The arithmeticsymmetry of colored crystals: classification of 2-color 2-lattices

In recent years, a framework for the detailed classification of crystal structures based on an arithmetic criterion has been developed, aiming to systematically categorize different forms based on their symmetry properties. This study applies this established method to investigate colored crystals, specifically focusing on the enumeration of 2-color 2-lattices in both 2-dimensional and 3-dimensional spaces. By systematically cataloging these lattices, we identify 51 unique types of 2-color 2-lattices in three dimensions and 10 types in two dimensions. This comprehensive catalog covers the simplest theoretical structures possible for 2-element compounds. Notably, the research also reconciles our findings with existing literature, retrieving all structurally characterized configurations that have been named in previous studies, as well as the 22 black-and-white lattices relevant to magnetic crystal theory. Furthermore, we delve into the implications of symmetry hierarchies and symmetry-breaking phenomena for the categorized 2-color 2-lattices in both dimensions. This methodological advance is crucial for future experimental and theoretical investigations of new crystalline materials.