  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.
