Non-repetitive tilings
Summary: In 1906 Axel Thue showed how to construct an infinite non-repetitive (or square-free) word on an alphabet of size 3. Since then this result has been rediscovered many times and extended in many ways. We present a two-dimensional version of this result. We show how to construct a rectangular tiling of the plane using 5 symbols which has the property that lines of tiles which are horizontal, vertical or have slope \(+1\) or \(-1\) contain no repetitions. As part of the construction we introduce a new type of word, one that is non-repetitive up to \(\text{mod }k\), which is of interest in itself. We also indicate how our results might be extended to higher dimensions.
scientific article; zbMATH DE number 1770058