The Dodecahedron as a Voronoi Cell and its (minor) importance for the Kepler conjecture

TL;DR

This paper investigates the role of the dodecahedron as a Voronoi cell in sphere packing and concludes it has minor importance for the proof of the Kepler conjecture, despite initial geometric considerations.

Contribution

The paper demonstrates that the dodecahedral configuration leads to larger tetrahedra than fcc packing, showing its limited relevance to the Kepler conjecture.

Findings

Dodecahedral Voronoi cell has minor importance for Kepler conjecture.

Icosahedral configuration from dodecahedron results in larger tetrahedra.

Tetrahedral perspective suggests dodecahedron's limited role in sphere packing density.

Abstract

The regular dodecahedron has a 2% smaller volume than the rhombic dodecahedron which is the Voronoi cell of a fcc packing. From this point of view it seems possible that the dodecahedral aspect which is the core of the so-called dodecahedral conjecture, will play a major part for an elementary proof of the Kepler conjecture. In this paper we will show that the icosahedral configuration caused by dodecahedron leads to tetrahedra with significantly larger volume than the fcc fundamental parallelotope tessellation tetrahedra. Therefore on the basis of a tetrahedral based point of view for sphere packing densities we will demonstrate the minor importance of the dodecahedron as a Voronoi cell for the Kepler conjecture.