On colourings of cubic lattices

2025/10/21 by Igor A. Baburin, Baburin, Igor A.
Materials Science · Computer Science · Mathematics · #Quasicrystal Structures and Properties #Cellular Automata and Applications #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2510.18765

Abstract

Given the integral lattice Λ^d in d-dimensional Euclidean space, partitions of the lattice nodes into orbits of finite-index subgroups of Aut(Λ^d) have been computed for d ≤ 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.

Citations

• The chromatic number of 4-dimensional lattices
• Perfect colorings of Z²: Nine colors

Related

• Uniform partitions of 3-space, their relatives and embedding
• Crystallographic Restrictions for Colour Lattices with Modular Sublattices
• Factorizations in Geometric Lattices
• Groups of piecewise isometric permutations of lattice points or finitary rearrangements of tessellations
• Finite diamond-colored modular and distributive lattices with\n applications to combinatorial Lie representation theory