Periodic double tilings of the plane

TL;DR

This paper classifies minimal interface configurations in periodic double tilings of the plane, revealing three distinct patterns depending on tile area ratios and optimal lattices.

Contribution

It provides a comprehensive classification of isoperimetric configurations in double tilings with two tile areas, including explicit profiles and lattice optimizations.

Findings

Three distinct isoperimetric configurations identified.

Explicit computation of isoperimetric profiles for each configuration.

Different optimal lattices depend on the area ratio.

Abstract

We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both in the case of a fixed lattice or for an arbitrary periodic lattice. We find three different configurations depending on the ratio between the assigned areas of the two tiles and compute the isoperimetric profile. The three different configurations are composed of tiles with a different number of circular edges, moreover, different configurations exhibit a different optimal lattice. Finally, we raise some open problems related to our investigation.