Non-Repetitive Tilings

In 1906 Axel Thue showed how to construct an infinite non-repetitive (or squarefree) 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 mod k, which is of interest in itself. We also indicate how our results might be extended to higher dimensions.

• Type: Journal
• Year: 2002
• Where: COMBINATORICS
• Authors: James D. Currie, Jamie Simpson
• Keywords: Axel Thue, COMBINATORICS 2002, Rectangular Tiling, Two-dimensional Version