Densest plane group packings of regular polygons

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. Formerly most of the results were on densest lattice or double-lattice configurations. Here we extend these results to all 2-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 2-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 the means of the Entropic Trust Region Packing Algorithm that solves this problem approximately, we examine some known as well as unknown densest packings of n-gons in all 17 plane groups and state conjectures about common symmetries of the densest plane group packings of all n-gons.