in this class; for example, calcite, chalybite, calamine, corundum
(ruby and sapphire), haematite, chabazite; the elements arsenic,
antimony, bismuth, selenium, tellurium and perhaps graphite;
also ice, sodium nitrate, thymol, &c.
Ditrigonal Pyramidal Class
Here there are three similar planes of symmetry intersecting in the triad axis; there are no dyad axes and no centre of symmetry. The triad axis is uniterminal and polar, and the crystals are differently developed at the two ends; crystals of this class are therefore pyro-electric. The forms are all open forms:—
 |
| Fig. 73.—Crystal of Tourmaline. |
Trigonal pyramid {hkk}, consisting of the three faces which correspond to the three upper or the three lower faces of a rhombohedron of the holosymmetric class.
Ditrigonal pyramid {hkl}, of six faces, corresponding to the six upper or lower faces of the scalenohedron.
Hexagonal pyramid (hkl) where (h − 2k + l = 0), of six faces, corresponding to the six upper or lower faces of the hexagonal bipyramid.
Trigonal prism {211} or {211}, two forms each consisting of three faces parallel to principal axis and perpendicular to the planes of symmetry.
Hexagonal prism {101}, which is geometrically the same as in the last class.
Ditrigonal prism {hkl} (where h + k + l = 0), of six faces parallel to the principal axis, and with two sets of angles between them.
Basal pedion (111) or (111), each consisting of a single plane perpendicular to the principal axis.
Fig. 73 represents a crystal of tourmaline with the trigonal prism (211), hexagonal prism (101), and a trigonal pyramid at each end. Other substances crystallizing in this class are pyrargyrite, proustite, iodyrite (AgI), greenockite, zincite, spangolite, sodium lithium sulphate, tolylphenylketone.
Trapezohedral Class
Here there are three similar dyad axes inclined to one another at 60° and perpendicular to the triad axis. There are no planes or centre of symmetry. The dyad axes are uniterminal, and are pyro-electric axes. Crystals of most substances of this class rotate the plane of polarization of a beam of light.
 | |
| Fig. 74.—Trigonal Trapezohedron. |
Fig. 75.—Trigonal Bipyramid. |
In this class the rhombohedra {hkk}, the hexagonal prism {211}, and the basal pinacoid {111} are geometrically the same as in the holosymmetric class; the trigonal prism {101} and the ditrigonal prisms are as in the ditrigonal pyramidal class. The remaining simple forms are:—
Trigonal trapezohedron (fig. 74), bounded by six trapezoidal faces. There are two complementary and enantiomorphous trapezohedra, {hkl} and {hlk}, derivable from the scalenohedron.
Trigonal bipyramid (fig. 75), bounded by six isosceles triangles; the indices are {hkl}, where h − 2k + l = 0, as in the hexagonal bipyramid.
The only minerals crystallizing in this class are quartz (q.v.) and cinnabar, both of which rotate the plane of a beam of polarized light transmitted along the triad axis. Other examples are dithionates of lead (PbS2O6·4H2O), calcium and strontium, and of potassium (K2S2O6), benzil, matico-stearoptene.
Rhombohedral Class
The only elements of symmetry are the triad axis and a centre of symmetry. The general form {hkl} is a rhombohedron, and is a hemihedral form, with parallel faces, of the scalenohedron. The form {hkl}, where h − 2k + l = 0, is also a rhombohedron, being the hemihedral form of the hexagonal bipyramid. The dihexagonal prism {hkl} of the holosymmetric class becomes here a hexagonal prism. The rhombohedra (hkk), hexagonal prisms {211} and {101}, and the basal pinacoid {111} are geometrically the same in this class as in the holosymmetric class.
Fig. 76 represents a crystal of dioptase with the fundamental rhombohedron r {100} and the hexagonal prism of the second order m {101} combined with the rhombohedron s {031}.
Examples of minerals which crystallize in this class are phenacite, dioptase, willemite, dolomite, ilmenite and pyrophanite: amongst artificial substances is ammonium periodate ((NH4)4I2O9·3H2O).
Trigonal Pyramidal Class
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| Fig. 76.—Crystal of Dioptase. |
Here there is only the triad axis of symmetry, which is uniterminal. The general form {hkl} is a trigonal pyramid consisting of three faces at one end of the crystal. All other forms, in which the faces are neither parallel nor perpendicular to the triad axis, are trigonal pyramids. All the prisms are trigonal prisms; and perpendicular to these are two pedions.
The only substance known to crystallize in this class is sodium periodate (NaIO4·3H2O), the crystals of which are circularly polarizing.
Trigonal Bipyramidal Class
Here there is a plane of symmetry perpendicular to the triad axis. The trigonal pyramids of the last class are here trigonal bipyramids (fig. 75); the prisms are all trigonal prisms, and parallel to the plane of symmetry is the basal pinacoid. No example is known for this class.
Ditrigonal Bipyramidal Class
Here there are three similar planes of symmetry intersecting in the triad axis, and perpendicular to them is a fourth plane of symmetry; at the intersection of the three vertical planes with the horizontal plane are three similar dyad axes; there is no centre of symmetry.
 |
| Fig. 77.—Dihexagonal Bipyramid. |
The general form is bounded by twelve scalene triangles and is a ditrigonal bipyramid. Like the general form of the last class, this has two sets of indices {hkl, pqr}, (hkl) for faces above the equatorial plane of symmetry and (pqr) for faces below: with hexagonal axes there would be only one set of indices. The hexagonal bipyramids, the hexagonal prism {101} and the basal pinacoid {111} are geometrically the same in this class as in the holosymmetric class. The trigonal prism {211} and ditrigonal prisms {hkl} are the same as in the ditrigonal pyramidal class.
The only representative of this type of symmetry is the mineral benitoite (q.v.).
In crystals of this division of the hexagonal system the principal axis is a hexad axis of symmetry. Hexagonal axes of reference are used: if rhombohedral axes be used many of the simple forms will have two sets of indices.
Holosymmetric Class
Intersecting in the hexad axis are six planes of symmetry of two kinds, and perpendicular to them is an equatorial plane of symmetry. Perpendicular to the hexad axis are six dyad axes of two kinds and each perpendicular to a vertical plane of symmetry. The seven simple forms are:—
Dihexagonal bipyramid, bounded by twenty-four scalene triangles (fig. 77; v in fig. 80). The indices are {2131}, &c., or in general {hikl}. This form may be considered as a combination of two scalenohedra, a direct and an inverse.
 | ||
| Fig. 78. | Fig. 79. | Fig. 80. |
| Combinations of Hexagonal forms. | ||
Hexagonal bipyramid of the first order, bounded by twelve isosceles triangles (fig. 70; p and u in fig. 80); indices {1011}, {2021} ... (hohl). The hexagonal bipyramid so common in quartz is geometrically similar to this form, but it really is a combination of two rhombohedra, a direct and an inverse, the faces of which differ in surface characters and often also in size.
