On optimal $λ$-separable packings in the plane

TL;DR

The paper develops λ-separable packings of congruent disks in planes of constant curvature, bridging ordinary packings and totally separable packings. It harnesses refined Molnár decompositions, built from Delaunay tilings, to reduce density, tightness, and contact-number questions to extremal local configurations across Euclidean, spherical, and hyperbolic geometries. In the Euclidean plane, it delivers sharp density and tightness bounds, realized by lattice packings, while in spherical and hyperbolic geometries the bounds are near-optimal and tied to congruence classes of Delaunay cells; the work also provides a hyperbolic Molnár decomposition as a versatile tool. Overall, it extends classical packing theory to λ-separable packings with a unified, geometrically grounded framework and explicit extremal configurations.

Abstract

Let $\mathcal{P}$ be a packing of circular disks of radius $ρ>0$ in the Euclidean, spherical, or hyperbolic plane. Let $0\leqλ\leqρ$. We say that $\mathcal{P}$ is a $λ$-separable packing of circular disks of radius $ρ$ if the family $\mathcal{P'}$ of disks concentric to the disks of $\mathcal{P}$ having radius $λ$ form a totally separable packing, i.e., any two disks of $\mathcal{P'}$ can be separated by a line which is disjoint from the interior of every disk of $\mathcal{F'}$. This notion bridges packings of circular disks of radius $ρ$ (with $λ=0$) and totally separable packings of circular disks of radius $ρ$ (with $λ=ρ$). In this note we extend several theorems on the density, tightness, and contact numbers of disk packings and totally separable disk packings to $λ$-separable packings of circular disks of radius $ρ$ in the Euclidean, spherical, and hyperbolic plane. In particular, our upper bounds (resp., lower bounds) for the density (resp., tightness) of $λ$-separable packings of unit disks in the Euclidean plane are sharp for all $0\leqλ\leq 1$ with the extremal values achieved by $λ$-separable lattice packings of unit disks. On the other hand, the bounds of similar results in the spherical and hyperbolic planes are not sharp for all $0\leqλ\leqρ$ although they do not seem to be far from the relevant optimal bounds either. The proofs use local analytic and elementary geometry and are based on the so-called refined Molnár decomposition, which is obtained from the underlying Delaunay decomposition and as such might be of independent interest.

Figures (6)

• Figure 1: A densest $\lambda$-separable packing of unit disks in $\mathbb{E}^2$ with $\lambda=0.93$. The unit disks and the sides of the Delaunay triangles are denoted by solid lines. The concentric disks of radius $\lambda$ and the lines separating them are drawn with dotted and dashed lines, respectively. The bases of the Delaunay triangles are horizontal, and their length is greater than $2$. The legs of the Delaunay triangles are of length $2$.
• Figure 2: The Delaunay triangle $T$ of three $\lambda$-separable unit disks in $\mathbb{E}^2$ for $\lambda =0.5<\frac{\sqrt{3}}{2}$ (case (a)), $\lambda=0,9 \in \left( \frac{\sqrt{3}}{2}, \frac{2\sqrt{2}}{3} \right)$ (case (b)) and $\lambda=0.97 > \frac{2\sqrt{2}}{3}$ (case (c)). The disks of radius $\lambda$ centered at the vertices of the triangles, and the lines separating them are drawn by dotted and dashed lines, respectively. The triangle $T$ in case (a) is a regular triangle of edge length $2$, the one in case (b) is an isosceles triangle of edge lengths $2$, $2$ and $2\sqrt{2-2\sqrt{1-\lambda^2}}$, and the one in case (c) is an isosceles triangle with edge lengths $\frac{3\lambda}{\sqrt{2}}$, $\frac{3 \lambda}{\sqrt{2}}$ and $\sqrt{6} \lambda$.
• Figure 3: The union of the graphs of the functions $y_s^s$ and $y_b^s$ decomposes the region consisting of the points with the possible values of $(\lambda, y)$ into three connected components, described in the previous list, which determine the relative position of $y$, $x_1^s(y)$ and $x_2^s(y)$. The region of the possible points is bounded by black curves. The red curve is the union of the graphs of $y_s^s$ and $y_b^s$. The green segment is the curve $y=\frac{\pi}{4}$.
• Figure 4: An illustration for Subcase 2.2. The boundary of the region of the possible parameter values is drawn with black color. The red curve is the union of the graphs of $y_s^s$ and $y_b^s$. The green curves are the segment $y=\frac{\pi}{4}$, and the curve $y=\arcsin(\sqrt{2}\sin (\lambda))$. Observe that the last curve passes through the intersection of the segment $y=\frac{\pi}{2}-\lambda$ and the curve $y=y_b(\lambda)$.
• Figure 5: An illustration for the curves $\lambda$ (yellow), $x^h(y_{min}^h(\lambda))$ (red), $y_s^h(\lambda)$ (blue), $y_{min}^h(\lambda)$ (green) and $\mathop{\mathrm{arcsinh}}\nolimits(\sqrt{2}\sinh \lambda)$ (black) for a hyperbolic triangle.

Theorems & Definitions (38)

• Definition 1
• Remark 1
• Remark 2
• Definition 2
• Theorem 1
• Definition 3
• Theorem 2
• Definition 4
• Theorem 3
• Remark 3