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Finite geometry
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===History=== Individual examples can be found in the work of [[Thomas Penyngton Kirkman]] (1847) and the systematic development of finite projective geometry given by [[Karl Georg Christian von Staudt|von Staudt]] (1856). The first axiomatic treatment of finite projective geometry was developed by the [[Italians|Italian]] mathematician [[Gino Fano]]. In his work<ref>{{citation|first=G.|last=Fano|title=Sui postulati fondamentali della geometria proiettiva|year=1892|journal=Giornale di Matematiche|volume= 30|pages=106β132}}</ref> on proving the independence of the set of axioms for [[Projective space|projective ''n''-space]] that he developed,<ref>{{harvnb|Collino|Conte|Verra|2013|loc=p. 6}}</ref> he considered a finite three dimensional space with 15 points, 35 lines and 15 planes (see diagram), in which each line had only three points on it.<ref>{{harvnb|Malkevitch}} Finite Geometries? an AMS Featured Column</ref> In 1906 [[Oswald Veblen]] and W. H. Bussey described [[projective geometry]] using [[homogeneous coordinates]] with entries from the [[Galois field]] GF(''q''). When ''n'' + 1 coordinates are used, the ''n''-dimensional finite geometry is denoted PG(''n, q'').<ref>[[Oswald Veblen]] (1906) [https://www.ams.org/journals/tran/1906-007-02/S0002-9947-1906-1500747-6/S0002-9947-1906-1500747-6.pdf Finite Projective Geometries], [[Transactions of the American Mathematical Society]] 7: 241β59</ref> It arises in [[synthetic geometry]] and has an associated transformation [[group (mathematics)|group]].
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