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...Projective geometry kristin dean abstract this paper investigates the nature of nite geometries it will focus on known as planes and conclude with example fano plane contents basic denitions axioms linear algebra quotient finite spaces references definitions first we must begin a few relating to can be thought set objects relation those elements denition is denoted g i where which both symmetric reexive therelation called an incidence for consider tradional euclidean in are points lines point incident line if lies that two they have all common only when same there often natural division into dierent kinds such suppose then ag mutually no element outside f added also maximal has rank r partitioned sets every contains exactly one each type thus these divisions give idea most examples dealt consist certain structures date july lemma let distinct proof not exist by form now some but contradiction because saw truly types termed case henceforth p l many ge ometries satisfy following axiom th...