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A systematic approach is presented for the construction of families of polytopes associated with a given group G and subgroup H. This procedure is applied to the icosahedral group and its three dihedral subgroups yielding three families of polyhedra in 
3. These families form the cells of three types of quasilattices associated with the icosahedral group, and hence are candidates for modelling quasicrystal structures. Some of the polyhedra are illustrated.
3. These families form the cells of three types of quasilattices associated with the icosahedral group, and hence are candidates for modelling quasicrystal structures. Some of the polyhedra are illustrated.
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