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On the emergence of almost-honeycomb structures in low-energy planar clusters

Marco Caroccia, Kenneth DeMason, Francesco Maggi

Abstract

Several commonly observed physical and biological systems are arranged in shapes that closely resemble an honeycomb cluster, that is, a tessellation of the plane by regular hexagons. Although these shapes are not always the direct product of energy minimization, they can still be understood, at least phenomenologically, as low-energy configurations. In this paper, explicit quantitative estimates on the geometry of such low-energy configurations are provided, showing in particular that the vast majority of the chambers must be generalized polygons with six edges, and be closely resembling regular hexagons. Part of our arguments is a detailed revision of the estimates behind the global isoperimetric principle for honeycomb clusters due to Hales (T. C. Hales. The honeycomb conjecture. Discrete Comput. Geom., 25(1):1-22, 2001).

On the emergence of almost-honeycomb structures in low-energy planar clusters

Abstract

Several commonly observed physical and biological systems are arranged in shapes that closely resemble an honeycomb cluster, that is, a tessellation of the plane by regular hexagons. Although these shapes are not always the direct product of energy minimization, they can still be understood, at least phenomenologically, as low-energy configurations. In this paper, explicit quantitative estimates on the geometry of such low-energy configurations are provided, showing in particular that the vast majority of the chambers must be generalized polygons with six edges, and be closely resembling regular hexagons. Part of our arguments is a detailed revision of the estimates behind the global isoperimetric principle for honeycomb clusters due to Hales (T. C. Hales. The honeycomb conjecture. Discrete Comput. Geom., 25(1):1-22, 2001).
Paper Structure (11 sections, 6 theorems, 238 equations, 3 figures)

This paper contains 11 sections, 6 theorems, 238 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Isoperimetric clusters and Hales' theorem
  4. Low-energy clusters and main result
  5. Hales' hexagonal isoperimetric inequality and strategy of proof
  6. Organization of the paper
  7. From the quantitative Hales' inequality to the main theorem
  8. The arc function
  9. Proof of the quantitative Hales' hexagonal inequality, part one
  10. Proof of the quantitative Hales' hexagonal inequality, part two
  11. The isoperimetric inequality for immersed polygons

Key Result

Theorem 1.2

There exists a computable constant $C_0$ with the following property. If $N\ge 2$ and ${\mathcal{E}}\in\mathcal{C}(N,M)$, then with and Finally, there is at least one $k\le 5$ such that ${\rm Ch}_k({\mathcal{E}})\ne\varnothing$.

Figures (3)

  • Figure 1.1: The idea behind the low-energy condition \ref{['M low']} is that it identifies unit-area $N$-clusters whose "internal perimeter" is comparable to that of an $\sqrt{N}\times\sqrt{N}$-chunk of ideal honeycomb, and whose "external perimeter" is comparable to $\sqrt{N}$ (i.e., the square root of the area of the bulk of the cluster). Unit-area locally minimizing clusters may fail to satisfy this condition. For example, the $N$-cluster depicted here satisfies, for some $r_0>0$, the local isoperimetric property $P({\mathcal{E}})\le P({\mathcal{F}})$ for every ${\mathcal{F}}$ with $|{\mathcal{F}}|=|{\mathcal{E}}|$ with $\mathop{\mathrm{diam}}\limits({\mathcal{F}}(h)\Delta{\mathcal{E}}(h))\le r_0$, $h=1,...,N$; but it does not satisfy \ref{['M low']} -- as it can be seen, for example, by looking at its external perimeter, which is ${\rm O}(N)$, compare with \ref{['external perimeter']} in Theorem \ref{['thm main']}.
  • Figure 1.2: In the example in the picture, the quantity $\alpha(\gamma,s,t)$ defined in \ref{['alfa']} is obtained by subtracting the areas depicted in light grey from the areas depicted in dark grey.
  • Figure 3.1: (a) ${\rm arc}(\ell,x)$ is defined as the length of a circular arc (depicted in bold) subtending a chord of length $\ell$ and including a secant area $x$; (b) An implicit formula for ${\rm arc}_1$ on the interval $[0,\pi/2]$ can be obtained by referring to this picture. The second argument of ${\rm arc}$ in \ref{['implicit']} is obtained by subtracting the area of a rectangle with sidelengths $R\,\sin\theta$ and $R\,\cos\theta$ from the area $\theta\,R^2$ of an angular sector whose amplitude equals $2\,\theta$ radians.

Theorems & Definitions (17)

  • Remark 1.1: The class $\mathcal{C}(N,M)$ and isoperimetric clusters
  • Theorem 1.2: Honeycomb-like structure of low-energy clusters
  • Remark 1.3
  • Remark 1.4: Connection with the connectedness conjecture
  • Theorem A: Hales' hexagonal isoperimetric inequality, hales
  • Remark 1.5
  • Remark 1.6: Proof of \ref{['hales lower bound']} and assumption \ref{['Agamma hales bound']}
  • Theorem 1.7: A quantitative Hales' hexagonal isoperimetric inequality
  • Remark 1.8
  • proof : Proof of Theorem \ref{['thm main']}
  • ...and 7 more