An axis passing through the centre O and the middle points d of two
opposite edges of the octahedron (fig. 4), i.e. parallel to the edges of
the octahedron, is a dyad axis of symmetry. About this axis there may be
rotation of 180 deg., and only twice in a complete revolution of 360
deg. (= 180 deg. X 2) is the crystal brought into interchangeable
positions. There being six pairs of parallel edges on an octahedron,
there are consequently six dyad axes of symmetry.
A regular octahedron thus possesses thirteen axes of symmetry (of three
kinds), and there are the same number in the cube. Fig. 5 shows the
three tetrad (or tetragonal) axes (aa), four triad (or trigonal) axes
(pp), and six dyad (diad or diagonal) axes (dd).
Although not represented in the cubic system, there is still another
kind of axis of symmetry possible in crystals. This is the hexad axis or
hexagonal axis, for which the angle of rotation is 60 deg., or one-sixth
of 360 deg. There can be only one hexad axis of symmetry in any crystal
(see figs. 77-80).
[Illustration: FIG. 5.--Axes of Symmetry of a Cube.]
_Planes of Symmetry._--A regular octahedron can be divided into two
equal and similar halves by a plane passing through the corners
a1a3a'1a'3 and the centre O (fig. 3). One-half is the mirror reflection
of the other in this plane, which is called a plane of symmetry.
Corresponding planes on either side of a plane of symmetry are inclined
to it at equal angles. The octahedron can also be divided by similar
planes of symmetry passing through the corners a1a2a'1a'2 and
a2a3a'2a'3. These three similar planes of symmetry are called the cubic
planes of symmetry, since they are parallel to the faces of the cube
(compare figs. 6-8, showing combinations of the octahedron and the
cube).
A regular octahedron can also be divided symmetrically into two equal
and similar portions by a plane passing through the corners a3 and a'3,
the middle points d of the edges a1a'2 and a'1a2, and the centre O (fig.
4). This is called a dodecahedral plane of symmetry, being parallel to
the face of the rhombic dodecahedron which truncates the edge a1a2
(compare fig. 14, showing a combination of the octahedron and rhombic
dodecahedron). Another similar plane of symmetry is that passing through
the corners a3a'3 and the middle points of the edges a1a2 and a'1a'2,
and altogether there are six dodecahedral planes of symmetry, two
through each of the corners a1, a2, a3 of the octahedron.
A regular octahedron and a cube are thus each symmetrical with respect
to the following elements of symmetry: a centre of symmetry, thirteen
axes of symmetry (of three kinds), and nine planes of symmetry (of two
kinds). This degree of symmetry, which is the type corresponding to one
of the classes of the cubic system, is the highest possible in crystals.
As will be pointed out below, it is possible, however, for both the
octahedron and the cube to be associated with fewer elements of symmetry
than those just enumerated.
(b) _Simple Forms and Combinations of Forms._
A single face a1a2a3 (figs. 3 and 4) may be repeated by certain of the
elements of symmetry to give the whole eight faces of the octahedron.
Thus, by rotation about the vertical tetrad axis a3a'3 the four upper
faces are obtained; and by rotation of these about one or other of the
horizontal tetrad axes the eight faces are derived. Or again, the same
repetition of the faces may be arrived at by reflection across the three
cubic planes of symmetry. (By reflection across the six dodecahedral
planes of symmetry a tetrahedron only would result, but if this is
associated with a centre of symmetry we obtain the octahedron.) Such a
set of similar faces, obtained by symmetrical repetition, constitutes a
"simple form." An octahedron thus consists of eight similar faces, and a
cube is bounded by six faces all of which have the same surface
characters, and parallel to each of which all the properties of the
crystal are identical.
[Illustration: FIG. 6.--Cube in combination with Octahedron.]
[Illustration: FIG. 7.--Cubo-octahedron.]
[Illustration: FIG. 8.--Octahedron in combination with Cube.]
Examples of simple forms amongst crystallized substances are octahedra
of alum and spinel and cubes of salt and fluorspar. More usually,
however, two or more forms are present on a crystal, and we then have a
combination of forms, or simply a "combination." Figs. 6, 7 and 8
represent combinations of the octahedron and the cube; in the first the
faces of the cube predominate, and in the third those of the octahedron;
fig. 7 with the two forms equally developed is called a cubo-octahedron.
Each of these combined forms has all the elements of symmetry proper to
the simple forms.
The simple forms, though referable to the same type of symmetry and axes
of reference, are quite independent, and cannot be derived one from the
other by symmetrical repetition, but, after the manner of Rome de
l'Isle, they may be derived by replacing edges or corners by a face
equally inclined to the faces forming the edges or corners; this is
known as "truncation" (Lat. _truncare_, to cut off). Thus in fig. 6 the
corners of the cube are symmetrically replaced or truncated by the faces
of the octahedron, and in fig. 8 those of the octahedron are truncated
by the cube.
(c) _Law of Rational Intercepts._
For axes of reference, OX, OY, OZ (fig. 9), take any three edges formed
by the intersection of three faces of a crystal. These axes are called
the crystallographic axes, and the planes in which they lie the axial
planes. A fourth face on the crystal intersecting these three axes in
the points A, B, C is taken as the parametral plane, and the lengths OA
: OB : OC are the parameters of the crystal. Any other face on the
crystal may be referred to these axes and parameters by the ratio of
the intercepts
OA OB OC
-- : -- : --.
h k l
Thus for a face parallel to the plane A Be the intercepts are in the
ratio OA : OB : Oe, or
OA OB OC
-- : -- : --
1 1 2
and for a plane fgC' they are Of : Og : OC' or
OA OB OC'
-- : -- : ---.
2 3 1
Now the important relation existing between the faces of a crystal is
that the denominators h, k and l are always rational whole numbers,
rarely exceeding 6, and usually 0, 1, 2 or 3. Written in the form (hkl),
h referring to the axis OX, k to OY, and l to OZ, they are spoken of as
the indices (Millerian indices) of the face. Thus of a face parallel to
the plane ABC the indices are (111), of A Be they are (112), and of fgC'
(231'). The indices are thus inversely proportional to the intercepts,
and the law of rational intercepts is often spoken of as the "law of
rational indices."
The angular position of a face is thus completely fixed by its indices;
and knowing the angles between the axial planes and the parametral plane
all the angles of a crystal can be calculated when the indices of the
faces are known.
[Illustration: FIG. 9.--Crystallographic axes of reference.]
Although any set of edges formed by the intersection of three planes may
be chosen for the crystallographic axes, it is in practice usual to
select certain edges related to the symmetry of the crystal, and usually
coincident with axes of symmetry; for then the indices will be simpler
and all faces of the same simple form will have a similar set of
indices. The angles between the axes and the ratio of the lengths of the
parameters OA: OB: OC (usually given as a: b: c) are spoken of as the
"elements" of a crystal, and are constant for and characteristic of all
crystals of the same substance.
The six systems of crystal forms, to be enumerated below, are defined by
the relative inclinations of the crystallographic axes and the lengths
of the parameters. In the cubic system, for example, the three
crystallographic axes are taken parallel to the three tetrad axes of
symmetry, i.e. parallel to the edges of the cube (fig. 5) or joining the
opposite corners of the octahedron (fig. 3), and they are therefore all
at right angles; the parametral plane (111) is a face of the octahedron,
and the parameters are all of equal length. The indices of the eight
faces of the octahedron will then be (111), (1'11), (11'1), (1'1'1),
(111'), (1'11'), (11'1'), (1'1'1'). The symbol {111} indicates all the
faces belonging to this simple form. The indices of the six faces of the
cube are (100), (010), (001), (1'00), (01'0), (001'); here each face is
parallel to two axes, i.e. intercepts them at infinity, so that the
corresponding indices are zero.
(d) _Zones._
An important consequence of the law of rational intercepts is the
arrangement of the faces of a crystal in zones. All faces, whether they
belong to one or more simple forms, which intersect in parallel edges
are said to lie in the same zone. A line drawn through the centre O of
the crystal parallel to these edges is called a zone-axis, and a plane
perpendicular to this axis is called a zone-plane. On a cube, for
example, there are three zones each containing four faces, the zone-axes
being coincident with the three tetrad axes of symmetry. In the crystal
of zircon (fig. 88) the eight prism-faces a, m, &c. constitute a zone,
denoted by [a, m, a', &c.], with the vertical tetrad axis of symmetry
as zone-axis. Again the faces [a, x, p, e', p', x"', a"] lie in
another zone, as may be seen by the parallel edges of intersection of
the faces in figs. 87 and 88; three other similar zones may be traced on
the same crystal.
The direction of the line of intersection (i.e. zone-axis) of any two
planes (hkl) and (h1k1l1) is given by the zone-indices [uvw], where u =
kl1 - lk1, v = lh1 - hl1, and w = hk1 - kh1, these being obtained from
the face-indices by cross multiplication as follows:--
h k l h k l
X X X
h1 k1 l1 h1 k1 l1.