Pragmatic Application of Abstract Algebra to Two-Dimensional Lattice Matching

Abstract

We investigated the lattice matching condition of two-dimensional (2D) lattices based on the group isomorphism of 2D Euclidean space to complex plane. This isomorphism enables us to avoid the inconvenience derived from the algebraic structure of 2D vectors and provides the systematic analysis. We found that the lattice matching is closely connected with ideal class group which is an invariant in the algebraic number field. We also provide an algorithm to construct a structure model for a superstructure formed by overlapping two 2D lattices, which is helpful for making trial models in the structure analysis. [DOI: 10.1380/ejssnt.2015.361]