On the maximum size packings of disks with kissing radius 3

TL;DR

This paper confirms a conjecture about the maximum number of congruent disks in a three-family packing with no interior overlaps, extending the classical kissing number problem.

Contribution

It proves that the maximum number of disks in a three-family packing touching a central disk is 37, confirming a conjecture by Fejes Tóth and Heppes.

Findings

Maximum packing size for three families is 37 disks.

Confirms the conjecture by Fejes Tóth and Heppes.

Extends the understanding of kissing number generalizations.

Abstract

L\'{a}szl\'{o} Fejes T\'{o}th and Alad\'{a}r Heppes proposed the following generalization of the kissing number problem. Given a ball in $\mathbb{R}^d$, consider a family of balls touching it, and another family of balls touching the first family. Find the maximal possible number of balls in this arrangement, provided that no two balls intersect by interiors, and all balls are congruent. They showed that the answer for disks on the plane is $19$. They also conjectured that if there are three families of disks instead of two, the answer is $37$. In this paper we confirm this conjecture.