Mathematics of 2-Dimensional Lattices

Abstract A periodic lattice in Euclidean space is the infinite set of all integer linear combinations basis vectors. Any can be generated by infinitely many different bases. This ambiguity was partially resolved, but standard reductions remain discontinuous under perturbations modelling atomic displacements. paper completes a continuous classification 2-dimensional lattices up to isometry (or congruence), rigid motion (without reflections), and similarity (with uniform scaling). The new homogeneous invariants allow easily computable metrics on considered equivalences above. are especially non-trivial settle remaining questions (dis)continuity These lead real-valued chiral distances that continuously measure deviations from higher-symmetry neighbours. geometric methods extend past work Delone, Conway, Sloane.