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.
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; 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° 55′, or the angle between the normal to a face of this rhombohedron and the principal axis is 44° 3612′.
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.
In the rhombohedral or trigonal division of the hexagonal system there are seven symmetry-classes, all of which possess a single triad axis of symmetry.
Holosymmetric Class
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. |
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| 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.
Hexagonal bipyramid (fig. 70), bounded by twelve isosceles triangles each of which are equally inclined to two planes of symmetry. The indices are {210}, {412}, &c., or in general (hkl), where h − 2k + l = 0.
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Hexagonal prism of the first order (211), consisting of six faces parallel to the principal axis and perpendicular to the planes of symmetry; the angles between (the normals to) the faces are 60°.
Hexagonal prism of the second order (101), consisting of six faces parallel to the principal axis and parallel to the planes of symmetry. The faces of this prism are inclined to 30° to those of the last prism.
Dihexagonal prism, consisting of twelve faces parallel to the principal axis and inclined to the planes of symmetry. There are two sets of angles between the faces. The indices are {321}, {532} ... {hkl}, where h + k + l = 0.
Basal pinacoid {111}, consisting of a pair of parallel faces perpendicular to the principal axis.
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| Fig. 72.—Stereographic Projection of a Holosymmetric Rhombohedral Crystal. |
Fig. 71 shows a combination of a hexagonal prism (m) with the basal pinacoid (c). For figures of other combinations see Calcite and Corundum. The relation between rhombohedral forms and their indices are best studied with the aid of a stereographic projection (fig. 72); in this figure the thicker lines are the projections of the three planes of symmetry, and on these lie the poles of the rhombohedra (six of which are indicated).
Numerous substances, both natural and artificial, crystallize
