ic ▁poly hed ra ▁with ▁the ▁same ▁volume ▁and ▁the ▁same ▁De hn ▁invariant ▁can ▁always ▁be ▁cut ▁and ▁re ass emble d ▁into ▁each ▁other . ▁Every ▁hyper bol ic ▁manifold ▁with ▁finite ▁volume ▁can ▁be ▁cut ▁along ▁ge odes ic ▁surfaces ▁into ▁a ▁hyper bol ic ▁poly hed ron , ▁which ▁necessarily ▁has ▁zero ▁De hn ▁invariant . ▁ ▁The ▁De hn ▁invariant ▁also ▁controls ▁the ▁ability ▁of ▁a ▁poly hed ron ▁to ▁tile ▁space ▁( part ▁of ▁the ▁subject ▁of ▁Hil bert ' s ▁eigh teenth ▁problem ). ▁Every ▁space - fill ing ▁tile ▁has ▁De hn ▁invariant ▁zero , ▁like ▁the ▁cube . ▁The ▁reverse ▁of ▁this ▁is ▁not ▁true ▁– ▁there ▁exist ▁poly hed ra ▁with ▁De hn ▁invariant ▁zero ▁that ▁do ▁not ▁tile ▁space , ▁but ▁they ▁can ▁always ▁be ▁disse cted ▁into ▁another ▁shape ▁( the ▁cube ) ▁that ▁does ▁tile ▁space . ▁ ▁More ▁generally , ▁if ▁some ▁combination ▁of ▁poly hed ra ▁joint ly ▁t iles ▁space , ▁then ▁the ▁sum ▁of ▁their ▁De hn ▁invari ants ▁( t aken ▁in ▁the ▁same ▁proportion ) ▁must ▁be ▁zero . ▁For ▁instance , ▁the ▁t etra hed ral - oct ah ed ral ▁h oney comb ▁is ▁a ▁t iling ▁of ▁space ▁by ▁t etra hed ra ▁and ▁oct ah ed ra ▁( with ▁twice ▁as ▁many ▁t etra hed ra ▁as ▁oct ah ed ra ), ▁corresponding ▁to ▁the ▁fact ▁that ▁the ▁sum ▁of ▁the ▁De hn ▁invari ants ▁of ▁an ▁oct ah ed