Densest plane group packings of regular polygons

TL;DR

This paper investigates the densest arrangements of regular polygons within all 17 two-dimensional symmetry groups, formulating the problem as a nonlinear optimization and proposing conjectures on their symmetries.

Contribution

It extends the study of dense polygon packings to all plane groups using a novel optimization approach and introduces conjectures on their symmetry properties.

Findings

Identified densest packings for various polygons across all plane groups.

Formulated the packing problem as a nonlinear constrained optimization.

Proposed conjectures on the symmetry of densest packings.

Abstract

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.