Uniform Partitions of 3-space, their Relatives and Embedding

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 lattice Zn . 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.