Constructing aperiodicity

Abstract

Aperiodic Order may be regarded as an extension of Symbolic Dynamics to higher dimensions with added geometry. Instead of studying infinite strings of symbols, one may also assign each symbol a length, for example. In higher dimensions one considers infinite point patterns, or tilings by tiles of a variety of shapes. The analogue of a shift of finite type for tilings is a set of tiles with rules on their relative placements in a valid tiling. One of the most enticing questions in this area is the Monotile Problem: is there a single shape of tile in the plane whose rigid motions may tile the plane, but only non-periodically? Taylor and Socolar gave a satisfying answer to this question with a certain marked hexagonal tile, although to enforce aperiodicity restrictions are needed for neighbours as well as next nearest neighbours of tiles, or alternatively a tile with disconnected interior must be used. I will give an introductory overview of the field of Aperiodic Order through examples, explaining the main constructions of interesting non-periodic patterns. I will then explain recent work with Mike Whittaker on a single aperiodic tile in the plane with edge-to-edge matching rules. Unfortunately these rules cannot be enforced by shape alone, since only symbols in certain orientations enforce matching restrictions. However, the proof of aperiodicity turns out to be remarkably simple and exploits a very different type of structure to that used for the Taylor-Socolar tile. I will explain the proof of aperiodicity of the tile and which valid tilings it permits.