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° 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.
Holosymmetric Class
(Holohedral; Ditrigonal scalenohedral).
In this class, which presents the commonest type of symmetry of the hexagonal system, the triad axis is associated with three similar planes of symmetry inclined to one another at 60° and intersecting in the triad axis; there are also three similar dyad axes, each perpendicular to a plane of symmetry, and a centre of symmetry. The seven simple forms are:—
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| Fig. 66. | Fig. 67. |
| Direct and Inverse Rhombohedra. |
| Fig. 68.—Scalenohedron. |
Rhombohedron (figs. 66 and 67), consisting of six rhomb-shaped faces with the edges all of equal lengths: the faces are perpendicular to the planes of symmetry. There are two sets of rhombohedra, distinguished respectively as direct and inverse; those of one set (fig. 66) are brought into the orientation of the other set (fig. 67) by a rotation of 60° or 180° about the principal axis. For the fundamental rhombohedron, parallel to the edges of which are the crystallographic axes of reference, the indices are {100}. Other rhombohedra may have the indices {211}, {411}, {110}, {221}, {111}, &c., or in general {hkk}. (Compare fig. 72; for figures of other rhombohedra see [Calcite].)
Scalenohedron (fig. 68), bounded by twelve scalene triangles, and with the general indices {hkl}. The zig-zag lateral edges coincide with the similar edges of a rhombohedron, as shown in fig. 69; if the indices of the inscribed rhombohedron be {100}, the indices of the scalenohedron represented in the figure are {201}. The scalenohedron {201} is a characteristic form of calcite, which for this reason is sometimes called “dog-tooth-spar.” The angles over the three edges of a face of a scalenohedron are all different; the angles over three alternate polar edges are more obtuse than over the other three polar edges. Like the two sets of rhombohedra, there are also direct and inverse scalenohedra, which may be similar in form and angles, but different in orientation and indices.