On colourings of cubic lattices – a tribute to Heinrich Heesch

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 permutations of colours. Complete results are obtained for d = 2 up to 64 orbits, for d = 3 up to eight orbits, and for two 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 Heesch [ Z. Kristallogr. (1933), 85 , 335–344] who overlooked one partition.