▁. ▁ ▁Definition ▁The ▁definition ▁of ▁the ▁De hn ▁invariant ▁requires ▁a ▁notion ▁of ▁a ▁poly hed ron ▁for ▁which ▁the ▁lengths ▁and ▁di hed ral ▁angles ▁of ▁edges ▁are ▁well ▁defined . ▁Most ▁commonly , ▁it ▁applies ▁to ▁the ▁poly hed ra ▁whose ▁boundaries ▁are ▁manif olds , ▁embedded ▁on ▁a ▁finite ▁number ▁of ▁plan es ▁in ▁E uclidean ▁space . ▁However , ▁the ▁De hn ▁invariant ▁has ▁also ▁been ▁considered ▁for ▁poly hed ra ▁in ▁spher ical ▁geometry ▁or ▁in ▁hyper bol ic ▁space , ▁and ▁for ▁certain ▁self - cross ing ▁poly hed ra ▁in ▁E uclidean ▁space . ▁ ▁The ▁values ▁of ▁the ▁De hn ▁invariant ▁belong ▁to ▁an ▁ab elian ▁group ▁defined ▁as ▁the ▁tensor ▁product ▁ ▁The ▁left ▁factor ▁of ▁this ▁tensor ▁product ▁is ▁the ▁set ▁of ▁real ▁numbers ▁( in ▁this ▁case ▁representing ▁lengths ▁of ▁edges ▁of ▁poly hed ra ) ▁and ▁the ▁right ▁factor ▁represents ▁di hed ral ▁angles ▁in ▁rad ians , ▁given ▁as ▁numbers ▁mod ulo ▁ 2 . ▁( Some ▁sources ▁take ▁the ▁angles ▁mod ulo ▁ ▁instead ▁of ▁mod ulo ▁ 2 , ▁or ▁divide ▁the ▁angles ▁by ▁ ▁and ▁use ▁ ▁in ▁place ▁of ▁ ▁but ▁this ▁makes ▁no ▁difference ▁to ▁the ▁resulting ▁tensor ▁product , ▁as ▁any ▁rational ▁multiple ▁of ▁ ▁in ▁the ▁right ▁factor ▁becomes ▁zero ▁in ▁the ▁product .) ▁ ▁The ▁De hn ▁invariant ▁of ▁a ▁poly hed ron ▁with ▁edge ▁lengths ▁ ▁and ▁edge ▁di hed ral ▁angles ▁ ▁is ▁the ▁sum ▁ ▁An ▁alternative ▁but