Dissections of Polyhedra and Hilbert's third problem

During the second international congress of mathematicians, held in Paris, at one of the sessions, the famous mathematician David Hilbert read his well-known list of mathematical problems. Hilbert is without doubt one of the most prominent mathematicians in history, he had a very broad and profound knowledge of mathematics and physics. He thought of his 23 problems as good challenges, that could motivate mathematical research for the new century. Most of the problems he proposed ask to develop general programs. The third problem was, first of all, looked for a counterexample rather than a large theory. Second it simple to state understandable to anybody with a rudimentary acquaintance with geometry.

In Euclidean plane geometry, areas of polygons can be computed through a finite process of cutting and pasting (scissors congruence). Hilbert did not believe that a general theory of volume can be based on the idea of cutting and pasting. Here is a partial translation of Hilbert's statement of the problem: polyhedra.

``Gauss mentions in particular the theorem of Euclid that tetrahedra of equal altitudes their volumes are proportional as their bases. ... Gerling succeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probably that a general proof of the kind for the Euclid's theorem is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained as soon as we specify two tetrahedra of equal bases and altitudes which can not be split up into congruent tetrahedra and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra.''

Curiously the following two tetrahedra with vertices {(0,0,0),(1,0,0),(0,1,0),(0,0,1)} and {(0,0,0),(1,0,0),(0,1,0),(0,1,1)} can be seen to be counterexamples to the problem. Max Dehn found the solution a few months after it was posed. Dehn's work extended and corrected earlier works of Bricard, he constructed an algebraic invariant associated to a polyhedron that essentially tells apart those that are congruent by scissors cuts.