Regular Totally Separable Sphere Packings

Abstract

The topic of totally separable sphere packings is surveyed with a focus on regular constructions, uniform tilings, and contact number problems. An enumeration of all regular totally separable sphere packings in ${\mathbb{R}}^2$, ${\mathbb{R}}^3$, and ${\mathbb{R}}^4$ which are based on convex uniform tessellations, honeycombs, and tetracombs, respectively, is presented, as well as a construction of a family of regular totally separable sphere packings in ${\mathbb{R}}^d$ that is not based on a convex uniform $d$-honeycomb for $d\ge 3$.

10 pages, 1 figure