Penny graphs in the hyperbolic plane

TL;DR

This paper investigates the maximum number of touching pairs in packings of congruent circles in the hyperbolic plane, providing bounds and conjectures on extremal configurations related to hyperbolic tilings.

Contribution

It introduces new bounds for touching pairs in hyperbolic circle packings and conjectures the extremal configuration based on hyperbolic tilings.

Findings

Upper bounds for touching pairs depending on circle diameter.

Lower bounds showing linear growth in the number of touching pairs.

Identification of specific diameter ranges where certain configurations are optimal.

Abstract

We consider the problem of finding the maximum number $e_d(n)$ of pairs of touching circles in a packing of $n$ congruent circles of diameter $d$ in the hyperbolic plane of curvature $-1$. In the Euclidean plane, the maximum comes from a spiral construction of the tiling of the plane with equilateral triangles (Harborth 1974), with a similar result in the hyperbolic plane for the values of $d$ corresponding to the order-$k$ triangular tilings (Bowen 2000). We present various upper and lower bounds for $e_d(n)$ for all values of $d > 0$. In particular, we prove that if $d > 0.66114\dots$ except for $d=0.76217\dots$, then the number of touching pairs is less than the one coming from a spiral construction in the order-$7$ triangular tiling, which we conjecture to be extremal. We also give a lower bound $e_d(n) > (2+\varepsilon_d)n$ where $\varepsilon_d > 1$ for all $d > 0$.