Polar Space

In mathematics, in the field of geometry, 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 projective 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. Finite polar spaces (where P is a finite set) are also studied as combinatorial objects.