Our starting point is the infinite triangular grid. Deleting an edge of this grid (or, in other words, merging two adjacent triangles) produces a rhombus (lozenge). The vertices of this grid form a lattice L0 (isomorphic to Z^2), and we consider a cofinite sublattice L1 of L0, whose embedding into L0 is given by an invertible 2*2 matrix with integer entries. Is it possible to find an L1-periodic lozenge tiling of the plane? Can we find them all? In my talk I will address these questions and show how to obtain the answers to these questions just by looking at the matrix B.