CRYSTALLOGRAPHY, GEOMETRY AND PHYSICS IN HIGHER DIMENSIONS .14. FILIATION FROM ONE-DIMENSIONAL, 2-DIMENSIONAL AND 3-DIMENSIONAL CRYSTAL FAMILIES AND POINT GROUPS TO THE MONOINCOMMENSURATE, DI-INCOMMENSURATE AND TRI-INCOMMENSURATE CRYSTAL FAMILIES IN 4-DIMENSIONAL, 5-DIMENSIONAL AND 6-DIMENSIONAL SPACES

The previous paper in this series [Phan & Veysseyre (1994). Acta Cryst. A50, 438-444] mainly compared the mono-, di- and tri-incommensurate point-symmetry operations, their number and their symbols. In this paper, the filiation from the gZ-irreducible crystal families of the one-, two- and three-dimensional spaces to the mono-, di- and tri-incommensurate families of the four-, five- and six-dimensional spaces is established. The holohedries and the different point groups of these crystal families are compared. The paper begins with a list of the incommensurate families; then a series of nine further tables establishes the connection between the different families and their point groups. It is proved that there are 30 mono-incommensurate (MI) point groups, 47 di-incommensurate (DI) point groups and 57 tri-incommensurate (TI) point groups belonging to the six MI crystal families of four-dimensional space, to the 11 DI crystal families of five-dimensional space and to the 14 TI crystal families of the six-dimensional space.