@article {15329, title = {Geometric units in hexagonal and rhombohedral space groups}, journal = {Acta Cryst. (1984). A40, 567-571}, year = {1984}, abstract = {Hexagonal space groups,\ i.e.\ those with an\ hP\ lattice, are classified from the geometric-unit viewpoint by considering hexagonal crystal structures as combinations and permutations of some basic hexagonal prisms. Geometric units are the Dirichlet domains of the Wyckoff positions with the highest point-group symmetry in the space group. In this classification, there are six types of hexagonal space groups. Type\ h1 consists of two independent geometric units of the same symmetry per crystallographic cell; in type\ h2, the two units are identical, but differently oriented. Type\ h3 has six independent geometric units, again of the same point-group symmetry, but the six units can be made up of three pairs, each consisting of two identical units, thus giving rise to type\ h4. There are subclasses in types\ hl and\ h3. Centers of geometric units in\ hl(a) and\ h3(a) are uniquely defined by intersections of point-group symmetry elements, whereas those in\ h1(b) and\ h3(b) are not because the space groups in these subtypes are hemimorphic. Therefore, the two units along the polar axis may be combined as one. Type\ h5 consists of three units, each turned 120{\textdegree} from its neighbors owing to the screw axis 31, 32, 62 and 64. Similarly, type\ h6 has six units due to screw axes 61 and 65, and adjacent units are 60{\textdegree} apart. Rhombohedral space groups show two types of patterns: type\ r1 has two independent, and type\ r2 two identical, units. The h.c.p. and related structures are used to demonstrate the application of geometric units to crystal-structure descriptions.}, author = {C. Chieh} }