On Limits of Dense Packing of Equal Spheres in a Cube

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 $n\le \lceil p^3/2\rceil -2$. We introduce an optimization method to reveal improvements of these packings, and present many new improvements for $n\le 4629$.