and the eye is then raised so that a third cube face a1 may
be seen.
(f) Crystal Systems and Classes.
According to the mutual inclinations of the crystallographic axes of reference and the lengths intercepted on them by the parametral plane, all crystals fall into one or other of six groups or systems, in each of which there are several classes depending on the degree of symmetry. In the brief description which follows of these six systems and thirty-two classes of crystals we shall proceed from those in which the symmetry is most complex to those in which it is simplest.
1. CUBIC SYSTEM
In this system the three crystallographic axes of reference are all at right angles to each other and are equal in length. They are parallel to the edges of the cube, and in the different classes coincide either with tetrad or dyad axes of symmetry. Five classes are included in this system, in all of which there are, besides other elements of symmetry, four triad axes.
In crystals of this system the angle between any two faces P and Q with the indices (hkl) and (pqr) is given by the equation
| COS PQ = | hp + kq + lr | . |
| √(h² + k² + l²) (p² + q² + r²) |
The angles between faces with the same indices are thus the same in all substances which crystallize in the cubic system: in other systems the angles vary with the substance and are characteristic of it.
Holosymmetric Class
Crystals of this class possess the full number of elements of symmetry already mentioned above for the octahedron and the cube, viz. three cubic planes of symmetry, six dodecahedral planes, three tetrad axes of symmetry, four triad axes, six dyad axes, and a centre of symmetry.
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| Fig. 13.—Rhombic Dodecahedron. | Fig. 14.—Combination of Rhombic Dodecahedron and Octahedron. |
There are seven kinds of simple forms, viz.:—
Cube (fig. 5). This is bounded by six square faces parallel to the cubic planes of symmetry; it is known also as the hexahedron. The angles between the faces are 90°, and the indices of the form are {100}. Salt, fluorspar and galena crystallize in simple cubes.
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| Fig. 15.—Triakis-octahedron. | Fig. 16.—Combination of Triakis-octahedron and Cube. |
Octahedron (fig. 3). Bounded by eight equilateral triangular faces perpendicular to the triad axes of symmetry. The angles between the faces are 70° 32′ and 109° 28′, and the indices are {111}. Spinel, magnetite and gold crystallize in simple octahedra. Combinations of the cube and octahedron are shown in figs. 6-8.
Rhombic dodecahedron (fig. 13). Bounded by twelve rhomb-shaped faces parallel to the six dodecahedral planes of symmetry. The angles between the normals to adjacent faces are 60°, and between other pairs of faces 90°; the indices are {110}. Garnet frequently crystallizes in this form. Fig. 14 shows the rhombic dodecahedron in combination with the octahedron.
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| Fig. 17.—Icositetrahedron. | Fig. 18.—Combination of Icositetrahedron and Cube. |
In these three simple forms of the cubic system (which are shown in combination in fig. 11) the angles between the faces and the indices are fixed and are the same in all crystals; in the four remaining simple forms they are variable.
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| Fig. 19.—Combination of Icositetrahedron and Octahedron. |
Fig. 20.—Combination of Icositetrahedron {211} and Rhombic Dodecahedron. |
Triakis-octahedron (three-faced octahedron) (fig. 15). This solid is bounded by twenty-four isosceles triangles, and may be considered as an octahedron with a low triangular pyramid on each of its faces. As the inclinations of the faces may vary there is a series of these forms with the indices {221}, {331}, {332}, &c. or in general {hhk}.
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| Fig. 21.—Tetrakis-hexahedron. | Fig. 22.—Tetrakis-hexahedron. |
Icositetrahedron (fig. 17). Bounded by twenty-four trapezoidal faces, and hence sometimes called a “trapezohedron.” The indices are {211}, {311}, {322}, &c., or in general {hkk}. Analcite, leucite and garnet often crystallize in the simple form {211}. Combinations are shown in figs. 18-20. The plane ABe in fig. 9 is one face (112) of an icositetrahedron; the indices of the remaining faces in this octant being (211) and (121).
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| Fig. 23.—Combination of Tetrakis-hexahedron and Cube. |
Tetrakis-hexahedron (four-faced cube) (figs. 21 and 22). Like the triakis-octahedron this solid is also bounded by twenty-four isosceles triangles, but here grouped in fours over the cubic faces. The two figures show how, with different inclinations of the faces, the form may vary, approximating in fig. 21 to the cube and in fig. 22 to the rhombic dodecahedron. The angles over the edges lettered A are different from the angles over the edges lettered C. Each face is parallel to one of the crystallographic axes and intercepts the two others in different lengths; the indices are therefore {210}, {310}, {320}, &c., in general {hko}. Fluorspar sometimes crystallizes in the simple form {310}; more usually, however, in combination with the cube (fig. 23).
Hexakis-octahedron (fig. 24). Here each face of the octahedron is replaced by six scalene triangles, so that altogether there are
