On Limits of Dense Packing of Equal Spheres in a Cube

TL;DR

This paper investigates the densest arrangements of equal spheres in a cube near the cubic close-packed configuration, introducing an optimization method that improves known packings for a wide range of sphere counts.

Contribution

It presents a new optimization approach that finds denser sphere packings in a cube for all n up to a certain limit, surpassing previous arrangements.

Findings

Better sphere packings found for all n ≤ ⌈p³/2⌉ - 2

New arrangements improve previous configurations for n ≤ 4629

Optimization method reveals denser packings beyond known structures

Abstract

We examine packing of $n$ congruent spheres in a cube when $n$ is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangement of $\lceil p^{3}/2\rceil$ spheres. For this family of packings, the previous best-known arrangements were usually derived from a ccp by omission of a certain number of spheres without changing the initial structure. In this paper, we show that better arrangements exist for all $n\leq\lceil p^{3}/2\rceil-2$. We introduce an optimization method to reveal improvements of these packings, and present many new improvements for $n\leq4629$.