On Limits of Dense Packing of Equal Spheres in a Cube

Milos Tatarevic

doi:10.37236/3784

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 $n\leq4629$.

Highlights

We consider the problem of finding the densest packings of congruent, non-overlapping, spheres in a cube
We provide a lower bound for these improvements
We show that the described procedure can be used as a good packing method when n is slightly smaller than g(p)

Summary

Introduction

We consider the problem of finding the densest packings of congruent, non-overlapping, spheres in a cube. Optimality of dn was conjectured for an infinite family of packings where dp3 /2e spheres are arranged in a cubic close-packed (ccp) structure [3]. √ of spheres in these packings, with a maximum separation distance denoted by dp = 2/(p − 1). We examine arrangements when n is close, but less than g(p) For this family of packings dn = d0p is often assumed, to mean that the densest known arrangements are derived from ccp by omission of a certain number of spheres without changing the initial structure. We provide a lower bound for these improvements.

Existence of improved packings

Now we can give Tp as

We denote the lower bound of improvements by

Optimization Approach