Note on minimal number of skewed unit cells for periodic distance calculation

TL;DR

This paper determines the minimal number of skewed unit cells needed for periodic distance calculations in Euclidean space, providing explicit constructions in 2D and 3D.

Contribution

It introduces a method to find the smallest domain ensuring distance equivalence across lattice points and describes how to obtain all primitive cells realizing this minimal number.

Findings

Constructed the smallest domain fulfilling the distance condition in Euclidean space.

Provided explicit descriptions of primitive cells in 2D and 3D.

Identified the minimal number of copies needed for periodic distance calculations.

Abstract

How many copies of a parallelepiped are needed to ensure that for every point in the parallelepiped a copy of each other point exists, such that the distance between them equals the distance of the pair of points when the opposite sites of the parallelepiped are identified? This question is answered in Euclidean space by constructing the smallest domain that fulfills the above condition. We also describe how to obtain all primitive cells of a lattice (i.e., closures of fundamental domains) that realise the smallest number of copies needed and give them explicitly in 2D and 3D.