Finding the Best Face on a Voronoi Polyhedron – The Strong Dodecahedral Conjecture Revisited

Abstract.

In this paper we prove the following theorem. The surface area density of a unit ball in any face cone of a Voronoi cell in an arbitrary packing of unit balls of Euclidean 3-space is at most ${-9\pi + 30\,{\rm arccos}\left({\sqrt{3}\over 2}{\rm sin}\,\left({\pi\over 5}\right) \right)\over 5\, {\rm tan}\left({\pi\over 5}\right)}=0.77836\ldots,$ and so the surface area of any Voronoi cell in a packing with unit balls in Euclidean 3-space is at least ${20\pi\cdot\,{\rm tan}\,\left( {\pi \over 5}\right) \over -9\pi + 30\,{\rm arccos}\left({\sqrt{3}\over 2}{\rm sin}\,\left({\pi\over 5}\right) \right)}=16.1445\ldots\ .$

This result and the ideas of its proof support the Strong Dodecahedral Conjecture according to which the surface area of any Voronoi cell in a packing with unit balls in Euclidean 3-space is at least as large as 16.6508..., the surface area of a regular dodecahedron of inradius 1.