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 into a set is a relation such that for each element there is a unique element which is assigned to . The element is called the image of .
Fig. 2.1.1 The relation of the point to the points and is not a mapping because the image point is not uniquely defined (there are two image points). 
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.
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 rotoinversions, screw rotations, and glide reflections.
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 and around its center, reflections
through the three thick lines intersecting the center, and the identity
operation, see Section 3.2.

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