only if there is a projective plane of order N. While the classification of all projective planes is far from complete, results are known for small orders: 2 : all isomorphic to PG(2, 2) 3 : all isomorphic to PG(2, 3) 4 : all isomorphic to PG(2, 4) 5 : all isomorphic to PG(2, 5) 6 : impossible as the order of a projective plane, proved by Tarry who showed that Euler's thirty-six officers problem has no solution. However, the connection between these problems was not known until Bose proved it in 1938. 7 : all isomorphic to PG(2, 7) 8 : all isomorphic to PG(2, 8) 9 : PG(2, 9), and three more different (non-isomorphic) non-Desarguesian planes: a Hughes plane, a Hall plane, and the dual of this Hall plane. All are described in . 10 : impossible as an order of a projective plane, proved by heavy computer calculation. 11 : at least PG(2, 11), others are not known but possible. 12 : it is conjectured to be impossible as an order of a projective plane. Projective planes in higher-dimensional projective spaces Projective planes may be thought of as projective geometries of "geometric" dimension two. 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. 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 planes), these planes can not be