Solution of the coincidence problem in dimensions $d\le 4$

TL;DR

This paper addresses the algebraic solution of the coincidence problem for point sets like lattices and quasicrystals in dimensions up to 4, providing explicit parametrizations and statistical analysis of coincidence isometries.

Contribution

It introduces a mathematical framework for the coincidence problem and solves it algebraically in dimensions 2, 3, and 4, including explicit examples and generating functions.

Findings

Derived parametrizations of all linear coincidence isometries.

Determined the coincidence index and its relation to point density.

Developed Dirichlet series generating functions for coincidence statistics.

Abstract

Discrete point sets $\mathcal{S}$ such as lattices or quasiperiodic Delone sets may permit, beyond their symmetries, certain isometries $R$ such that $\mathcal{S}\cap R\mathcal{S}$ is a subset of $\mathcal{S}$ of finite density. These are the so-called coincidence isometrie. 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 $H_2$, $H_3$ and $H_4$. We derive parametrizations of all linear coincidence isometries, determine the corresponding coincidence index (the reciprocal of the density of coinciding points, also called $\varSigma$-factor), and finally encapsulate their statistics in suitable Dirichlet series generating functions.