▁configurations , ▁configurations ▁of ▁points ▁and ▁lines ▁( in ▁non - E uclidean ▁spaces ) ▁with ▁no ▁two - point ▁line . ▁For ▁example , ▁the ▁F ano ▁plane ▁and ▁the ▁H esse ▁configuration ▁give ▁rise ▁to ▁Syl v ester ▁mat roid s ▁with ▁seven ▁and ▁nine ▁elements ▁respectively , ▁and ▁may ▁be ▁interpreted ▁either ▁as ▁Ste iner ▁triple ▁systems ▁or ▁as ▁Syl v ester – G alla i ▁configurations . ▁ ▁Properties ▁A ▁Syl v ester ▁mat roid ▁with ▁rank ▁ ▁must ▁have ▁at ▁least ▁ ▁elements ; ▁this ▁bound ▁is ▁tight ▁only ▁for ▁the ▁project ive ▁spaces ▁over ▁G F ( 2 ), ▁of ▁which ▁the ▁F ano ▁plane ▁is ▁an ▁example . ▁ ▁In ▁a ▁Syl v ester ▁mat roid , ▁every ▁independent ▁set ▁can ▁be ▁augment ed ▁by ▁one ▁more ▁element ▁to ▁form ▁a ▁circuit ▁of ▁the ▁mat roid . ▁ ▁Syl v ester ▁mat roid s ▁cannot ▁be ▁represented ▁over ▁the ▁real ▁numbers ▁( this ▁is ▁the ▁Syl v ester – G alla i ▁theorem ), ▁nor ▁can ▁they ▁be ▁orient ed . ▁ ▁History ▁Syl v ester ▁mat roid s ▁were ▁studied ▁and ▁named ▁by ▁ ▁after ▁James ▁Joseph ▁Syl v ester , ▁because ▁they ▁viol ate ▁the ▁Syl v ester – G alla i ▁theorem ▁( for ▁points ▁and ▁lines ▁in ▁the ▁E uclidean ▁plane , ▁or ▁in ▁higher - dimensional ▁E uclidean ▁spaces ) ▁that ▁for ▁every ▁finite ▁set ▁of ▁points ▁there ▁is ▁a ▁line ▁containing ▁only ▁two ▁of ▁the ▁points . ▁ ▁References ▁