Dehn invariant

In geometry, the Dehn invariant of a polyhedron is a value used to determine whether polyhedra can be dissected into each other or whether they can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem on whether all polyhedra with equal volume could be dissected into each other.

Two polyhedra have a dissection into polyhedral pieces that can be reassembled into either one, if and only if their volumes and Dehn invariants are equal. A polyhedron can be cut up and reassembled to tile space if and only if its Dehn invariant is zero, so having Dehn invariant zero is a necessary condition for being a space-filling polyhedron. It is also an open problem wheather the Dehn invariant of a self-intersection free flexible polyhedron is invariant as it flexes.

The Dehn invariant is zero for the cube but nonzero for the other Platonic solids, implying that the other solids cannot tile space and that they cannot be dissected into a cube. All of the Archimedean solids have Dehn invariants that are rational combinations of the invariants for the Platonic solids.

The Dehn invariants of polyhedra are elements of an infinite-dimensional vector space. As an abelian group, this space is part of an exact sequence involving group homology. Similar invariants can also be defined for some other dissection puzzles, including as the problem of dissecting rectilinear polygons into each other by axis-parallel cuts and translations.

In two dimensions, the Wallace–Bolyai–Gerwien theorem states that any two polygons of equal area can be cut up into polygonal pieces and reassembled into each other. David Hilbert became interested in this result as a way to axiomatize area, in connection with Hilbert's axioms for Euclidean geometry. In Hilbert's third problem, he posed the question of whether two polyhedra of equal volumes can always be cut into polyhedral pieces and reassembled into each other. Hilbert's student Max Dehn, in his 1900 habilitation thesis, invented the Dehn invariant in order to provide a negative solution to Hilbert's problem. Although Dehn formulated his invariant differently, the modern approach is to describe it as a value in a tensor product, following Jessen (1968).

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