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- Polar_space abstract "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 is isomorphic to a projective geometry Rd(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 andeach line l ∈ L, the set of points of l collinear to p, is either a singleton or the whole l.A polar space of rank two is a generalized quadrangle; in this case in the latter definition the set of points of a line l collinear to a point p is the whole l only if p ∈ l. One recovers the former definition from the latter under assumptions that lines have more than 2 points, points lie on more than 2 lines, and there exist a line l and a point p not on l so that p is collinear to all points of l.Finite polar spaces (where P is a finite set) are also studied as combinatorial objects.".
- Polar_space wikiPageExternalLink buekenhout3.pdf.
- Polar_space wikiPageExternalLink mathscinet-getitem?mr=3014979.
- Polar_space wikiPageExternalLink pps.
- Polar_space wikiPageID "3948557".
- Polar_space wikiPageLength "4104".
- Polar_space wikiPageOutDegree "14".
- Polar_space wikiPageRevisionID "706464289".
- Polar_space wikiPageWikiLink Category:Projective_geometry.
- Polar_space wikiPageWikiLink Category:Set_families.
- Polar_space wikiPageWikiLink Combinatorics.
- Polar_space wikiPageWikiLink Division_ring.
- Polar_space wikiPageWikiLink Generalized_quadrangle.
- Polar_space wikiPageWikiLink Geometry.
- Polar_space wikiPageWikiLink Hermitian_variety.
- Polar_space wikiPageWikiLink Isomorphism.
- Polar_space wikiPageWikiLink Jacques_Tits.
- Polar_space wikiPageWikiLink Mathematics.
- Polar_space wikiPageWikiLink Partial_linear_space.
- Polar_space wikiPageWikiLink Projective_space.
- Polar_space wikiPageWikiLink Quadric.
- Polar_space wikiPageWikiLink Symplectic_geometry.
- Polar_space wikiPageWikiLinkText "Polar space".
- Polar_space wikiPageWikiLinkText "dual polar spaces".
- Polar_space wikiPageWikiLinkText "polar space".
- Polar_space wikiPageUsesTemplate Template:Citation.
- Polar_space subject Category:Projective_geometry.
- Polar_space subject Category:Set_families.
- Polar_space type Combinatoric.
- Polar_space type Concept.
- Polar_space comment "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 is isomorphic to a projective geometry Rd(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.".
- Polar_space label "Polar space".
- Polar_space sameAs Q3058226.
- Polar_space sameAs Espace_polaire.
- Polar_space sameAs מרחב_פולרי.
- Polar_space sameAs m.0b84yh.
- Polar_space sameAs Q3058226.
- Polar_space wasDerivedFrom Polar_space?oldid=706464289.
- Polar_space isPrimaryTopicOf Polar_space.