(Tetrahedral-hemihedral; Hexakis-tetrahedral).
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.
[Illustration: FIG. 26.--Tetrahedron.]
[Illustration: 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."
[Illustration: FIG. 28.--Triakis-tetrahedron.]
[Illustration: 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 deg. 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.
[Illustration: FIG. 30.--Combination of two Tetrahedra.]
[Illustration: 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.
[Illustration: FIG. 32.--Combination of Tetrahedron, Cube and Rhombic
Dodecahedron.]
[Illustration: 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 {1'11}, 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.