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 (3)

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

Cited by in corpus (15)

• Planar coincidences for N-fold symmetry
• Coincidence rotations of the root lattice $A_4$
• What Symmetries are Preserved by a Fermion Boundary State?
• Spectral Properties of Grain Boundaries at Small Angles of Rotation
• Combinatorial problems of (quasi-)crystallography
• A Note on Coincidence Isometries of Modules in Euclidean Space
• Coincidence site modules in 3-space
• Similarity and Coincidence Isometries for Modules
• Coincidence isometries of a shifted square lattice
• Invariant Submodules and Semigroups of Self-Similarities for Fibonacci Modules
• A note on the relation between fixed point and orbit count sequences
• Similar Sublattices and Coincidence Rotations of the Root Lattice A4 and its Dual
• Discrete Tomography of F-Type Icosahedral Model Sets
• On the Existence of Similar Sublattices
• Geometric enumeration problems for lattices and embedded $\mathbb{Z}$-modules