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 showed that two interdissectable polyhedra must have equal Dehn invariants, settling the third of Hilbert's problems. Later, Sydler 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 is necessary (but not sufficient) for a polyhedron to be space-filling. In general, as a result of the above, a polyhedron is either itself space-filling or else can be cut up and reassembled into a space-filling polyhedron iff its Dehn invariant is zero.
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