Dehn Invariant

An invariant defined using the angles of a three-dimensional polyhedron. It remains constant under solid dissection and reassembly. Solids with the same volume can have different Dehn invariants.

Two polyhedra can be dissected into each other only if they have the same volume and the same Dehn invariant. In 1902, Dehn showed that two interdissectable polyhedra must have equal Dehn invariants, settling the third of Hilbert's problems, and Sydler (1965) showed that two polyhedra with the same Dehn invariants are interdissectable.

REFERENCES:

Dehn, M. "Über raumgleiche Polyeder." Nachr. Königl. Ges. der Wiss. zu Göttingen f. d. Jahr 1900, 345-354, 1900.

Dehn, M. "Über den Rauminhalt." Math. Ann. 55, 465-478, 1902.

Kagan, B. "Über die Transformation der Polyeder." Math. Ann. 57, 421-424, 1903.

Sydler, J.-P. "Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidean à trois dimensions." Comment. Math. Helv. 40, 43-80, 1965.