In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem by proving that certain polyhedra with equal volume cannot 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. Having Dehn invariant zero is a necessary (but not sufficient) condition for being a space-filling polyhedron, and a polyhedron can be cut up and reassembled into a space-filling polyhedron if and only if its Dehn invariant is zero. The Dehn invariant of a self-intersection-free flexible polyhedron is invariant as it flexes. Dehn invariants are also an invariant for dissection in higher dimensions, and (with volume) a complete invariant in four dimensions.

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. In particular, the truncated octahedron also tiles space and has Dehn invariant zero like the cube.

The Dehn invariants of polyhedra are not numbers. Instead, they are elements of an infinite-dimensional tensor space. This space, viewed as an abelian group, is part of an exact sequence involving group homology. Similar invariants can also be defined for some other dissection puzzles, including the problem of dissecting rectilinear polygons into each other by axis-parallel cuts and translations.

In two dimensions, the Wallace–Bolyai–Gerwien theorem from the early 19th century states that any two polygons of equal area can be cut up into polygonal pieces and reassembled into each other. In the late 19th century, David Hilbert became interested in this result. He used it as a way to axiomatize the area of two-dimensional polygons, in connection with Hilbert's axioms for Euclidean geometry. This was part of a program to make the foundations of geometry more rigorous, by treating explicitly notions like area that Euclid's Elements had handled more intuitively. Naturally, this raised the question of whether a similar axiomatic treatment could be extended to solid geometry.

At the 1900 International Congress of Mathematicians, Hilbert formulated Hilbert's problems, a set of problems that became very influential in 20th-century mathematics. One of those, Hilbert's third problem, addressed this question on the axiomatization of solid volume. Hilbert's third problem asked, more specifically, whether every two polyhedra of equal volumes can always be cut into polyhedral pieces and reassembled into each other. If this were the case, then the volume of any polyhedron could be defined, axiomatically, as the volume of an equivalent cube into which it could be reassembled. However, the answer turned out to be negative: not all polyhedra can be dissected into cubes.

Unlike some of the other Hilbert problems, the answer to the third problem came very quickly. In fact, Raoul Bricard had already claimed it as a theorem in 1896, but with a proof that turned out to be incomplete. Hilbert's student Max Dehn, in his 1900 habilitation thesis, invented the Dehn invariant in order to solve this problem. Dehn proved that, to be reassembled into each other, two polyhedra of equal volume should also have equal Dehn invariant, but he found two tetrahedra of equal volume whose Dehn invariants differed. This provided a negative solution to the problem. Although Dehn formulated his invariant differently, the modern approach to Dehn's invariant is to describe it as a value in a tensor product, following Jessen (1968).

Defining the Dehn invariant in a way that can apply to all polyhedra simultaneously involves infinite-dimensional vector spaces (see § Full definition, below). However, when restricted to any particular example consisting of finitely many polyhedra, such as the Platonic solids, it can be defined in a simpler way, involving only a finite number of dimensions, as follows: