  We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic
d-space and Coxeter's regular hyperbolic honeycombs (with infinite or
star-shaped cells or vertex figures) with respect of possible embedding,
isometric up to a scale, of their skeletons into a m-cube or m-dimensional
cubic lattice. In section 2 the last remaining 2-dimensional case is decided:
for any odd m>6, star-honeycombs {m, m/2} are embeddable while {m/2, m} are not
(unique case of non-embedding for dimension 2). As a spherical analogue of
those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing
all nine regular polyhedra on the sphere. In section 4, non-embeddability of
all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved.
In the last section 5, all cases of embedding for dimension d>2 are identified.
Besides hyper-simplices and hyper-octahedra, they are exactly those with
bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1 tilings of
hyperbolic 3-, 4-, 5-space (only two, {435} and {4335}, of those 11 are
compact).
