The Strong Dodecahedral Conjecture and Fejes Toth's Conjecture on Sphere Packings with Kissing Number Twelve

TL;DR

This paper presents proofs of two conjectures in sphere packings in three-dimensional space, establishing lower bounds on Voronoi cell surface area and characterizing packings with twelve contacts per sphere, using computer-assisted methods.

Contribution

It provides the first complete proofs of the strong dodecahedral conjecture and Fejes Toth's contact conjecture in 3D sphere packings, employing computer-assisted techniques.

Findings

Voronoi cells have surface area at least that of a regular dodecahedron.

Packings with each sphere touched by twelve others form hexagonal layers.

Both theorems are proved with computer assistance, confirming longstanding conjectures.

Abstract

This article sketches the proofs of two theorems about sphere packings in Euclidean 3-space. The first is K. Bezdek's strong dodecahedral conjecture: the surface area of every bounded Voronoi cell in a packing of balls of radius 1 is at least that of a regular dodecahedron of inradius 1. The second theorem is L. Fejes Toth's contact conjecture, which asserts that in 3-space, any packing of congruent balls such that each ball is touched by twelve others consists of hexagonal layers. Both proofs are computer assisted. Complete proofs of these theorems appear in the author's book "Dense Sphere Packings" and a related preprint