Polar space | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-03

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Polar_space


00:01:31 1 Generalized quadrangles
00:02:07 2 Finite classical polar spaces
00:05:18 3 Classification



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SUMMARY
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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 called the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms:

Every subspace is isomorphic to a projective geometry Pd(K) with −1 ≤ d ≤ (n − 1) and K a division ring.  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.It is possible to define and study a slightly bigger class of objects using only relationship between points and lines: a polar space is a partial linear space  (P,L), so that for each point p ∈ P and
each line l ∈ L, the set of points of l collinear to p, is either a singleton or the whole l.
Finite polar spaces (where P is a finite set) are also studied as combinatorial objects.