Abstract
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 (Q) except at a dense set of points of measure 0 in Q. 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 re ected versions of the tile by gray tiles. It turns out that the Taylor Socolar tiling is mutually locally derivable from this parity tiling.
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