Densest plane group packings of regular polygons

Torda, Miloslav ORCID: 0000-0002-2115-7811, Goulermas, John Y, Kurlin, Vitaliy ORCID: 0000-0001-5328-5351 and Day, Graeme M
(2022) Densest plane group packings of regular polygons. [Preprint]

[thumbnail of 2207.08959v4.pdf] PDF
2207.08959v4.pdf - Published version

Download (383kB) | Preview


Packings of regular convex polygons ($n$-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Former results were mainly regarding densest lattice or double-lattice configurations. Here we consider all two-dimensional crystallographic symmetry groups (plane groups) by restricting the configuration space of the general packing problem of congruent copies of a compact subset of the two-dimensional Euclidean space to particular isomorphism classes of the discrete group of isometries. We formulate the plane group packing problem as a nonlinear constrained optimization problem. By means of the Entropic Trust Region Packing Algorithm that approximately solves this problem, we examine some known and unknown densest packings of various $n$-gons in all $17$ plane groups and state conjectures about common symmetries of the densest plane group packings for every $n$-gon.

Item Type: Preprint
Uncontrolled Keywords: math.MG, math.MG, cond-mat.soft, cond-mat.stat-mech, math-ph, math.MP
Depositing User: Symplectic Admin
Date Deposited: 28 Nov 2022 12:16
Last Modified: 14 Mar 2024 17:32
DOI: 10.48550/arxiv.2207.08959
Related URLs: