Upper Bounds for Non-Congruent Sphere Packings

TL;DR

This paper establishes upper bounds for the average kissing number, contact number, and packing density in finite and infinite non-congruent sphere packings, advancing understanding of their geometric constraints.

Contribution

It provides the first known upper bounds for these parameters specifically for non-congruent sphere packings, extending classical results to more general configurations.

Findings

Upper bounds on average kissing number for finite packings

Upper bounds on contact number for finite packings

Upper bounds on packing density for infinite packings

Abstract

We prove upper bounds on the average kissing number $k(\mathcal{P})$ and contact number $C(\mathcal{P})$ of an arbitrary finite non-congruent sphere packing $\mathcal{P}$, and prove an upper bound on the packing density $\delta(\mathcal{P})$ of an arbitrary infinite non-congruent sphere packing $\mathcal{P}$.