Densest plane group packings of regular polygons

Area of Science:

• Discrete Geometry
• Computational Geometry
• Crystallography

Background:

• Dense packings of regular polygons are crucial for modeling physical and biological systems.
• Previous research focused primarily on lattice and double-lattice configurations.
• A comprehensive analysis across all 17 two-dimensional crystallographic symmetry groups (plane groups) is lacking.

Purpose of the Study:

• To investigate densest packing configurations of congruent regular convex polygons (n-gons) within all 17 plane groups.
• To formulate the general packing problem as a nonlinear constrained optimization problem.
• To identify and conjecture common symmetries in these densest packings.

Main Methods:

• Formulation of the plane group packing problem as a nonlinear constrained optimization problem.
• Application of the Entropic Trust Region Packing Algorithm for approximate solution.
• Systematic examination of known and unknown densest packings for various n-gons across all 17 plane groups.

Main Results:

• Identification of densest packing configurations for various n-gons under all 17 plane groups.
• Discovery of previously unknown optimal packing arrangements.
• Observation of recurring symmetries in the densest packings across different n-gons and plane groups.

Conclusions:

• The study provides a systematic approach to solving dense packing problems within crystallographic constraints.
• The findings suggest universal principles governing the symmetry of optimal polygon packings.
• Conjectures on common symmetries offer a foundation for future theoretical and computational research in packing problems.