▁the ▁ax iom ▁of ▁choice , ▁this ▁can ▁be ▁avoided ▁( when ▁considering ▁any ▁specific ▁finite ▁set ▁of ▁poly hed ra ) ▁by ▁restrict ing ▁attention ▁to ▁the ▁finite - dimensional ▁vector ▁space ▁generated ▁over ▁ ▁by ▁the ▁di hed ral ▁angles ▁of ▁the ▁poly hed ra . ▁This ▁alternative ▁form ulation ▁shows ▁that ▁the ▁values ▁of ▁the ▁De hn ▁invariant ▁can ▁be ▁given ▁the ▁additional ▁structure ▁of ▁a ▁real ▁vector ▁space . ▁ ▁Ex amples ▁The ▁Plat onic ▁sol ids ▁each ▁have ▁uniform ▁edge ▁lengths ▁and ▁di hed ral ▁angles , ▁none ▁of ▁which ▁are ▁rational ▁multi ples ▁of ▁each ▁other . ▁The ▁di hed ral ▁angle ▁of ▁a ▁cube , ▁/ 2 , ▁is ▁a ▁rational ▁multiple ▁of ▁, ▁but ▁the ▁rest ▁are ▁not . ▁The ▁di hed ral ▁angles ▁of ▁the ▁regular ▁t etra hed ron ▁and ▁regular ▁oct ah ed ron ▁are ▁supp lement ary : ▁they ▁sum ▁to ▁. ▁ ▁In ▁the ▁Ham el ▁basis ▁form ulation ▁of ▁the ▁De hn ▁invariant , ▁one ▁can ▁choose ▁four ▁of ▁these ▁di hed ral ▁angles ▁as ▁part ▁of ▁the ▁Ham el ▁basis . ▁The ▁angle ▁of ▁the ▁cube , ▁/ 2 , ▁is ▁the ▁basis ▁element ▁that ▁is ▁disc arded ▁in ▁the ▁formula ▁for ▁the ▁De hn ▁invariant , ▁so ▁the ▁De hn ▁invariant ▁of ▁the ▁cube ▁is ▁zero . ▁More ▁generally , ▁the ▁De hn ▁invariant ▁of ▁any ▁par alle le pip ed ▁is ▁also ▁zero . ▁Only ▁one ▁of ▁the ▁two ▁angles ▁of ▁the ▁t etra hed ron