Crystallography, geometry and physics in higher dimensions. XIV. `Filiation' from one-, two- and three-dimensional crystal families and point groups to the mono-, di- and tri-incommensurate crystal families in four-, five- and six-dimensional spaces
Abstract
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.