Anna Persson Nygårdh: Hilberts tredje problem

Abstract.

The German mathematician David Hilbert (1862-1943) presented a list with 23 unsolved problems in mathematics year 1900. This paper addresses Hilbert’s third problem and the solution to the problem that was presented by Max Dehn, a student to David Hilbert, using the Dehn invariant. Hilbert’s third problem is a geometric question and concerns whether two tetrahedra with the same base and height can be divided into a finite number of pieces that form congruent tetrahedra, or if they can be combined with other tetrahedra to create new polyhedral that can, in turn, be divided in a similar way. This paper discusses mathematical concepts such as rational numbers, irrational numbers, vector spaces, polyhedra and dihedral angels, and explains how these concepts are used to define and calculate the Dehn invariant. A theorem presented in this paper shows how the Dehn invariant can be used to analyze whether two polyhedra are equicomplementary, which is crucial for understanding Hilbert’s third problem. In the end, examples are provided where the Dehn invariant is calculated; these examples give a negative solution to Hilbert’s third problem.