▁from ▁being ▁the ▁Le vi ▁graph ▁of ▁the ▁M ö b ius – K ant or ▁configuration . ▁It ▁has ▁one ▁vertex ▁per ▁point ▁and ▁one ▁vertex ▁per ▁triple , ▁with ▁an ▁edge ▁connecting ▁two ▁vertices ▁if ▁they ▁correspond ▁to ▁a ▁point ▁and ▁to ▁a ▁triple ▁that ▁contains ▁that ▁point . ▁ ▁The ▁points ▁and ▁lines ▁of ▁the ▁M ö b ius – K ant or ▁configuration ▁can ▁be ▁described ▁as ▁a ▁mat roid , ▁whose ▁elements ▁are ▁the ▁points ▁of ▁the ▁configuration ▁and ▁whose ▁n ont rivial ▁fl ats ▁are ▁the ▁lines ▁of ▁the ▁configuration . ▁In ▁this ▁mat roid , ▁a ▁set ▁S ▁of ▁points ▁is ▁independent ▁if ▁and ▁only ▁if ▁either ▁ ▁or ▁S ▁consists ▁of ▁three ▁non - col linear ▁points . ▁As ▁a ▁mat roid , ▁it ▁has ▁been ▁called ▁the ▁Mac L ane ▁mat roid , ▁after ▁the ▁work ▁of ▁ ▁prov ing ▁that ▁it ▁cannot ▁be ▁orient ed ; ▁it ▁is ▁one ▁of ▁several ▁known ▁minor - min imal ▁non - orient able ▁mat roid s . ▁ ▁Rel ated ▁configurations ▁The ▁solution ▁to ▁M ö b ius ' ▁problem ▁of ▁mut ually ▁ins cribed ▁pol yg ons ▁for ▁values ▁of ▁p ▁greater ▁than ▁four ▁is ▁also ▁of ▁interest . ▁In ▁particular , ▁one ▁possible ▁solution ▁for ▁ ▁is ▁the ▁Des arg ues ▁configuration , ▁a ▁set ▁of ▁ten ▁points ▁and ▁ten ▁lines , ▁three ▁points ▁per ▁line ▁and ▁three ▁lines ▁per ▁point , ▁that ▁does ▁admit ▁a ▁E uclidean ▁real ization . ▁ ▁The ▁M