ö b ius ▁configuration ▁is ▁a ▁three - dimensional ▁analog ue ▁of ▁the ▁M ö b ius – K ant or ▁configuration ▁consisting ▁of ▁two ▁mut ually ▁ins cribed ▁t etra hed ra . ▁ ▁The ▁M ö b ius – K ant or ▁configuration ▁can ▁be ▁augment ed ▁by ▁adding ▁four ▁lines ▁through ▁the ▁four ▁pairs ▁of ▁points ▁not ▁already ▁connected ▁by ▁lines , ▁and ▁by ▁adding ▁a ▁nin th ▁point ▁on ▁the ▁four ▁new ▁lines . ▁The ▁resulting ▁configuration , ▁the ▁H esse ▁configuration , ▁shares ▁with ▁the ▁M ö b ius – K ant or ▁configuration ▁the ▁property ▁of ▁being ▁realiz able ▁with ▁complex ▁coordinates ▁but ▁not ▁with ▁real ▁coordinates . ▁De let ing ▁any ▁one ▁point ▁from ▁the ▁H esse ▁configuration ▁produces ▁a ▁copy ▁of ▁the ▁M ö b ius – K ant or ▁configuration . ▁Both ▁configurations ▁may ▁also ▁be ▁described ▁algebra ically ▁in ▁terms ▁of ▁the ▁ab elian ▁group ▁ ▁with ▁nine ▁elements . ▁This ▁group ▁has ▁four ▁sub groups ▁of ▁order ▁three ▁( the ▁subsets ▁of ▁elements ▁of ▁the ▁form ▁, ▁, ▁, ▁and ▁ ▁respectively ), ▁each ▁of ▁which ▁can ▁be ▁used ▁to ▁partition ▁the ▁nine ▁group ▁elements ▁into ▁three ▁cos ets ▁of ▁three ▁elements ▁per ▁cos et . ▁These ▁nine ▁elements ▁and ▁twelve ▁cos ets ▁form ▁the ▁H esse ▁configuration . ▁Rem oving ▁the ▁zero ▁element ▁and ▁the ▁four ▁cos ets ▁containing ▁zero ▁gives ▁rise ▁to ▁the ▁M ö b ius – K ant or ▁configuration . ▁ ▁Notes ▁ ▁References ▁. ▁. ▁.