## 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)