Editing Desargues's theorem (section)

==Relation to Pappus's theorem==
[[Pappus's hexagon theorem]] states that, if a [[hexagon]] {{math|''AbCaBc''}} is drawn in such a way that vertices {{math|''a'', ''b''}} and {{math|''c''}} lie on a line and vertices {{math|''A'', ''B''}} and {{math|''C''}} lie on a second line, then each two opposite sides of the hexagon lie on two lines that meet in a point and the three points constructed in this way are collinear. A plane in which Pappus's theorem is universally true is called ''Pappian''.
{{harvtxt|Hessenberg|1905}}<ref>According to {{harv|Dembowski|1968|loc= pg. 159, footnote 1}}, Hessenberg's original proof is not complete; he disregarded the possibility that some additional incidences could occur in the Desargues configuration. A complete proof is provided by {{harvnb|Cronheim|1953}}.</ref> showed that Desargues's theorem can  be deduced from three applications of Pappus's theorem.<ref>{{harvnb|Coxeter|1969|loc=p. 238, section 14.3}}</ref>

The [[Theorem#Converse|converse]] of this result is not true, that is, not all Desarguesian planes are Pappian. Satisfying Pappus's theorem universally is equivalent to having the underlying coordinate system be [[commutative]]. A plane defined over a non-commutative division ring (a division ring that is not a field) would therefore be Desarguesian but not Pappian. However, due to [[Wedderburn's little theorem]], which states that all ''finite'' division rings are fields, all ''finite'' Desarguesian planes are Pappian. There is no known completely geometric proof of this fact, although {{harvtxt|Bamberg|Penttila|2015}} give a proof that uses only "elementary" algebraic facts (rather than the full strength of Wedderburn's little theorem).