An aperiodic tiling using a dynamical system and Beatty sequences

Summary

Abstract. Wang tiles are square unit tiles with colored edges. A finite set of Wang tiles is a valid tile set if the collection tiles the plane (using an unlimited number of copies of each tile), the only requirements being that adjacent tiles must have common edges with matching colors and each tile can be put in place only by translation. In 1995 Kari and Culik gave examples of tile sets with 14 and 13 Wang tiles respectively, which only tiled the plane aperiodically. Their tile sets were constructed using a piecewise multiplicative function of an interval. The fact the sets tile only aperiodically is derived from properties of the function.

Introduction

There is a vast literature connecting dynamical systems and tilings of the plane. In this paper, we give an exposition of the work of Kari and Culik to show how by starting with a piecewise multiplicative function f, with rational multiplicands defined on a finite interval, we can produce a finite set of Wang tiles which tiles the plane. Further, a choice of multiplicands and interval, so that the dynamical system f has no periodic points, results in a set of Wang tiles that can only tile the plane aperiodically. In this manner, Kari and Culik produce a set of 13 Wang tiles. This is currently, the smallest known set of Wang tiles which only tiles the plane aperiodically.