Georgia Institute of Technology College of Sciences

Lattice patterns are commonly observed in material sciences where microscopic structural nuances induce distinct macroscopic physical or chemical properties. Provided with two lattices of the same dimension, how do we measure their differences in a visually consistent way? Mathematically, any n-D lattice is determined by a set of n independent vectors. Since such basis-representation is non-unique, a direct comparison among basis-representations in Euclidean space is highly ambiguous. In this talk, I will focus on 2-D lattices and introduce the lattice metric space proposed in my earlier work. This geometric space was constructed mainly based on integrating the Modular group theory and the Poincaré metric. In the lattice metric space, each point represents a unique lattice pattern, and the visual difference between two patterns is measured by the shortest path connecting them. Some applications of the lattice metric space will be presented. If time allows, I will briefly discuss potential extensions to 3D-lattices.