Dehn invariant

2020 Mathematics Subject Classification: Primary: 52B10 Secondary: 52B45 [MSN][ZBL]

An invariant of polyhedra in three-dimensional space that decides whether two polyhedra of the same volume are "scissors congruent" (see Equal content and equal shape, figures of; Hilbert problems; Polyhedron).

Quite generally, a scissors-congruence invariant assigns to a polytope $P$ in space an element $D(P)$ in a group such that $D(P\cap P') + D(P \cup P') = D(P) + D(P')$, $D(P) = 0$ if $P$ is degenerate, and $D(P) = D(P')$ if there is a motion $g$ of the space such that $P' = gP$.

For the Dehn invariant, the group chosen is the tensor product $\mathbb{R} \otimes_{\mathbb{Z}} \mathbb{R} / 2\pi \mathbb{Z}$. To a polytope $P$ with edges $L_i$ one associates the element $D(P) = \sum_i |L_i| \otimes \delta_i$, where $|L_i|$ is the length of $L_i$ and $\delta_i$ is the dihedral angle of the planes meeting at $L_i$. The Sydler theorem states that two polytopes in three-dimensional space are scissors equivalent if and only if they have equal volume and the same Dehn invariant, thus solving Hilbert's third problem in a very precise manner (cf. also Hilbert problems).

For higher dimensions there is a generalization, called the Hadwiger invariant or Dehn–Hadwiger invariant. This has given results in dimension four, [a4], and for the case when the group consists of translations only, [a2].

References

[a1]	P. Cartier, "Decomposition des polyèdres: le point sur le troisième problème de Hilbert" Sem. Bourbaki , 1984/5 (1986) pp. 261–288
[a2]	C.H. Sah, "Hilbert's third problem: scissors congruence" , Pitman (1979)
[a3]	V.G. Boltianskii, "Hilbert's third problem" , Wiley (1978)
[a4]	B. Jessen, "Zur Algebra der Polytope" Göttinger Nachrichte Math. Phys. (1972) pp. 47–53