HEXAGONAL SYSTEM 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 divi sion, 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 com mon use; rhombohedral axes being usually used for crystals of the rhombohedral di vision, and hexagonal axes for those of the hexagonal division ; though sometimes either one or the other set is employed in both divisions.
Rhombohedral 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; fur ther, they are all of equal length and are in terchangeable. With such a set of axes there can be no statement of an axial ratio, but the angle a 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° 55' or the angle between the normal to a face of this rhombohedron and the principal axis is 44° 36-i'; whilst the angle a between the axes is IoI° 55', this being the plane angle of the rhomb-shaped faces.
Hexagonal axes are four in number, viz., a vertical axis coin tiding with the principal axis of the crystal, and three horizontal axes inclined to one another at 6o° 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. 7o), 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-Fi+k=o) 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 = i : o.4989 (often written c = o.4989) ; for zinc c