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If you look back at Fig. 1.1, you will see that it combines elements of symmetry
with a repeating pattern. We call such a repeating pattern of motifs an
*array* ; the smallest convenient parallelogram that can be repeated without
change of orientation to produce the pattern is called the *unit cell* . A
two-dimensional unit cell is outlined in Fig. 1.1; note that while the choice of
origin is somewhat arbitrary the shape is not. (Check this by identifying other
ways of defining the repeating unit.)

In a three-dimensional array, such as a crystal, the unit cell is a
parallelipeped defined by intercepts *a, b, c* on three axes (*x, y,
z* ) and the angles between them, , , , as shown in Fig.
3.1. If the contents have no symmetry or a centre of symmetry only, the unit
cell can have this quite general shape; more symmetrical cell contents restrict
the values of the interaxial angles and the relative sizes of the edges in the
manner given in Table 1. Thus a single direction of twofold symmetry
(monoclinic system) makes two of the angles into right-angles but places no
restrictions on the third or on the edge dimensions; at the other extreme, cubic
symmetry produces a cell whose edges are all equal and whose angles are all
90 (i.e. a cube!).

An array can have any of the symmetry elements that we have discussed already,
including no symmetry at all (Fig. 3.2), and it can also have additional types
of symmetry not possible in finite objects. Consider Fig. 3.3 which shows an
aerial view of a boat rowed by eight crew; provided that they have been well
coached they present a symmetrical appearance, but it is not one that can be
exactly described by any of the symmetry elements introduced so far. It is
obviously related to a mirror plane, but in Fig. 3.3 each rower is the mirror
image of one rowing immediately in front or behind. Any figure is related to
the next by moving one place along the boat and then reflecting across a mirror
plane. A symmetry operation of this type is called -- very descriptively --
*a glide plane* . Because a glide plane combines the operation of
reflection with that of translation it occurs only in extended arrays.

An analogous operation combining rotation and translation is called -- equally
descriptively -- a *screw axis* . As an actual example of an object
possessing this type of symmetry, a bolt is really better than a screw, since
most screws taper to a point, but the action of driving a screw -- or using a
corkscrew! -- illustrates very vividly the operation of a screw axis. Other
familiar objects having screw axes are spiral staircases, springs, and some
climbing plants. Formal examples are shown in Fig. 3.4, together with a formal
representation of a glide plane.

The general symbol for a screw axis is *N*_{n}, where *N* is the order (2, 3, 4 or
6) of the axis, and *n* /*N* the translation distance expressed as a fraction of the
repeat unit. Thus, 4_{1}, shown in Fig. 3.4b, means that the asymmetric unit
moves of a repeat unit along the axis for each of a
revolution about that axis. Glide planes are symbolised by a letter indicating
the direction of the glide: the letters *a, b* and *c* mean that the
direction of glide is parallel to the *a, b* and *c* axes,
respectively, while *n* and *d* refer to glide planes in which the
direction of glide is diagonally across a face of the unit cell or along a body
diagonal.

A combination of parallel translational and non-translational symmetry elements
produces an interesting effect on the way in which the pattern repeats.
Consider Fig. 3.5. The pattern in 3.5a contains mirror planes: note that two
mirror planes are associated with each repeat unit across the page. The pattern
in 3.5b is based on a similar motif related by glide planes (*g* ); again two are
associated with each repeat unit across the page. In 3.5c there are parallel
mirror and glide planes, and as a result the grouping of motifs at the cente of
the rectangular cell is identical with that at the corners. Such a pattern is
called *centred* , while those of 3.5a and 3.5b are said to be
*primitive* . It is always possible to define a smaller primitive cell for
a centred pattern, such as the diamond-shaped cell outlined at the lower right
of 3.5c. However, this is not normally done, partly because such a cell is a
less convenient shape, but more importantly because its axes no longer bear the
correct relationship to the symmetry elements of the pattern.

**Copyright © 1984, 1997 International Union of
Crystallography**