Regular Totally Separable Sphere Packings

TL;DR

This paper surveys regular totally separable sphere packings, providing classifications in low dimensions and constructing new packings in higher dimensions beyond known uniform tessellations.

Contribution

It offers a complete enumeration of regular totally separable sphere packings in dimensions 2 to 4 and introduces a novel family of such packings in higher dimensions.

Findings

Enumerated all regular totally separable packings in $ ext{R}^2$, $ ext{R}^3$, and $ ext{R}^4$.

Constructed a new family of packings in $ ext{R}^d$ not based on convex uniform tessellations.

Analyzed contact number problems for these 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 $d \geq 3$.