▁and ▁oct ah ed ron ▁can ▁be ▁included , ▁as ▁the ▁other ▁one ▁is ▁a ▁rational ▁combination ▁of ▁the ▁one ▁that ▁is ▁included ▁and ▁the ▁angle ▁of ▁the ▁cube . ▁The ▁De hn ▁invari ants ▁of ▁each ▁of ▁the ▁other ▁Plat onic ▁sol ids ▁will ▁be ▁a ▁vector ▁in ▁ ▁formed ▁by ▁multip lying ▁the ▁unit ▁vector ▁for ▁that ▁solid ' s ▁angle ▁by ▁the ▁length ▁and ▁number ▁of ▁edges ▁of ▁the ▁solid . ▁No ▁matter ▁how ▁they ▁are ▁scale d ▁by ▁different ▁edge ▁lengths , ▁the ▁t etra hed ron , ▁ icos ah ed ron , ▁and ▁d ode c ah ed ron ▁all ▁have ▁De hn ▁invari ants ▁that ▁form ▁vectors ▁pointing ▁in ▁different ▁directions , ▁and ▁therefore ▁are ▁une qual ▁and ▁non zero . ▁ ▁The ▁neg ated ▁di hed ral ▁angle ▁of ▁the ▁oct ah ed ron ▁diff ers ▁from ▁the ▁angle ▁of ▁a ▁t etra hed ron ▁by ▁an ▁integer ▁multiple ▁of ▁, ▁and ▁in ▁addition ▁the ▁oct ah ed ron ▁has ▁two ▁times ▁as ▁many ▁edges ▁as ▁the ▁t etra hed ron ▁( tw elve ▁instead ▁of ▁six ). ▁Therefore , ▁the ▁De hn ▁invariant ▁of ▁the ▁oct ah ed ron ▁is ▁− 2 ▁times ▁the ▁De hn ▁invariant ▁of ▁a ▁t etra hed ron ▁of ▁the ▁same ▁edge ▁length . ▁The ▁De hn ▁invari ants ▁of ▁the ▁other ▁Arch im ede an ▁sol ids ▁can ▁also ▁be ▁expressed ▁as ▁rational ▁combinations ▁of ▁the ▁invari ants ▁of ▁the ▁Plat onic ▁sol ids . ▁ ▁App lications