Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices

Abstract

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 $m \ge 7$, 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 $d \gt 2$ 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, $\{4, 3, 5\}$ and $\{4, 3, 3, 5\}$, of those 11 have compact both, facets and vertex figures).