The coincidence problem for shifted lattices and crystallographic point packings

TL;DR

This paper extends the concept of coincidence site lattices to include affine isometries, analyzing shifted lattices and crystallographic packings with detailed examples like the shifted square lattice and diamond packing.

Contribution

It introduces a general framework for coincidence isometries involving affine transformations, broadening the scope beyond linear isometries for lattices and packings.

Findings

Derived results for coincidence isometries of shifted lattices

Analyzed crystallographic point packings including the diamond packing

Provided detailed case studies for specific lattice types

Abstract

A coincidence site lattice is a sublattice formed by the intersection of a lattice $\Gamma$ in $\mathbb{R}^d$ with the image of $\Gamma$ under a linear isometry. Such a linear isometry is referred to as a linear coincidence isometry of $\Gamma$. Here, we consider the more general case allowing any affine isometry. Consequently, general results on coincidence isometries of shifted copies of lattices, and of crystallographic point packings are obtained. In particular, we discuss the shifted square lattice and the diamond packing in detail.