PC Seminar: Periodic lozenge tilings of the plane

Abstract: We start with the tiling of the plane by equilateral triangles. Their vertices form a lattice which we will call L0. By merging two adjacent triangles of this tiling we obtain a rhombus, also known as a lozenge. It is clear that there exist 3 different orientations of them, and that the plane can be tiled with lozenges. Now let L1 be a cofinite sublattice of L0 whose embedding into L0 is given by an invertible integer 2*2 matrix B. We are interested in lozenge tilings of the plane which are invariant under the translation by any element in L1. Since L0/L1 is finite, we are using only finitely many lozenges in our tiling (mod L1). To each L1-periodic tiling one can attach a vector (called the type of the tiling) storing the information about the number of lozenges of each orientation used in the tiling (mod L1). This way we can split all the L1-periodic tilings into groups of different types.

The main focus of the talk is to address the following questions:

1. For a given cofinite sublattice L1 of L0, which types of L1-periodic tilings exist?
2. For a given type, can we list all the L1-periodic tilings of this type?

I will address these questions and show how one can obtain the answers

just by looking at the 2*2 matrix B.

This is a seminar in our seminar series on Probability and Combinatorics (PC).