Math Capstone

West Virginia University Mathematics Capstone Project

May 2023

Abstract

This project traces a modern take on a solution to Hilbert’s Third Problem. It was shown in 1832 by Farkas Bolyai that any polygon can be cut into triangles and rearranged without any deformation into any other polygon of equal area. In 1900, David Hilbert published his third question: can any polyhedron be decomposed into any other polyhedron of equal volume? Only two years later, Hilbert’s student, Max Dehn, created an invariant that was necessary to be the same between polyhedra to be decomposed into one another however did not fully determine which polyhedra could be decomposed into each other. This problem remained unsolved until 1965 when J.P. Sydler showed that having equal volume and equal Dehn Invariants is necessary and sufficient to determine that two polyhedra are decomposable into one another. Børge Jessen simplified this proof in 1968. This is the proof that I trace in this project with some clarifying steps from a 1986 paper by Pierre Cartier in Seminaire Bourbaki.

Full Paper