Hilbert's third problem

Solved, 1900GeometryHilbert #3

Statement

Given any two polyhedra of equal volume in three-dimensional Euclidean space $\mathbb{R}^3$ , is it always possible to cut the first into finitely many polyhedral pieces and reassemble them by rigid motions to form the second? Hilbert conjectured that the answer is no, and asked for two polyhedra of equal volume that are not scissors-congruent.

Max Dehn, a student of Hilbert, solved the problem in 1900 — making it the first of Hilbert's 23 problems to be resolved — by introducing the Dehn invariant $\operatorname{Dehn}(P) = \sum_e \ell(e) \otimes [\theta(e)] \in \mathbb{R} \otimes_{\mathbb{Q}} (\mathbb{R}/\pi\mathbb{Q})$ , where $\ell(e)$ is the length and $\theta(e)$ the dihedral angle of each edge $e$ . Any finite polyhedral dissection preserves $\operatorname{Dehn}(P)$ ; a cube has $\operatorname{Dehn} = 0$ because its dihedral angles are $\pi/2$ , whereas a regular tetrahedron has $\theta = \arccos(1/3)$ , which is not a rational multiple of $\pi$ , giving $\operatorname{Dehn} \ne 0$ . Dehn's paper appeared in Mathematische Annalen in 1901.

Posed by, 1900David Hilbert

Sydler's 1965 theorem — simplified in 1968 by Børge Jessen using homological algebra — showed that volume and the Dehn invariant completely classify polyhedra up to scissors congruence in $\mathbb{R}^3$ , and Jessen–Thorup (1978) extended the result to $\mathbb{R}^4$ . In three-dimensional hyperbolic and spherical geometries, Dupont and Sah (1982) discovered deep connections between scissors congruence groups, the group homology of $\mathrm{SL}_2(\mathbb{C})$ , algebraic $K$ -theory, and the dilogarithm via the Bloch group; whether volume and the Dehn invariant suffice in hyperbolic 3-space remains an active question tied to Goncharov's conjectures on mixed Tate motives.

References

David Hilbert (1900). Mathematische Probleme
Max Dehn (1901). Ueber den Rauminhalt · DOI:10.1007/bf01448001
Jean-Pierre Sydler (1965). Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions · DOI:10.1007/bf02564364

History of the problem

1833Farkas Bolyai and Paul Gerwien complete the proof of the Wallace–Bolyai–Gerwien theorem: any two polygons of equal area in the plane are scissors-congruent.
1844Carl Friedrich Gauss writes to Christian Ludwig Gerling regretting that the volume of a pyramid in elementary geometry still depends on the method of exhaustion (an infinite limiting process) rather than a finite dissection.Carl Friedrich Gauss
1900David Hilbert poses the third problem at the Paris ICM; before the proceedings are even printed, his student Max Dehn proves that a cube and a regular tetrahedron of equal volume are not scissors-congruent.David Hilbert
1965Jean-Pierre Sydler proves the converse to Dehn's theorem: two polyhedra in $\mathbb{R}^3$ are scissors-congruent if and only if they have the same volume and the same Dehn invariant.