Dehn invariant

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.

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.

[1] Naturally, this raised the question of whether a similar axiomatic treatment could be extended to solid geometry.

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.

In fact, Raoul Bricard had already claimed it as a theorem in 1896, but with a proof that turned out to be incomplete.

[5][6] 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:[7] Although this method involves arbitrary choices of basis elements, these choices affect only the coefficients by which the Dehn invariants are represented.

The vector space spanned by the Dehn invariants of any finite set of polyhedra forms a finite-dimensional subspace of the infinite-dimensional vector space in which the Dehn invariants of all polyhedra are defined.

Each set of four parallel edges in a parallelepiped have the same length and have dihedral angles summing to

If a new edge is introduced in this cutting process, then either it is interior to the polyhedron, and surrounded by dihedral angles totaling

The new dihedral angles on that edge combine to the same sum, and the same contribution to the Dehn invariant, that they had before.

[14][15] The reverse of this is not true – there exist polyhedra with Dehn invariant zero that do not tile space.

[17] More generally, if some combination of polyhedra jointly tiles space, then the sum of their Dehn invariants (taken in the same proportion) must be zero.

[b] The definition of the Dehn invariant requires a notion of a polyhedron for which the lengths and dihedral angles of edges are well defined.

Most commonly, it applies to the polyhedra whose boundaries are piecewise linear manifolds, embedded on a finite number of planes in Euclidean space.

[18] The values of the Dehn invariant belong to an abelian group[19] defined as the tensor product

The left factor of this tensor product is the set of real numbers (in this case representing lengths of edges of polyhedra) and the right factor represents dihedral angles in radians, given as numbers modulo rational multiples of 2π.

Its structure as a tensor gives the Dehn invariant additional properties that are geometrically meaningful.

[22] An alternative but equivalent description of the Dehn invariant involves the choice of a Hamel basis, an infinite subset

[23] This alternative formulation shows that the values of the Dehn invariant can be given the additional structure of a real vector space.

[24] Although, in general, the construction of Hamel bases involves the axiom of choice, this can be avoided (when considering any specific finite set of polyhedra) by restricting attention to the finite-dimensional vector space generated over

[4] For an ideal polyhedron in hyperbolic space, the edge lengths are infinite, making the usual definition of the Dehn invariant inapplicable.

The result does not depend on the choice of horospheres for the truncation, as long as each one cuts off only a single vertex of the given polyhedron.

The Dehn invariants of Euclidean polyhedra form a real linear subspace of

is the group of Möbius transformations; the superscript minus-sign indicates the (−1)-eigenspace for the involution induced by complex conjugation.

[22] This invariant can be used to prove another result of Dehn from 1903: two rectangles of the same area can be dissected into each other if and only if their aspect ratios are rational multiples of each other.

For this version of the Dehn invariant, the tensor rank equals the minimum number of rectangles into which a polygon can be dissected.

[33] However, for more complicated flexible polyhedra with self-intersections the Dehn invariant may change continuously as the polyhedron flexes.

Dissection of a square and equilateral triangle into each other. No such dissection exists for the cube and regular tetrahedron .
Dissection of a cube into orthoschemes . In the cube, each new edge introduced in this dissection is surrounded by dihedral angles that sum to (for the face diagonals) or (for the body diagonal), so the total contribution to the Dehn invariant from these edges is zero.
Three-piece dissection of a Greek cross to a rectangle, using only axis-parallel cuts and translations. A Dehn-like invariant shows that neither of these shapes can be dissected to a square, with this kind of restricted dissection.