On Hilbert’s third problem

Extract

The year 2000 was the centenary of not only Hubert’s Problems but also Dehn’s solution of the Third Problem, which was the first to be solved. The Third Problem is concerned with the Euclidean theorem that two tetrahedra with equal base and height have equal volume [5, Book XII, Proposition 5].

Today one proves this theorem by integration, showing that the volume of a tetrahedron is a third base times height. This 3-dimensional theorem is the analogue of the 2-dimensional theorem that the area of a triangle is a half base times height. But Euclid did not have access to integration, nor the real numbers, nor the product of real numbers, so he did not use the concept of an area being the product of two lengths or a volume being the product of a numerical area and a length. For instance when he wanted to state the triangle theorem he said that two copies of the triangle were ‘equal’ to a rectangle with the same base and height, where by ‘equal’ he meant here that the two triangles could be cut into pieces and reassembled into the rectangle. So Hilbert’s problem was equivalent to asking whether the same could be said about a tetrahedron. Can three copies of the tetrahedron be cut into pieces and reassembled into a rectangular prism with the same base and height? Dehn proved that it was impossible, and later Boltianskii simplified his proof. Dehn’s proof invented a real number invariant of a solid, that was invariant under cutting and reassembling, and he showed that two particular tetrahedra with the same base and height had different values of his invariant.