▁group , ▁this ▁space ▁is ▁part ▁of ▁an ▁exact ▁sequence ▁involving ▁group ▁hom ology . ▁Similar ▁invari ants ▁can ▁also ▁be ▁defined ▁for ▁some ▁other ▁dis section ▁puzz les , ▁including ▁the ▁problem ▁of ▁dis sect ing ▁rect il ine ar ▁pol yg ons ▁into ▁each ▁other ▁by ▁axis - parallel ▁cut s ▁and ▁transl ations . ▁ ▁Background ▁ ▁In ▁two ▁dimensions , ▁the ▁Wal lace – B oly ai – G er w ien ▁theorem ▁states ▁that ▁any ▁two ▁pol yg ons ▁of ▁equal ▁area ▁can ▁be ▁cut ▁up ▁into ▁pol yg onal ▁pieces ▁and ▁re ass emble d ▁into ▁each ▁other . ▁David ▁Hil bert ▁became ▁interested ▁in ▁this ▁result ▁as ▁a ▁way ▁to ▁ax iom at ize ▁area , ▁in ▁connection ▁with ▁Hil bert ' s ▁ax i oms ▁for ▁E uclidean ▁geometry . ▁In ▁Hil bert ' s ▁third ▁problem , ▁he ▁pos ed ▁the ▁question ▁of ▁whether ▁two ▁poly hed ra ▁of ▁equal ▁volumes ▁can ▁always ▁be ▁cut ▁into ▁poly hed ral ▁pieces ▁and ▁re ass emble d ▁into ▁each ▁other . ▁Hil bert ' s ▁student ▁Max ▁De hn , ▁in ▁his ▁ 1 9 0 0 ▁hab il itation ▁th esis , ▁invent ed ▁the ▁De hn ▁invariant ▁in ▁order ▁to ▁prove ▁that ▁this ▁is ▁not ▁always ▁possible , ▁providing ▁a ▁negative ▁solution ▁to ▁Hil bert ' s ▁problem . ▁Although ▁De hn ▁form ulated ▁his ▁invariant ▁differently , ▁the ▁modern ▁approach ▁is ▁to ▁describe ▁it ▁as ▁a ▁value ▁in ▁a ▁tensor ▁product , ▁following