Crystals of this system are characterized by the presence of a single
axis of either triad or hexad symmetry, which is spoken of as the
"principal" or "morphological" axis. Those with a triad axis are
grouped together in the rhombohedral or trigonal division, and those
with a hexad axis in the hexagonal division. By some authors these two
divisions are treated as separate systems; or again the rhombohedral
forms may be considered as hemihedral developments of the hexagonal.
On the other hand, hexagonal forms may be considered as a combination
of two rhombohedral forms.
Owing to the peculiarities of symmetry associated with a single triad
or hexad axis, the crystallographic axes of reference are different in
this system from those used in the five other systems of crystals. Two
methods of axial representation are in common use; rhombohedral axes
being usually used for crystals of the rhombohedral division, and
hexagonal axes for those of the hexagonal division; though sometimes
either one or the other set is employed in both divisions.
Rhomobohedral axes are taken parallel to the three sets of edges of a
rhombohedron (fig. 66). They are inclined to one another at equal
oblique angles, and they are all equally inclined to the principal
axis; further, they are all of equal length and are interchangeable.
With such a set of axes there can be no statement of an axial ratio,
but the angle between the axes (or some other angle which may be
calculated from this) may be given as a constant of the substance.
Thus in calcite the rhombohedral angle (the angle between two faces of
the fundamental rhombohedron) is 74 deg. 55', or the angle between the
normal to a face of this rhombohedron and the principal axis is 44
deg. 36(1/2)'.
Hexagonal axes are four in number, viz. a vertical axis coinciding
with the principal axis of the crystal, and three horizontal axes
inclined to one another at 60 deg. in a plane perpendicular to the
principal axis. The three horizontal axes, which are taken either
parallel or perpendicular to the faces of a hexagonal prism (fig. 71)
or the edge of a hexagonal bipyramid (fig. 70), are equal in length
(a) but the vertical axis is of a different length (c). The indices of
planes referred to such a set of axes are four in number; they are
written as {hikl}, the first three (h + i + k = 0) referring to the
horizontal axes and the last to the vertical axis. The ratio a : c of
the parameters, or the axial ratio, is characteristic of all the
crystals of the same substance. Thus for beryl (including emerald) a :
c = 1 : 0.4989 (often written c = 0.4989); for zinc c = 1.3564.
_Rhombohedral Division._
In the rhomobohedral or trigonal division of the hexagonal system
there are seven symmetry-classes, all of which possess a single triad
axis of symmetry.