Hexagonal parquet tilings: k-isohedral monotiles with arbitrarily large k
| dc.creator | Socolar, Joshua E. S. | |
| dc.date | 2007-08-20 | |
| dc.date.accessioned | 2026-07-07T08:24:40Z | |
| dc.date.available | 2026-07-07T08:24:40Z | |
| dc.description | This paper addresses the question of whether a single tile with nearest neighbor matching rules can force a tiling in which the tiles fall into a large number of isohedral classes. A single tile is exhibited that can fill the Euclidean plane only with a tiling that contains k distinct isohedral sets of tiles, where k can be made arbitrarily large. It is shown that the construction cannot work for a simply connected 2D tile with matching rules for adjacent tiles enforced by shape alone. It is also shown that any of the following modifications allows the construction to work: (1) coloring the edges of the tiling and imposing rules on which colors can touch; (2) allowing the tile to be multiply connected; (3) requiring maximum density rather than space-filling; (4) allowing the tile to have a thickness in the third dimension. | |
| dc.description | 7 pages, 8 figures. Published in The Mathematical Intelligencer. NOTE: The MI mistakenly published an earlier draft | |
| dc.identifier | https://arxiv.org/abs/0708.2663 | |
| dc.identifier | http://arxiv.org/abs/0708.2663 | |
| dc.identifier | Mathematical Intelligencer, Volume 29, Number 2, pages 33-38 (2007) | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/136413 | |
| dc.subject | Other Condensed Matter | |
| dc.subject | General Mathematics | |
| dc.subject | Metric Geometry | |
| dc.title | Hexagonal parquet tilings: k-isohedral monotiles with arbitrarily large k | |
| dc.type | text |