On colourings of cubic lattices

TL;DR

This paper classifies and analyzes the partitions of cubic lattices into orbits under automorphism groups, providing comprehensive results for low-dimensional cases and correcting previous inaccuracies in the literature.

Contribution

It offers complete classifications of lattice partitions into orbits for dimensions up to 4, including automorphism groups, and corrects historical errors in the case of three dimensions.

Findings

Classified lattice partitions for dimensions 2 to 4.

Determined automorphism groups of the partitions.

Corrected previous results on three-dimensional lattice partitions.

Abstract

Given the integral lattice $\Lambda^d$ in $d$-dimensional Euclidean space, partitions of the lattice nodes into orbits of finite-index subgroups of $Aut(\Lambda^d)$ have been computed for $d \leq 4$. These partitions can be interpreted as colourings of orbits defined up to permutation of colours. Complete results are obtained for $d=2$ up to 64 orbits, for $d=3$ up to 8 orbits, and for 2 orbits in dimension 4. The automorphism groups of the partitions are also determined. Our results for two orbits in dimension 3 correct the old result of H. Heesch [Z. Kristallogr., (1933), 85, 335--344] who overlooked one partition.