Classification of quasi-periodic structure with local configurations of Archimedes tiling

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We study details of local configurations around vertices of non-periodic tilings. We can construct an uncountable family of non-periodic tilings with 7-fold rotational symmetry which have just three kinds of local configurations around vertices in Archimedean tilings. These non-periodic tilings have singular local configurations around vertices. The Danzer tiling with 7-fold rotational symmetry has a singular local configurations around vertices. This implies that the Danzer tiling with 7-fold rotational symmetry cannot be obtained as a limit of sequence of canonical tilings.