h2u + k2v + l2w = 0.
This important relation connecting the indices of a face lying in a zone
with the zone-indices is known as Weiss's zone-law, having been first
enunciated by C. S. Weiss. It may be pointed out that the indices of a
face may be arrived at by adding together the indices of faces on either
side of it and in the same zone; thus, (311) in fig. 12 lies at the
intersections of the three zones [210, 101], [201, 110] and [211, 100],
and is obtained by adding together each set of indices.
(e) _Projection and Drawing of Crystals._
The shapes and relative sizes of the faces of a crystal being as a rule
accidental, depending only on the distance of the faces from the centre
of the crystal and not on their angular relations, it is often more
convenient to consider only the directions of the normals to the faces.
For this purpose projections are drawn, with the aid of which the zonal
relations of a crystal are more readily studied and calculations are
simplified.
[Illustration: FIG. 10.--Stereographic Projection of a Cubic Crystal.]
The kind of projection most extensively used is the "stereographic
projection." The crystal is considered to be placed inside a sphere from
the centre of which normals are drawn to all the faces of the crystal.
The points at which these normals intersect the surface of the sphere
are called the poles of the faces, and by these poles the positions of
the faces are fixed. The poles of all faces in the same zone on the
crystal will lie on a great circle of the sphere, which are therefore
called zone-circles. The calculation of the angles between the normals
of faces and between zone-circles is then performed by the ordinary
methods of spherical trigonometry. The stereographic projection,
however, represents the poles and zone-circles on a plane surface and
not on a spherical surface. This is achieved by drawing lines joining
all the poles of the faces with the north or south pole of the sphere
and finding their points of intersection with the plane of the
equatorial great circle, or primitive circle, of the sphere, the
projection being represented on this plane. In fig. 10 is shown the
stereographic projection, or stereogram, of a cubic crystal; a^1, a^2,
&c. are the poles of the faces of the cube. o^1, o^2, &c. those of the
octahedron, and d^1, d^2, &c. those of the rhombic dodecahedron. The
straight lines and circular arcs are the projections on the equatorial
plane of the great circles in which the nine planes of symmetry
intersect the sphere. A drawing of a crystal showing a combination of
the cube, octahedron and rhombic dodecahedron is shown in fig. 11, in
which the faces are lettered the same as the corresponding poles in the
projection. From the zone-circles in the projection and the parallel
edges in the drawing the zonal relations of the faces are readily seen:
thus [a^1o^1d^5], [a^1d^1a^5], [a^5o^1d^2], &c. are zones. A
stereographic projection of a rhombohedral crystal is given in fig. 72.
[Illustration: FIG. 11.--Clinographic Drawing of a Cubic Crystal.]
Another kind of projection in common use is the "gnomonic projection"
(fig. 12). Here the plane of projection is tangent to the sphere, and
normals to all the faces are drawn from the centre of the sphere to
intersect the plane of projection. In this case all zones are
represented by straight lines. Fig. 12 is the gnomonic projection of a
cubic crystal, the plane of projection being tangent to the sphere at
the pole of an octahedral face (111), which is therefore in the centre
of the projection. The indices of the several poles are given in the
figure.
[Illustration: FIG. 12.--Gnomonic Projection of a Cubic Crystal.]
In drawing crystals the simple plans and elevations of descriptive
geometry (e.g. the plans in the lower part of figs. 87 and 88) have
sometimes the advantage of showing the symmetry of a crystal, but they
give no idea of solidity. For instance, a cube would be represented
merely by a square, and an octahedron by a square with lines joining the
opposite corners. True perspective drawings are never used in the
representation of crystals, since for showing the zonal relations it is
important to preserve the parallelism of the edges. If, however, the
eye, or point of vision, is regarded as being at an infinite distance
from the object all the rays will be parallel, and edges which are
parallel on the crystal will be represented by parallel lines in the
drawing. The plane of the drawing, in which the parallel rays joining
the corners of the crystals and the eye intersect, may be either
perpendicular or oblique to the rays; in the former case we have an
"orthographic" ([Greek: orthos], straight; [Greek: graphein], to draw)
drawing, and in the latter a "clinographic" ([Greek: klinein], to
incline) drawing. Clinographic drawings are most frequently used for
representing crystals. In representing, for example, a cubic crystal
(fig. 11) a cube face a^5 is first placed parallel to the plane on which
the crystal is to be projected and with one set of edges vertical; the
crystal is then turned through a small angle about a vertical axis until
a second cube face a^2 comes into view, and the eye is then raised so
that a third cube face a^1 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.