Tiling Spaces of Taylor-Socolar Tilings

J.-Y. Lee Department of Mathematics Education, Kwandong University, Gangneung, Gyeonggi-do, 210-701, Korea

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The Taylor-Socolar tiling has been introduced as an aperiodic mono-tile tiling. We consider a tiling space which consists of all the tilings that are locally indistinguishable from a Taylor-Socolar tiling and study its structure. It turns out that there is a bijective map between the space of the Taylor-Socolar tilings and a compact Abelian group of a Q-adic space (Q̅) except at a dense set of points of measure 0 in Q̅. From this we can derive that the Taylor-Socolar tilings have quasicrystalline structures. We make a parity tiling from the Taylor-Socolar tiling identifying all the rotated versions of a tile in the Taylor-Socolar tiling by white tiles and all the reflected versions of the tile by gray tiles. It turns out that the Taylor-Socolar tiling is mutually locally derivable from this parity tiling.

DOI: 10.12693/APhysPolA.126.508 PACS numbers: 61.44.Br, 61.44.-n