Uniform Partitions of 3-space, their Relatives and Embedding

Abstract

We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic latticeZn. We also consider some relatives of those 28 partitions, including Archimedean 4-polytopes of Conway–Guy, non-compact uniform partitions, Kelvin partitions and those with a unique vertex figure (i.e., Delaunay star). Among the latter ones we indicate two continuums of aperiodic tilings by semi-regular 3-prisms with cubes or with regular tetrahedra and regular octahedra. In the process many new partitions are added to the incomplete cases considered here.