Coincidence isometries of a shifted square lattice

TL;DR

This paper investigates the coincidence isometries of a square lattice shifted by any vector, using Gaussian integers to characterize and compute the set of such isometries and their associated lattices.

Contribution

It provides a general framework for understanding coincidence isometries of shifted square lattices through Gaussian integer analysis, including explicit calculations and generating functions.

Findings

Characterization of coincidence isometries for shifted square lattices

Explicit formulas for coincidence site lattices and isometries

Generating functions for counting coincidence structures

Abstract

We consider the coincidence problem for the square lattice that is translated by an arbitrary vector. General results are obtained about the set of coincidence isometries and the coincidence site lattices of a shifted square lattice by identifying the square lattice with the ring of Gaussian integers. To illustrate them, we calculate the set of coincidence isometries, as well as generating functions for the number of coincidence site lattices and coincidence isometries, for specific examples.