forty-eight faces. This is the greatest number of faces possible for
any simple form in crystals. The faces are all oblique to the planes
and axes of symmetry, and they intercept the three crystallographic
axes in different lengths, hence the indices are all unequal, being in
general {hkl}, or in particular cases {321}, {421}, {432}, &c. Such
a form is known as the “general form” of the class. The interfacial
angles over the three edges of each triangle are all different. These
forms usually exist only in combination with other cubic forms
(for example, fig. 25), but {421} has been observed as a simple form
on fluorspar.
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| Fig. 24.—Hexakis-octahedron. | Fig. 25.—Combination of Hexakis-octahedron and Cube. |
Several examples of substances which crystallize in this class have been mentioned above under the different forms; many others might be cited—for instance, the metals iron, copper, silver, gold, platinum, lead, mercury, and the non-metallic elements silicon and phosphorus.
Tetrahedral Class
In this class there is no centre of symmetry nor cubic planes of symmetry; the three tetrad axes become dyad axes of symmetry, and the four triad axes are polar, i.e. they are associated with different faces at their two ends. The other elements of symmetry (six dodecahedral planes and six dyad axes) are the same as in the last class.
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| Fig. 26.—Tetrahedron. | Fig. 27.—Deltoid Dodecahedron. |
Of the seven simple forms, the cube, rhombic dodecahedron and tetrakis-hexahedron are geometrically the same as before, though on actual crystals the faces will have different surface characters. For instance, the cube faces will be striated parallel to only one of the diagonals (fig. 90), and etched figures on this face will be symmetrical with respect to two lines, instead of four as in the last class. The remaining simple forms have, however, only half the number of faces as the corresponding form in the last class, and are spoken of as “hemihedral with inclined faces.”
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| Fig. 28.—Triakis-tetrahedron. | Fig. 29.—Hexakis-tetrahedron. |
Tetrahedron (fig. 26). This is bounded by four equilateral triangles and is identical with the regular tetrahedron of geometry. The angles between the normals to the faces are 109° 28′. It may be derived from the octahedron by suppressing the alternate faces.
Deltoid[1] dodecahedron (fig. 27). This is the hemihedral form of the triakis-octahedron; it has the indices {hhk} and is bounded by twelve trapezoidal faces.
Triakis-tetrahedron (fig. 28). The hemihedral form {hkk} of the icositetrahedron; it is bounded by twelve isosceles triangles arranged in threes over the tetrahedron faces.
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| Fig. 30.—Combination of two Tetrahedra. |
Fig. 31.—Combination of Tetrahedron and Cube. |
Hexakis-tetrahedron (fig. 29). The hemihedral form {hkl} of the hexakis-octahedron; it is bounded by twenty-four scalene triangles and is the general form of the class.
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| Fig. 32.—Combination of Tetrahedron, Cube and Rhombic Dodecahedron. |
Fig. 33.—Combination of Tetrahedron and Rhombic Dodecahedron. |
Corresponding to each of these hemihedral forms there is another geometrically similar form, differing, however, not only in orientation, but also in actual crystals in the characters of the faces. Thus from the octahedron there may be derived two tetrahedra with the indices {111} and {111}, which may be distinguished as positive and negative respectively. Fig. 30 shows a combination of these two tetrahedra, and represents a crystal of blende, in which the four larger faces are dull and striated, whilst the four smaller are bright and smooth. Figs. 31-33 illustrate other tetrahedral combinations.
Tetrahedrite, blende, diamond, boracite and pharmacosiderite are substances which crystallize in this class.
Pyritohedral[2] Class
Crystals of this class possess three cubic planes of symmetry but no dodecahedral planes. There are only three dyad axes of symmetry, which coincide with the crystallographic axes; in addition there are three triad axes and a centre of symmetry.
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| Fig. 34. Pentagonal Dodecahedron. | Fig. 35. Dyakis-dodecahedron. |
Here the cube, octahedron, rhombic dodecahedron, triakis-octahedron and icositetrahedron are geometrically the same as in the first class. The characters of the faces will, however, be different; thus the cube faces will be striated parallel to one edge only (fig. 89), and triangular markings on the octahedron faces will be placed obliquely to the edges. The remaining simple forms are “hemihedral with parallel faces,” and from the corresponding holohedral forms two hemihedral forms, a positive and a negative, may be derived.
Pentagonal dodecahedron (fig. 34). This is bounded by twelve pentagonal faces, but these are not regular pentagons, and the angles over the three sets of different edges are different. The regular dodecahedron of geometry, contained by twelve regular pentagons, is not a possible form in crystals. The indices are {hko}: as a simple form {210} is of very common occurrence in pyrites.
Dyakis-dodecahedron (fig. 35). This is the hemihedral form of
