Affine extension of noncrystallographic Coxeter groups and quasicrystals

Unique affine extensions $H_2^{\text{aff}}$, $H_3^{\text{aff}}$ and $H_4^{\text{aff}}$ are determined for the noncrystallographic Coxeter groups $H_2$, $H_3$ and $H_4$. They are used for the construction of new mathematical models for quasicrystal fragments with 10-fold symmetry. The case of $H_2^{\text{aff}}$ corresponding to planar point sets is discussed in detail. In contrast to the cut-and-project scheme we obtain by construction finite point sets, which grow with a model specific growth parameter.