, 1 ), ▁( ω , ▁− 1 , ▁ 1 ), ▁( − 1 , ▁ 0 , ▁ 1 ), ▁( − 1 , ω 2 , 1 ), ▁( 1 , ω , 0 ), ▁( 0 , 1 , 0 ), ▁( 0 , − 1 , 1 ), ▁where ▁ ω ▁denotes ▁the ▁complex ▁cube ▁root ▁of ▁ 1 . ▁ ▁These ▁are ▁the ▁vertices ▁of ▁the ▁complex ▁polygon ▁ 3 { 3 } 3 ▁with ▁the ▁ 8 ▁vertices ▁and ▁ 8 ▁ 3 - ed ges . ▁C ox eter ▁named ▁it ▁a ▁M ö b ius – K ant or ▁polygon . ▁ ▁Abstract ▁inc idence ▁pattern ▁ ▁More ▁abstract ly , ▁the ▁M ö b ius – K ant or ▁configuration ▁can ▁be ▁described ▁as ▁a ▁system ▁of ▁eight ▁points ▁and ▁eight ▁tri ples ▁of ▁points ▁such ▁that ▁each ▁point ▁belongs ▁to ▁exactly ▁three ▁of ▁the ▁tri ples . ▁With ▁the ▁additional ▁conditions ▁( natural ▁to ▁points ▁and ▁lines ) ▁that ▁no ▁pair ▁of ▁points ▁belong ▁to ▁more ▁than ▁one ▁triple ▁and ▁that ▁no ▁two ▁tri ples ▁have ▁more ▁than ▁one ▁point ▁in ▁their ▁intersection , ▁any ▁two ▁systems ▁of ▁this ▁type ▁are ▁equivalent ▁under ▁some ▁permut ation ▁of ▁the ▁points . ▁That ▁is , ▁the ▁M ö b ius – K ant or ▁configuration ▁is ▁the ▁unique ▁project ive ▁configuration ▁of ▁type ▁( 8 3 8 3 ). ▁ ▁The ▁M ö b ius – K ant or ▁graph ▁der ives ▁its ▁name