Tiling Spaces of TaylorSocolar Tilings

The TaylorSocolar 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 TaylorSocolar tiling and study its structure. It turns out that there is a bijective map between the space of the TaylorSocolar 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 TaylorSocolar tilings have quasicrystalline structures. We make a parity tiling from the TaylorSocolar tiling identifying all the rotated versions of a tile in the TaylorSocolar tiling by white tiles and all the reflected versions of the tile by gray tiles. It turns out that the TaylorSocolar tiling is mutually locally derivable from this parity tiling