Penny graphs in the hyperbolic plane

TL;DR

This work analyzes penny graphs arising from packings of congruent circles in the hyperbolic plane, focusing on the maximum number $e_d(n)$ of touching pairs and its asymptotic density $c(d)=\lim_{n\to\infty} e_d(n)/n$. Using a combination of spiral constructions tied to hyperbolic tilings, Euler–Gauss–Bonnet type area bounds, and isoperimetric arguments, the authors derive a suite of upper bounds $\gamma_i(d)$ that sharpen the trivial $2\le c(d)\le 3$ and prove tight results for distances $d(k)$ corresponding to order-$k$ tilings; they establish a self-contained Bowen-type upper bound and extend the analysis to subgraphs of $\{p,q\}$-tilings. They also provide nontrivial lower bounds via constructive spiral packings, showing $c(d)\ge \gamma_6(d)$ with $\gamma_6(d)=2+\frac{1}{4q-14}=2+\frac{\alpha}{8\pi}+O(\alpha^2)$ as $d\to\infty$, and demonstrate that $\lim_{d\to\infty} c(d)=2$. The results imply that, except for a small set of distances, the order-$7$ tiling spiral is extremal among penny graphs in the hyperbolic plane, and they discuss the horocycle case as a complementary extremal scenario.

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 $d > 0$.

Figures (6)

• Figure 1: Upper and lower bounds for $c(d)$. The red upper bound $\gamma_1(d)$ holds for all $d$ and is tight for the order-$k$ triangular tilings (the dots on the red graph). The orange $\gamma_2(d)$ and green $\gamma_3(d)$ bounds hold for all $d\neq d(k)$. The bound $\gamma_3(\overline{d}(6))=2.39698265738619\dots$ is the best we have for $d=\overline{d}(6)$ (see the dot on the green curve), and the bounds $\gamma_2(\overline{d}(k))$ are the best for $d=\overline{d}(k)$, $k\geq 7$ (see the dots on the orange curve). The blue $\gamma_4(d)$ and purple $\gamma_5(d)$ upper bounds hold for all $d$ such that $d\neq d(k)$, $k\geq 7$ and $d\neq\overline{d}(k)$, $d\geq 6$. These curves intersect at $d=1.1128036956703866\dots$. Apart from the semiregular tiling $d=\overline{d}(6)$, we have that Conjecture \ref{['conj3']} holds for all $d > 0.6611380871710578\dots$, the solution of $\gamma_5(d)=3-1/\varphi$ (shown by the dot on the purple curve). The vertical gray lines are at the values where $d=\overline{d}(k)$. The horizontal gray lines are at $3-1/\varphi$ and $2.75$. The black step function is the lower bound $\gamma_6(d)$.
• Figure 2:
• Figure 3: A spiral graph in the order-7 triangular tiling on 17 vertices. (Adapted from a figure from Wikipedia Taxel)
• Figure 4: The arrow indicates which edge is associated to an angle
• Figure 5: First step in lower bound construction

Theorems & Definitions (36)

• Theorem 2.1
• Theorem 2.2
• Conjecture 2.3
• Proposition 2.4
• Conjecture 2.5
• Conjecture 2.6
• Theorem 2.7
• Proposition 2.8
• proof
• Lemma 3.1