What question did this study set out to answer?

The research aims to classify tiling configurations of the plane using two types of tiles based on their areas and to analyze their isoperimetric properties.

February 27, 2026

Periodic double tilings of the plane

Key Points

The research aims to classify tiling configurations of the plane using two types of tiles based on their areas and to analyze their isoperimetric properties.
Studied tilings of the plane with two repeating tiles of different assigned areas.
Classified isoperimetric configurations under fixed and arbitrary periodic lattices.
Computed the isoperimetric profile for different configurations.
Identified three distinct configurations based on the area ratio of the two tiles.
Noted variations in the number of circular edges among configurations.
Demonstrated differing optimal lattices for each configuration.

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.

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Periodic double tilings of the plane

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