According to the first law, the angles between corresponding faces of
all crystals of the same chemical substance are always the same and are
characteristic of the substance.
(a) _Symmetry of Crystals._
Crystals may, or may not, be symmetrical with respect to a point, a line
or axis, and a plane; these "elements of symmetry" are spoken of as a
centre of symmetry, an axis of symmetry, and a plane of symmetry
respectively.
_Centre of Symmetry._--Crystals which are centro-symmetrical have their
faces arranged in parallel pairs; and the two parallel faces, situated
on opposite sides of the centre (O in fig. 3) are alike in surface
characters, such as lustre, striations, and figures of corrosion. An
octahedron (fig. 3) is bounded by four pairs of parallel faces. Crystals
belonging to many of the hemihedral and tetartohedral classes of the six
systems of crystallization are devoid of a centre of symmetry.
_Axes of Symmetry._--Consider the vertical axis joining the opposite
corners a3 and a'3 of an octahedron (fig. 3) and passing through its
centre O: by rotating the crystal about this axis through a right angle
(90 deg.) it reaches a position such that the orientation of its faces
is the same as before the rotation; the face a'1a'2a'3, for example,
coming into the position of a1a'2a3. During a complete rotation of 360
deg. (= 90 deg. X 4), the crystal occupies four such interchangeable
positions. Such an axis of symmetry is known as a tetrad axis of
symmetry. Other tetrad axes of the octahedron are a2a'2 and a1a1.
An axis of symmetry of another kind is that which passing through the
centre O is normal to a face of the octahedron. By rotating the crystal
about such an axis Op (fig. 3) through an angle of 120 deg. those faces
which are not perpendicular to the axis occupy interchangeable
positions; for example, the face a1a3a2 comes into the position of
a'2a1a'3, and a'2a1a'3 to a3a'2a'1. During a complete rotation of 360
deg. (= 120 deg. X 3) the crystal occupies similar positions three
times. This is a triad axis of symmetry; and there being four pairs of
parallel faces on an octahedron, there are four triad axes (only one of
which is drawn in the figure).
[Illustration: FIG. 3.]
[Illustration: FIG. 4.