Editing Hilbert's third problem (section)

== Dehn's proof ==
{{main article|Dehn invariant}}
Dehn's proof is an instance in which [[abstract algebra]] is used to prove an impossibility result in [[geometry]]. Other examples are [[doubling the cube]] and [[trisecting the angle]].

Two polyhedra are called {{anchor|1=Scissors congruence|2=Scissors-congruent}}''scissors-congruent'' if the first can be cut into finitely many polyhedral pieces that can be reassembled to yield the second. Any two scissors-congruent polyhedra have the same volume. Hilbert asks about the [[Converse (logic)|converse]].

For every polyhedron <math>P</math>, Dehn defines a value, now known as the [[Dehn invariant]] <math>\operatorname{D}(P)</math>, with the property that,
if <math>P</math> is cut into polyhedral pieces <math>P_1, P_2, \dots P_n</math>, then
<math display=block>\operatorname{D}(P) = \operatorname{D}(P_1)+\operatorname{D}(P_2)+\cdots + \operatorname{D}(P_n).</math>
In particular, if two polyhedra are scissors-congruent, then they have the same Dehn invariant. He then shows that every [[cube]] has Dehn invariant zero while every regular [[tetrahedron]] has non-zero Dehn invariant. Therefore, these two shapes cannot be scissors-congruent.

A polyhedron's invariant is defined based on the lengths of its edges and the angles between its faces. If a polyhedron is cut into two, some edges are cut into two, and the corresponding contributions to the Dehn invariants should therefore be additive in the edge lengths. Similarly, if a polyhedron is cut along an edge, the corresponding angle is cut into two. Cutting a polyhedron typically also introduces new edges and angles; their contributions must cancel out. The angles introduced when a cut passes through a face add to <math>\pi</math>, and the angles introduced around an edge interior to the polyhedron add to <math>2\pi</math>. Therefore, the Dehn invariant is defined in such a way that integer multiples of angles of <math>\pi</math> give a net contribution of zero.

All of the above requirements can be met by defining <math>\operatorname{D}(P)</math> as an element of the [[tensor product]] of the [[real number]]s <math>\R</math> (representing lengths of edges) and the [[Quotient space (linear algebra)|quotient space]] <math>\R/(\Q\pi)</math> (representing angles, with all rational multiples of <math>\pi</math> replaced by zero).<ref name=eom>{{SpringerEOM|first=M.|last=Hazewinkel|authorlink= Michiel Hazewinkel |title=Dehn invariant|id=Dehn_invariant&oldid=35803}}</ref> For some purposes, this definition can be made using the [[tensor product of modules]] over <math>\Z</math> (or equivalently of [[abelian group]]s), while other aspects of this topic make use of a [[vector space]] structure on the invariants, obtained by considering the two factors <math>\R</math> and <math>\R/(\Q\pi)</math> to be vector spaces over <math>\Q</math> and taking the [[Tensor product|tensor product of vector spaces]] over <math>\Q</math>. This choice of structure in the definition does not make a difference in whether two Dehn invariants, defined in either way, are equal or unequal.

For any edge <math>e</math> of a polyhedron <math>P</math>, let <math>\ell(e)</math> be its length and let <math>\theta(e)</math> denote the [[dihedral angle]] of the two faces of <math>P</math> that meet at <math>e</math>, measured in [[radian]]s and considered modulo rational multiples of <math>\pi</math>. The Dehn invariant is then defined as
<math display=block>\operatorname{D}(P) = \sum_{e} \ell(e)\otimes \theta(e)</math>
where the sum is taken over all edges <math>e</math> of the polyhedron <math>P</math>.<ref name=eom /> It is a [[Valuation (geometry)|valuation]].