B₇ as a supergroup of crystal and quasicrystal symmetries

In sharp contrast to the generation of a finite group that includes all the 14 types of Bravais lattices as its subgroups [Hosoya (2000). Acta Cryst. A56, 259–263; Hosoya (2002). Acta Cryst. A58, 208], it was proved that a signed permutation group B_k may be interpreted as the supergroup of both crystal and quasicrystal symmetries. Minimal dimension k = 6 is adequate for lattices referred to their three non-coplanar shortest vectors, or for symmetry groups of most quasicrystal types. If one prefers complete, well defined semi-reduced lattice descriptions or needs a dodecagonal group, the B₇ supergroup is necessary. All considered matrix groups correspond to isometric transformations in extended k-bases and may be easily derived from B₇ and projected onto three-dimensional crystallographic space. Three models of extended bases are proposed: semi-reduced, cyclic and axial. In all cases additional basis vectors are strictly (functionally) related to three original basis vectors.