The Coincidence Problem

Discrete point sets S such as lattices or quasiperiodic Delone sets may permit, beyond their symmetries, certain isometries R such that S ∩ RS is a subset of S of finite density. These are the so-called coincidence isometries. They are important in understanding and classifying grain boundaries and twins in crystals and quasicrystals. It is the purpose of this contribution to introduce the corresponding coincidence problem in a mathematical setting and to demonstrate how it can be solved algebraically in dimensions 2, 3 and 4. Various examples both from crystals and quasicrystals are treated explicitly, in particular (hyper-)cubic lattices and quasicrystals with non-crystallographic point groups of type H2, H3 and H4. We derive parametrizations of all linear coincidence isometries, determine the corresponding coincidence index (the reciprocal of the density of coinciding points, also called Σ-factor), and finally encapsulate their statistics in suitable Dirichlet series generating functions.