Hexagonal parquet tilings: k-isohedral monotiles with arbitrarily large k

Details Hexagonal parquet tilings: k-isohedral monotiles with arbitrarily large k Hexagonal parquet tilings: k-isohedral monotiles with arbitrarily large k

2007-08-20

arxiv.org/abs/0708.2663v1

Mathematical Intelligencer, Volume 29, Number 2, pages 33-38 (2007)

Physics

Condensed Matter

Other Condensed Matter

7 pages, 8 figures. Published in The Mathematical Intelligencer. NOTE: The MI mistakenly published an earlier draft

Scientific paper

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.

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