[Illustration: 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 deg. or 180 deg. 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}, {41'1'}, {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.
[Illustration: FIG. 69.--Scalenohedron with inscribed Rhombohedron.]
[Illustration: FIG. 70.--Hexagonal Bipyramid.]
[Illustration: FIG. 71.--Hexagonal Prism and Basal Pinacoid.]
Hexagonal prism of the first order (21'1'), 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 deg.
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 deg. 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 {32'1'}, {53'2'} ...
{hk'l}, where h + k + l = 0.
Basal pinacoid {111}, consisting of a pair of parallel faces
perpendicular to the principal axis.
[Illustration: 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 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.