Hexagonal parquet tilings: k-isohedral monotiles with arbitrarily large k - Condensed Matter > Other Condensed MatterReport as inadecuate




Hexagonal parquet tilings: k-isohedral monotiles with arbitrarily large k - Condensed Matter > Other Condensed Matter - Download this document for free, or read online. Document in PDF available to download.

Abstract: This paper addresses the question of whether a single tile with nearestneighbor matching rules can force a tiling in which the tiles fall into a largenumber of isohedral classes. A single tile is exhibited that can fill theEuclidean plane only with a tiling that contains k distinct isohedral sets oftiles, where k can be made arbitrarily large. It is shown that the constructioncannot work for a simply connected 2D tile with matching rules for adjacenttiles enforced by shape alone. It is also shown that any of the followingmodifications allows the construction to work: 1 coloring the edges of thetiling and imposing rules on which colors can touch; 2 allowing the tile tobe multiply connected; 3 requiring maximum density rather than space-filling;4 allowing the tile to have a thickness in the third dimension.



Author: Joshua E. S. Socolar

Source: https://arxiv.org/







Related documents