Bounding the density of binary sphere packing

TL;DR

This paper establishes the best known upper bound on the density of a binary sphere packing with a specific size ratio, using advanced computational methods to analyze tetrahedral decompositions.

Contribution

It introduces a new upper bound for binary sphere packing density in 3D space, derived through computationally intensive interval arithmetic calculations.

Findings

Established the tightest known upper bound for binary sphere packing density.

Utilized additively-weighted Delaunay decomposition for geometric analysis.

Demonstrated the effectiveness of computer-assisted proofs in geometric bounds.

Abstract

This paper provides the currently best known upper bound on the density of a packing in three-dimensional Euclidean space of two types of spheres whose size ratio is the largest one that allows the insertion of a small sphere in each octahedral hole of a hexagonal compact packing of large spheres. This upper bound is obtained by bounding from above the density of the tetrahedra which can appear in the additively-weighted Delaunay decomposition of the sphere centers of such packings. The proof relies on challenging computer calculations in interval arithmetic and may be of interest by their own.