title: B7as a supergroup of crystal and quasicrystal symmetries.
abstract: 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 Bkmay 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 B7supergroup is necessary. All considered matrix groups correspond to isometric transformations in extended k-bases and may be easily derived from B7and 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.
authors.affiliation: Faculty of Computer Science and Material Sciences, University of Silesia, Katowice, Poland.
authors.forename: Kazimierz
authors.initials: K
authors.lastname: Stróż
keywords: crystal symmetry; extended bases; projection; quasicrystal symmetry; supergroups