Contact numbers for sphere packings

TL;DR

This paper surveys the contact number problem in sphere packings, exploring special cases, complexity issues, and open questions relevant to mathematics and materials science.

Contribution

It provides a comprehensive overview of contact numbers in sphere packings, emphasizing special cases and discussing recognition complexity and open problems.

Findings

Analysis of contact numbers in various sphere packing configurations

Discussion of complexity in recognizing contact graphs

Listing of conjectures and open problems in the field

Abstract

In discrete geometry, the contact number of a given finite number of non-overlapping spheres was introduced as a generalization of Newton's kissing number. This notion has not only led to interesting mathematics, but has also found applications in the science of self-assembling materials, such as colloidal matter. With geometers, chemists, physicists and materials scientists researching the topic, there is a need to inform on the state of the art of the contact number problem. In this paper, we investigate the problem in general and emphasize important special cases including contact numbers of minimally rigid and totally separable sphere packings. We also discuss the complexity of recognizing contact graphs in a fixed dimension. Moreover, we list some conjectures and open problems.