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In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are the and near n-gons with ''n'' = 4. They are also precisely the partial geometries pg(''s'',''t'',α) with α = 1. ==Definition== A generalized quadrangle is an incidence structure (''P'',''B'',I), with I ⊆ ''P'' × ''B'' an incidence relation, satisfying certain axioms. Elements of ''P'' are by definition the ''points'' of the generalized quadrangle, elements of ''B'' the ''lines''. The axioms are the following: * There is an ''s'' (''s'' ≥ 1) such that on every line there are exactly ''s'' + 1 points. There is at most one point on two distinct lines. * There is a ''t'' (''t'' ≥ 1) such that through every point there are exactly ''t'' + 1 lines. There is at most one line through two distinct points. * For every point ''p'' not on a line ''L'', there is a unique line ''M'' and a unique point ''q'', such that ''p'' is on ''M'', and ''q'' on ''M'' and ''L''. (''s'',''t'') are the ''parameters'' of the generalized quadrangle. The parameters are allowed to be infinite. If either ''s'' or ''t'' is one, the generalized quadrangle is called ''trivial''. For example, the 3x3 grid with ''P'' = and ''L'' = is a trivial GQ with ''s'' = 2 and ''t'' = 1. A generalized quadrangle with parameters (''s'',''t'') is often denoted by GQ(''s'',''t''). The smallest non-trivial generalized quadrangle is GQ(2,2), whose representation has been dubbed "the doily" by Stan Payne in 1973. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「generalized quadrangle」の詳細全文を読む スポンサード リンク
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