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Mappings and symmetry operations

In crystallography, mapping an object of point space, e.g. the atomic centers of a molecule or a crystal pattern, is one of the most basic procedures. Most crystallographic mappings are rather special. Nevertheless, the term `mapping' will be introduced first in a more general way. What is a mapping of, e.g., a set of points ?

Definition (D 2.1.1) A mapping of a set $A$ into a set $B$ is a relation such that for each element $a \in A$ there is a unique element $b\in B$ which is assigned to $a$. The element $b$ is called the image of $a$.

Fig. 2.1.1
Fig. 2.1.1 The relation of the point $X$ to the points $\tilde{X}_1$ and $\tilde{X}_2$ is not a mapping because the image point is not uniquely defined (there are two image points).

\psfig{figure=mapp.eps,width=5cm}%%(exported with 50 % size)
Fig. 2.1.2 The five regions of the set $A$ (the triangle) are mapped onto the five separated regions of the set $B$. No point of $A$ is mapped onto more than one image point. Region 2 is mapped on a line, the points of the line are the images of more than one point of $A$. Such a mapping is called a projection.

The mapping which is displayed in Fig. 2.1.2 is rather complicated and can hardly be described analytically. The mappings which are mainly used in crystallography are much simpler: In general they map closed regions onto closed regions. Although distances between points or angles between lines may be changed, parallel lines of the original figure are always parallel also in the image. Such mappings are called affine mappings. An affine mapping will in general distort an object, e.g. by a shearing action or by an (isotropic or anisotropic) shrinking, see Fig. 2.1.3. For example, in the space a cube may be distorted by an affine mapping into an arbitrary parallelepiped but not into an octahedron or tetrahedron.

Fig. 2.1.3 In an affine mapping parallel lines of the original figure (the rectangular triangle) are mapped onto parallel lines of the image (the nearly isoscale triangle). Lengths and angles may be distorted but relations of lengths on the same line are preserved.

If an affine mapping leaves all distances and thus all angles invariant, it is called isometric mapping, isometry, motion, or rigid motion. We shall use the name `isometry'. An isometry does not distort but moves the undistorted object through the point space. However, it may change the orientation of an object, e.g. transfer a right glove into an (otherwise identical) left one. Different types of isometries are distinguished: In the space these are translations, rotations, inversions, reflections, and the more complicated roto-inversions, screw rotations, and glide reflections.

\psfig{figure=isma.eps,width=4.5cm}%%(exported \lq Portrait', 35 % size)
Fig. 2.1.4 An isometry leaves all distances and angles invariant. An `isometry of the first kind', preserving the counter-clockwise sequence of the edges `short-middle-long' of the triangle is displayed in the upper mapping. An `isometry of the second kind', changing the counter-clockwise sequence of the edges of the triangle to a clockwise one is seen in the lower mapping.

Fig. 2.1.5 A parallel shift of the triangle is called a translation. Translations are special isometries. They play a distinguished role in crystallography.

One of the outstanding concepts in crystallography is `symmetry'. An object has symmetry if there are isometries which map the object onto itself such that the mapped object can not be distinguished from the object in the original state. The isometries which map the object onto itself are called symmetry operations of this object. The symmetry of the object is the set of all its symmetry operations. If the object is a crystal pattern, representing a real crystal, its symmetry operations are called crystallographic symmetry operations.

Fig. 2.1.6 The equilateral triangle allows six symmetry operations: rotations by $120^{\circ}$ and $240^{\circ}$ around its center, reflections through the three thick lines intersecting the center, and the identity operation, see Section 3.2.

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