Embedding of regular tilings and star-honeycomb
| dc.creator | Deza, M. | |
| dc.creator | Shtogrin, M. I. | |
| dc.date | 1999-06-10 | |
| dc.date.accessioned | 2026-07-07T05:29:25Z | |
| dc.date.available | 2026-07-07T05:29:25Z | |
| dc.description | 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). | |
| dc.description | 17 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/math/9906062 | |
| dc.identifier | http://arxiv.org/abs/math/9906062 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78634 | |
| dc.subject | Metric Geometry | |
| dc.subject | Combinatorics | |
| dc.title | Embedding of regular tilings and star-honeycomb | |
| dc.type | text |