High Quality Content by WIKIPEDIA articles! In mathematics, in the field of combinatorics, a polar space of rank n (n >= 3), or projective index n–1, consists of a set P, conventionally the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms : * Every subspace, together with its own subspaces, is isomorphic with a partial geometry PG(d,q) with –1 <= d <= (n–1) and q a prime power. By definition, for each subspace the corresponding d is its dimension. * The intersection of two subspaces is always a subspace. * For each point p not in a subspace A of dimension of n–1, there is a unique subspace B of dimension n–1 such that A B is (n–2)-dimensional. The points in A B are exactly the points of A that are in a common subspace of dimension 1 with p. * There are at least two disjoint subspaces of dimension n–1. A polar space of rank two is a generalized quadrangle.

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