebraic) vector subspace of Kd+1 ("goes through the origin"). The projective spaces in turn generalize to the Grassmannian spaces.
Projective planes may be thought of as projective geometries of dimension two.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. Higher-dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane. 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 Kd+1, i.e. PG(d, K). As in the construction given earlier, the points of the d-dimensional projective space PG(d, K) are the lines through the origin in Kd+1 and a line in PG(d, K) corresponds to a plane through the origin in Kd+1. In fact, each i-dimensional object in PG(d, K), with , is an -dimensional (algebraic) vector subspace of Kd+1 ("goes through the origin") The projective spaces in turn generalize to the Grassmannian spaces.