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

TL;DR

This paper provides quantitative geometric estimates for low-energy planar clusters, showing most chambers resemble regular hexagons, thus explaining the emergence of honeycomb-like structures in physical and biological systems.

Contribution

It offers explicit bounds and a detailed revision of the isoperimetric estimates underlying honeycomb structures in energy-minimizing configurations.

Findings

Most chambers are generalized polygons with six edges.

Chambers closely resemble regular hexagons.

Provides quantitative geometric estimates for low-energy configurations.

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).