A higher-dimensional geometrical approach for the classification of 2D square-triangle-rhombus tilings

Square-triangle-rhombus (STR) tilings are encountered in various self-organized multi-component systems. They exhibit a rich structural diversity, encompassing both periodic tilings and long-range ordered quasicrystals, depending on the proportions of the three tiles and their orientation distributions. We derive a general scheme for characterizing STR tilings based on their lift into a four-dimensional hyperspace. In this approach, the average hyperslope (2 2) matrix H of a patch defines its global composition with four real coefficients: X, Y, Z, and W. The matrix H can be computed either directly from the area-weighted average of the hyperslopes of individual tiles or indirectly from the border of the patch alone. The coefficient W plays a special role as it depends solely on the rhombus tiles and encapsulates a topological charge, which remains invariant upon local reconstructions in the tiling. For instance, a square can transform into a pair of rhombuses with opposite topological charges, giving rise to local modes with five degrees of freedom. We exemplify this classification scheme for STR tilings through its application to experimental structures observed in two-dimensional Ba-Ti-O films on metal substrates, demonstrating the hyperslope matrix H as a precise tool for structural analysis and characterization.