Dehn Invariant -- from Wolfram MathWorld

The Dehn invariant is a constant defined using the angles and edge lengths of a three-dimensional polyhedron. It is significant because it remains constant under polyhedron dissection and reassembly. Dehn (1902) showed that two interdissectable polyhedra must have equal Dehn invariants, settling the third of Hilbert's problems. Later, Sydler (1965) showed that two polyhedra can be dissected into each other iff they have the same volume and the same Dehn invariant. Having Dehn invariant zero...