Hilbert’s third problem: decomposing polyhedra

Abstract

This problem can be traced back to two letters of Carl Friedrich Gauss from 1844 (published in Gauss’ collected works in 1900). If tetrahedra of equal volume could be split into congruent pieces, then this would give one an “elementary” proof of Euclid’s theorem XII.5 that pyramids with the same base and height have the same volume. It would thus provide an elementary definition of the volume for polyhedra (that would not depend on continuity arguments).