T Hexagonal Parquet Tilings k - Isohedral Monotiles with Arbitrarily Large k JOSHUA

T he 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 Grün-baum 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. The discovery of a set of only two tiles that could fill space but only in a non-periodic way raised a host of interesting questions. The Penrose tilings have elegant geometric and algebraic properties [1, 5, 6]. One successful line of research has been the discovery of tile sets that have the Penrose properties but different point group symmetries in two [7, 8, 9] and three dimensions [7, 10] or in hyperbolic space [11]. In all of these cases, the rules one must follow to construct a tiling are strictly local. Any configuration in which adjacent tiles fit together to leave no holes is allowed. There is no explicit constraint on the relative positions of tiles that do not touch each other. Another question, which has proven more difficult, is the quest for a single tile (rather than a set of two) that forces a non-periodic, space-filling tiling of the plane. It may be fruitful to view this as a limiting case of the following more general problem. Any tiling can be classified according to its isohedral number k, defined as the size of the largest set of tiles for which no two can be brought into coincidence by a global symmetry (any reflection, rotation , translation, or any combination of these) that leaves the entire tiling invariant. A set of tiles for which the …