Hexagonal Parquet Tilings k - Isohedral Monotiles with Arbitrarily Large k

The interplay between local constraints and global structure of mathematical and physical systems is both subtle and important. The macroscopic physical properties of a system depend heavily on its global symmetries, but these are often difficult to predict given only information about local interactions between the components. A rich history of work on tilings of the Euclidean plane and higher dimensional or non-Euclidean spaces has brought to light numerous examples of finite sets of tiles with rules governing local configurations that lead to surprising global structures. Perhaps the most famous now is the set of two tiles discovered by Penrose that can be used to cover the plane with no overlap but only in a pattern whose symmetries are incompatible with any crystallographic space group. [1, 2] The Penrose tiles “improved” on previous examples due to Berger [3] and others (reviewed by Grünbaum and Shephard [4]) showing that larger sets of square tiles with colored edges (or several types of bumps and complementary nicks) could force the construction of a non-periodic pattern.