Neighbors

TL;DR

This paper surveys various mathematical problems related to the concept of neighboring members in packings, including maximum, average, and mutual neighbors, highlighting historical debates and open questions.

Contribution

It provides a comprehensive overview of classical and modern problems concerning neighbors in packings, consolidating known results and identifying future research directions.

Findings

Historical dispute between Newton and Gregory on maximum neighbors

Summary of known bounds and results for average neighbors

Discussion of maximum mutually neighboring members

Abstract

Two members of a packing are neighbors if they have a common boundary point. A multitude of problems arises in connection with neighbors in a packing. The oldest one concerns a dispute between Newton and Gregory about the maximum number of neighbors a member can have in a packing of congruent balls. Other problems ask for the average number of neighbors or the maximum number of mutually neighboring members in a packing. The present work gives a survey of these problems.