Hilbert's third problem
The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved.
The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?
Based on earlier writings by Carl Friedrich Gauss,[1] David Hilbert conjectured that this is not always possible.
This was confirmed within the year by his student Max Dehn, who proved that the answer in general is "no" by producing a counterexample.
[2] The answer for the analogous question about polygons in 2 dimensions is "yes" and had been known for a long time; this is the Wallace–Bolyai–Gerwien theorem.
Unknown to Hilbert and Dehn, Hilbert's third problem was also proposed independently by Władysław Kretkowski for a math contest of 1882 by the Academy of Arts and Sciences of Kraków, and was solved by Ludwik Antoni Birkenmajer with a different method than Dehn's.
Birkenmajer did not publish the result, and the original manuscript containing his solution was rediscovered years later.
[3] The formula for the volume of a pyramid, had been known to Euclid, but all proofs of it involve some form of limiting process or calculus, notably the method of exhaustion or, in more modern form, Cavalieri's principle.
Similar formulas in plane geometry can be proven with more elementary means.
Gauss regretted this defect in two of his letters to Christian Ludwig Gerling, who proved that two symmetric tetrahedra are equidecomposable.
[3] Gauss's letters were the motivation for Hilbert: is it possible to prove the equality of volume using elementary "cut-and-glue" methods?
Dehn's proof is an instance in which abstract algebra is used to prove an impossibility result in geometry.
Two polyhedra are called scissors-congruent if the first can be cut into finitely many polyhedral pieces that can be reassembled to yield the second.
A polyhedron's invariant is defined based on the lengths of its edges and the angles between its faces.
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
, and the angles introduced around an edge interior to the polyhedron add to
For some purposes, this definition can be made using the tensor product of modules over
(or equivalently of abelian groups), while other aspects of this topic make use of a vector space structure on the invariants, obtained by considering the two factors
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.
, measured in radians and considered modulo rational multiples of
Sydler (1965) showed that two polyhedra are scissors-congruent if and only if they have the same volume and the same Dehn invariant.
[4] Børge Jessen later extended Sydler's results to four dimensions.
[5] In 1990, Dupont and Sah provided a simpler proof of Sydler's result by reinterpreting it as a theorem about the homology of certain classical groups.
[6] Debrunner showed in 1980 that the Dehn invariant of any polyhedron with which all of three-dimensional space can be tiled periodically is zero.
However, it remains an open problem whether pairs of polyhedra with the same volume and the same Dehn invariant, in these geometries, are always scissors-congruent.
Dehn's invariant can be used to yield a negative answer also to this stronger question.
