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

TL;DR

This paper presents a single tile construction that can produce tilings with an arbitrarily large number of isohedral classes, demonstrating complex symmetry properties in plane tilings.

Contribution

It introduces a novel tile that enforces high isohedral diversity in tilings, extending understanding of tile design and symmetry constraints.

Findings

A single tile can generate tilings with arbitrarily many isohedral classes.

The construction cannot be achieved with simply connected 2D tiles using shape alone.

Modifications like coloring or allowing multiply connected tiles enable the construction.

Abstract

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.