ron ▁and ▁two ▁t etra hed ra ▁( with ▁the ▁same ▁side ▁lengths ) ▁is ▁zero . ▁ ▁Real iz ability ▁Although ▁the ▁De hn ▁invariant ▁takes ▁values ▁in ▁ ▁not ▁all ▁of ▁the ▁elements ▁in ▁this ▁space ▁can ▁be ▁realized ▁as ▁the ▁De hn ▁invari ants ▁of ▁poly hed ra . ▁The ▁De hn ▁invari ants ▁of ▁E uclidean ▁poly hed ra ▁form ▁a ▁linear ▁sub space ▁of ▁: ▁one ▁can ▁add ▁the ▁De hn ▁invari ants ▁of ▁poly hed ra ▁by ▁taking ▁the ▁dis joint ▁union ▁of ▁the ▁poly hed ra ▁( or ▁gl uing ▁them ▁together ▁on ▁a ▁face ), ▁neg ate ▁De hn ▁invari ants ▁by ▁making ▁holes ▁in ▁the ▁shape ▁of ▁the ▁poly hed ron ▁into ▁large ▁cub es , ▁and ▁multiply ▁the ▁De hn ▁invariant ▁by ▁any ▁scalar ▁by ▁scaling ▁the ▁poly hed ron ▁by ▁the ▁same ▁number . ▁The ▁question ▁of ▁which ▁elements ▁of ▁ ▁( or , ▁equival ently , ▁) ▁are ▁realiz able ▁was ▁clar ified ▁by ▁the ▁work ▁of ▁Dup ont ▁and ▁Sah , ▁who ▁showed ▁the ▁existence ▁of ▁the ▁following ▁short ▁exact ▁sequence ▁of ▁ab elian ▁groups ▁( not ▁vector ▁spaces ) ▁involving ▁group ▁hom ology : ▁ ▁Here , ▁the ▁notation ▁ ▁represents ▁the ▁free ▁ab elian ▁group ▁over ▁E uclidean ▁poly hed ra ▁mod ulo ▁certain ▁relations ▁derived ▁from ▁pairs ▁of ▁poly hed ra ▁that ▁can ▁be ▁disse cted ▁into ▁each ▁other . ▁ ▁is ▁the ▁subgroup ▁generated ▁in ▁this ▁group ▁by ▁the ▁tri angular ▁pr isms , ▁and ▁is