Dehn Invariant

In geometry, the Dehn invariant is a quantity associated with a polyhedron that helps determine whether it can be cut into pieces and reassembled into a different polyhedron, and whether the polyhedron or its pieces can tile space. Two polyhedra can be dissected into pieces that reassemble into either one exactly when both their volumes and their Dehn invariants are equal. A Dehn invariant of zero is a necessary, though not sufficient, condition for a polyhedron to be able to fill space on its own, and a polyhedron can be cut up and reassembled into a space-filling polyhedron precisely when its Dehn invariant is zero.