Publications [#248350] of Joshua Socolar
- Socolar, JES, The hexagonal parquet tiling k-isohedral monotiles with arbitrarily large k,
The Mathematical Intelligencer, vol. 29 no. 2
pp. 33-38, Springer Nature, ISSN 0343-6993 [pdf], [doi]
(last updated on 2019/07/16)
NOTE: An editorial mix-up resulted in the publication of the wrong version of this article. The link shown here for the PDF is for the correct version.
The interplay between local constraints and global structure of mathematical and physical systems is both subtle and important. This paper shows how to construct a single tile 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.