Note for the Third Hilbert Problem: a Fractal Construction

Abstract

Hilbert’s Third problem questioned whether, given two polyhedrons with the same volume, it is possible to decompose the first one into a finite number of polyhedral parts that can be put together to yield the second one. This finite equidecomposition process had already been shown to be possible between polygons of the same area. Dehn solved the problem by showing that a regular tetrahedron and a cube with equal volume were not equidecomposable. In this paper, we present an infinite fractal process that allows the cube to be visually reconstructed from a tetrahedron with equal volume. We have proved that, given two tetrahedrons with the same volume, the first one can be decomposed into an infinite number of polyhedral parts that can be put together to yield the second one. This process makes it possible to obtain the volume of a tetrahedron from the volume of the parallelepiped, without the use of formulas or the Cavallieri Principle