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Remarks on 2-dimensional quasiperiodic tilings with rotational symmetries. (English) Zbl 1175.52026

A sequentially compact space of patches is constructed. Using this construction it is proved:
Let \(S\) be a prototile set with a substitution rule \(\Phi\). If we have a prototile with a vertex of an angle \(\pi/n\), then there exists a tiling \(T_n\) with \(n\)-fold rotational symmetry, which is a periodic point of \(\widehat\Phi\).
Furthermore, for an integer \(n\) with \(n= 5\) or \(6< n\), its tiling \(T_n\) is nonperiodic and has non-trivial action of a non-crystallographic Coxeter group.

MSC:

52C23 Quasicrystals and aperiodic tilings in discrete geometry
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
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