BRAVAIS CLASSES AND SPACE-GROUPS FOR TRIGONAL AND HEXAGONAL QUASI-PERIODIC CRYSTALS OF ARBITRARY FINITE RANK

To demonstrate the power of the Fourier-space approach to crystallogra phy, the Bravais classes and space groups of hexagonal and trigonal qu asiperiodic crystals are derived for lattices of arbitrary finite rank . The specification of the space groups for each Bravais class is give n by an elementary extension of the rank-4 case. The conventional clas sification of incommensurately modulated hexagonal and trigonal crysta ls, previously derived using the superspace approach for Bravais class es up to rank (3 + 3) [Janner, Janssen & de Wolff (1993). Acta Cryst. A39, 658-666] and for superspace groups of rank (3 + 1) [de Wolff, Jan ssen & Janner (1981). Acta Cryst. A37, 625-636], is easily extracted f rom the general classification for modulations of any finite rank.