▁to ▁solve ▁Hil bert ' s ▁third ▁problem ▁on ▁whether ▁all ▁poly hed ra ▁with ▁equal ▁volume ▁could ▁be ▁disse cted ▁into ▁each ▁other . ▁ ▁Two ▁poly hed ra ▁have ▁a ▁dis section ▁into ▁poly hed ral ▁pieces ▁that ▁can ▁be ▁re ass emble d ▁into ▁either ▁one , ▁if ▁and ▁only ▁if ▁their ▁volumes ▁and ▁De hn ▁invari ants ▁are ▁equal . ▁A ▁poly hed ron ▁can ▁be ▁cut ▁up ▁and ▁re ass emble d ▁to ▁tile ▁space ▁if ▁and ▁only ▁if ▁its ▁De hn ▁invariant ▁is ▁zero , ▁so ▁having ▁De hn ▁invariant ▁zero ▁is ▁a ▁necessary ▁condition ▁for ▁being ▁a ▁space - fill ing ▁poly hed ron . ▁The ▁De hn ▁invariant ▁of ▁a ▁self - inter section ▁free ▁flexible ▁poly hed ron ▁is ▁invariant ▁as ▁it ▁flex es . ▁▁ ▁The ▁De hn ▁invariant ▁is ▁zero ▁for ▁the ▁cube ▁but ▁non zero ▁for ▁the ▁other ▁Plat onic ▁sol ids , ▁imp lying ▁that ▁the ▁other ▁sol ids ▁cannot ▁tile ▁space ▁and ▁that ▁they ▁cannot ▁be ▁disse cted ▁into ▁a ▁cube . ▁All ▁of ▁the ▁Arch im ede an ▁sol ids ▁have ▁De hn ▁invari ants ▁that ▁are ▁rational ▁combinations ▁of ▁the ▁invari ants ▁for ▁the ▁Plat onic ▁sol ids . ▁In ▁particular , ▁the ▁trunc ated ▁oct ah ed ron ▁also ▁t iles ▁space ▁and ▁has ▁De hn ▁invariant ▁zero ▁like ▁the ▁cube . ▁ ▁The ▁De hn ▁invari ants ▁of ▁poly hed ra ▁are ▁elements ▁of ▁an ▁infinite - dimensional ▁vector ▁space . ▁As ▁an ▁ab elian