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

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 $\Sigma$-factor), and finally encapsulate their statistics in suitable Dirichlet series generating functions.

30 pages, 3 figures; revised and updated version of a summary presented during a meeting on aperiodic order at the Fields Institute in 1995

References in corpus (2)

• Bravais colourings of planar modules with N-fold symmetry
• Coincidence site modules in 3-space

Cited by in corpus (7)

• Coincidence rotations of the root lattice $A_4$
• Combinatorial problems of (quasi-)crystallography
• A Note on Coincidence Isometries of Modules in Euclidean Space
• Coincidence site modules in 3-space
• Similar Sublattices and Coincidence Rotations of the Root Lattice A4 and its Dual
• On the Existence of Similar Sublattices
• Discrete Tomography of F-Type Icosahedral Model Sets