A note on energy minimization in dimension 2

TL;DR

This paper demonstrates that the hexagonal lattice has lower energy than certain natural periodic configurations and rules out the honeycomb as a lower-energy alternative, advancing understanding of energy minimization in two dimensions.

Contribution

It provides new results showing the hexagonal lattice's optimality among specific classes and excludes the honeycomb as a lower-energy configuration in 2D.

Findings

Hexagonal lattice outperforms certain natural periodic configurations.

Honeycomb cannot have lower energy than the hexagonal lattice at any scale.

Advances understanding of energy minimization in 2D configurations.

Abstract

Proving the universal optimality of the hexagonal lattice is one of the big open challenges of nowadays mathematics. We show that the hexagonal lattice outperforms certain "natural" classes of periodic configurations. Also, we rule out the option that the canonical non-lattice rival -- the honeycomb -- has lower energy than the hexagonal lattice at any scale.