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Unique affine extensions $H^{\aff}_2$, $H^{\aff}_3$ and $H^{\aff}_4$ aredetermined for the noncrystallographic Coxeter groups $H_2$, $H_3$ and $H_4$.They are used for the construction of new mathematical models for quasicrystalfragments with 10-fold symmetry. The case of $H^{\aff}_2$ corresponding toplanar point sets is discussed in detail. In contrast to the cut-and-projectscheme we obtain by construction finite point sets, which grow with a modelspecific growth parameter.Comment: (27 pages, to appear in J. Phys. A

Affine extension of noncrystallographic Coxeter groups and quasicrystals