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On colourings of cubic lattices

Igor A. Baburin

TL;DR

This work addresses orbit colourings of cubic lattices by partitioning lattice nodes into orbits of finite-index subgroups of $Aut(\Lambda^d)$, interpreted as colourings up to colour permutation. It develops a computational group-theoretic framework, implemented in GAP, to enumerate subgroups, compute orbit partitions, and determine automorphism groups of partitions, with a two-step strategy for the $d=4$ case. The paper provides complete two-orbit results for $d=2$ (up to 64 orbits) and $d=3$ (up to 8 orbits), and two-orbit results for $d=4$, including a corrected count in dimension three where it identifies 25 partitions (one previously overlooked by Heesch). It also delivers extensive data (Wyckoff-like descriptors and automorphism groups) and discusses the connections to prior work, including equitable partitions and the normalizer-based approach to Aut($\Pi$). The results advance the systematic enumeration of orbit colourings in crystallographic lattices and furnish a benchmark for computational group-theoretic methods in high-symmetry periodic graphs.

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 $d \leq 4$. 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.

On colourings of cubic lattices

TL;DR

This work addresses orbit colourings of cubic lattices by partitioning lattice nodes into orbits of finite-index subgroups of , interpreted as colourings up to colour permutation. It develops a computational group-theoretic framework, implemented in GAP, to enumerate subgroups, compute orbit partitions, and determine automorphism groups of partitions, with a two-step strategy for the case. The paper provides complete two-orbit results for (up to 64 orbits) and (up to 8 orbits), and two-orbit results for , including a corrected count in dimension three where it identifies 25 partitions (one previously overlooked by Heesch). It also delivers extensive data (Wyckoff-like descriptors and automorphism groups) and discusses the connections to prior work, including equitable partitions and the normalizer-based approach to Aut(). The results advance the systematic enumeration of orbit colourings in crystallographic lattices and furnish a benchmark for computational group-theoretic methods in high-symmetry periodic graphs.

Abstract

Given the integral lattice in -dimensional Euclidean space, partitions of the lattice nodes into orbits of finite-index subgroups of have been computed for . These partitions can be interpreted as colourings of orbits defined up to permutation of colours. Complete results are obtained for up to 64 orbits, for 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.
Paper Structure (4 sections, 1 theorem, 4 figures, 6 tables)

This paper contains 4 sections, 1 theorem, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Theory and computational methods
  3. Results
  4. Comments on earlier works

Key Result

Proposition 1

Let $X$ be an orbit of a finite group $G$, and let $H < G$ with index $i = |G:H|$. By the orbit-stabilizer lemma we have $|X| = |G|/|Stab_{G}(x)|$. Under the action of $H$, $X$ splits into $n$ orbits $X_1, ... , X_n$. Then $|X| = \sum_{k=1}^{n} |H|/|Stab_{H}(x_k)|$. By equating the two expressions f

Figures (4)

  • Figure 1: Two-orbit partitions of the square lattice.
  • Figure 2: Node neighbourhoods in the two-orbit partitions of $\Lambda^3$.
  • Figure 3: The two-orbit partitions of $\Lambda^3$ with symmetry $Im\bar{3}m$ (No. 2) and $Ia\bar{3}d$ (No. 20). Lattice nodes are drawn as cubes.
  • Figure 4: Types of node neighbourhoods in the two-orbit partitions of $\Lambda^4$.

Theorems & Definitions (1)

  • Proposition 1