Geometric mappings have the property that for each point P of the space, and thus of the object, there is a uniquely determined point , the image point. If also for each image point there is a uniquely determined preimage or original point P, then the mapping is called reversible. Nonreversible mappings are called projections, cf. Section 1.4.5
.
A mapping is called a motion, a rigid motion or an isometry if it leaves all distances invariant (and thus all angles, as well as the size and shape of an object). In this volume the term `isometry' is used.
Isometries are a special kind of affine mappings. In an affine mapping, parallel lines are mapped onto parallel lines; lengths and angles may be distorted but distances along the same line are preserved.
A mapping is called a symmetry operation of an object if (i) it is an isometry, and (ii) it maps the object onto itself. Instead of `maps the object onto itself' one frequently says `leaves the object invariant (as a whole)'.
Real crystals are finite objects in physical space, which because of the presence of impurities and structural imperfections such as disorder, dislocations etc. are not perfectly symmetric. In order to describe their symmetry properties, real crystals are modelled as blocks of ideal, infinitely extended periodic structures, known as ideal crystals or (ideal) crystal structures. Crystal patterns are models of crystal structures in point space. In other words, while the crystal structure is an infinite periodic spatial arrangement of the atoms (ions, molecules) of which the real crystal is composed, the crystal pattern is the related model of the ideal crystal (crystal structure) consisting of a strictly threedimensional periodic set of points in point space. If the growth of the ideal crystal is undisturbed, then it forms an ideal macroscopic crystal and displays its ideal shape with planar faces.
Both the symmetry operations of an ideal crystal and of a crystal pattern are called crystallographic symmetry operations. The symmetry operations of the ideal macroscopic crystal form the finite point group of the crystal, those of the crystal pattern form the (infinite) space group of the crystal pattern. Because of its periodicity, a crystal pattern always has translations among its symmetry operations.
The symmetry operations are divided into two main kinds depending whether they preserve or not the socalled handedness or chirality of chiral objects. Isometries of the first kind or proper isometries are those that preserve the handedness of chiral objects: e.g. if a right (left) glove is mapped by one of these isometries, then the image is also a right (left) glove of equal size and shape. Isometries that change the handedness, i.e. the image of a right glove is a left one, of a left glove is a right one, are called isometries of the second kind or improper isometries. Improper isometries cannot be performed in space physically but can nevertheless be observed as symmetries of objects.
The notion of fixed points is essential for the characterization of symmetry operations. A point P is a fixed point of a mapping if it is mapped onto itself, i.e. the image point is the same as the original point P: . The set of all fixed points of an isometry may be the whole space, a plane in the space, a straight line, a point, or the set may be empty (no fixed point).
Crystallographic symmetry operations are also characterized by their order: a symmetry operation is of order k if its application k times results in the identity mapping, i.e , where is the identity operation, and is the smallest number for which this equation is fulfilled.

There are eight different types of isometries that may be crystallographic symmetry operations:
(1) The identity operation maps each point of the space onto itself, i.e. the set of fixed points is the whole space. It is the only operation whose order is 1. The identity operation is a symmetry operation of the first kind. It is a symmetry operation of any object and although trivial, it is indispensable for the group properties of the set of symmetry operations of the object (cf. Section 1.1.2
).
(2) A translation is characterized by its translation vector t. Under translation every point of space is shifted by t, hence a translation has no fixed point. A translation is a symmetry operation of infinite order as there is no number such that with translation vector o. It preserves the handedness of any chiral object.
(3) A rotation is an isometry which leaves one line fixed pointwise. This line is called the rotation axis. The degree of rotation about this axis is described by its rotation angle . Because of the periodicity of crystals, the rotation angles of crystallographic rotations are restricted to , where N = 2, 3, 4 or 6 and k is an integer which is relative prime to N. A rotation of rotation angle is of order N and is called an Nfold rotation. A rotation preserves the handedness of any chiral object.
The rotations are also characterized by their sense of rotation. The adopted convention for positive (negative) sense of rotation follows the mathematical convention for positive (negative) sense of rotation: the sense of rotation is positive (negative) if the rotation is counterclockwise (clockwise) when viewed down the rotation axis.
(4) A screw rotation is a rotation coupled with a translation parallel to the rotation axis. The rotation axis is called the screw axis. The translation vector is called the screw vector or the intrinsic translation component (of the screw rotation), cf. Section 1.2.2.4. A screw rotation has no fixed points because of its translation component. However, the screw axis is invariant pointwise under the socalled reduced symmetry operation of the screw rotation: it is the rotation obtained from the screw rotation by removing its intrinsic translation component.
The screw rotation is a proper symmetry operation. If is the smallest rotation angle of a screw rotation, then the screw rotation is called Nfold. Owing to its translation component, the order of any screw rotation is infinite. Let u be the shortest lattice vector in the direction of the screw axis, and , with and integer, be the screw vector of the screw rotation by the angle . After N screw rotations with rotation angle the crystal pattern has its original orientation but is shifted parallel to the screw axis by the lattice vector .
(5) An Nfold rotoinversion is an Nfold rotation coupled with an inversion through a point on the rotation axis. This point is called the centre of the rotoinversion. For it is the only fixed point. The axis of the rotation is invariant as a whole under the rotoinversion and is called its rotoinversion axis. The restrictions on the angles of the rotational parts are the same as for rotations. The order of an Nfold rotoinversion is N for even N and 2N for odd N. A rotoinversion changes the handedness by its inversion component: it maps any righthand glove onto a lefthand one and vice versa. Special rotoinversions are those for and which are dealt with separately.
The rotoinversions can be described equally as rotoreflections . The Nfold rotation is now coupled with a reflection through a plane which is perpendicular to the rotation axis and cuts the axis in its centre. The following equivalences hold: , , , and . In this volume the description by rotoinversions is chosen.
(6) The inversion can be considered as a onefold rotoinversion () or equally as a twofold rotoreflection . The fixed point is called the inversion centre. The inversion is a symmetry operation of the second kind, its order is 2.
(7) A twofold rotoinversion () is equivalent to a reflection or a reflection through a plane and is simultaneously a onefold rotoreflection (). It is an isometry which leaves the plane perpendicular to the twofold rotoinversion axis fixed pointwise. This plane is called the reflection plane or mirror plane; it intersects the rotation axis in its centre. Its orientation is described by the direction of its normal vector, i.e. of the rotation axis. (Note that in the spacegroup tables of Part 2 the reflection planes are specified by their locations, and not by their normal vectors, cf. Section 1.4.2.1
.) The order of a reflection is 2. As for any rotoinversion, the reflection changes the handedness of a chiral object.
(8) A glide reflection is a reflection through a plane coupled with a translation parallel to this plane. The translation vector is called the glide vector (or the intrinsic translation component of the glide reflection, cf. Section 1.2.2.4). A glide reflection changes the handedness and has no fixed point. The set of fixed points of the related reduced symmetry operation (i.e. the reflection that is obtained by removing the glide component from the glide reflection) is called the glide plane. The glide vector of a glide reflection is 1/2 of a lattice vector t (including centring translations of centredcell lattice descriptions, cf. Table 2.1.1.2
). Whereas twice the application of a reflection restores the original position of the crystal pattern, applying a glide reflection twice results in a translation of the crystal pattern with the translation vector . The order of any glide reflection is infinite.

It is natural to combine the matrix part and the column part describing an affine mapping to form a (3 × 4) matrix, but such matrices cannot be multiplied by the usual matrix multiplication and cannot be inverted. However, if one supplements the (3 × 4) matrix by a fourth row `0 0 0 1', one obtains a (4 × 4) square matrix which can be combined with the analogous matrices of other mappings and can be inverted. These matrices are called augmented matrices, and here they are designated by openface letters. Similarly, the columns and x also have to be extended to the augmented columns and :The horizontal and vertical lines in the augmented matrices are useful to facilitate recognition of their coefficients; they have no mathematical meaning.
Equations (1.2.2.1), (1.2.2.7) and (1.2.2.8) then become In the usual description by columns, the vector coefficients cannot be distinguished from the point coordinates, but in the augmentedcolumn description the difference becomes visible.
If and are the augmented columns of coordinates of the points and , and , then is the augmented column of the coefficients of the vector v between P and Q. The last coefficient of is zero, because 1 − 1 = 0. Thus, the column of the coefficients of a vector is not augmented by `1' but by `0'. From the equation for the transformation of the vector coefficients it becomes clear that when the point P is mapped onto the point by according to equation (1.2.2.3), then the vector is mapped onto the vector by transforming its coefficients by . This is because the coefficients are multiplied by the number `0' augmenting the column . Indeed, the vector is not changed when the whole space is mapped onto itself by a translation.
Given the matrix–column pair (W, w) of a symmetry operation , the geometric interpretation of , i.e. the type of operation, screw or glide component, location etc., can be calculated provided the coordinate system to which (W, w) refers is known.
(1) Evaluation of the matrix part W:
(a) Type of operation: In general the coefficients of the matrix depend on the choice of the basis; a change of basis changes the coefficients, see Section 1.5.2
. However, there are geometric quantities that are independent of the basis.
The type of isometry: the types 1, 2, 3, 4, 6 or , , , , can be uniquely specified by the matrix invariants: the determinant and the trace tr():
tr(W)   
3  2  1  0  −1  −3  −2  −1  0  1 
Type 
1 
6 
4 
3 
2 





Order 
1 
6 
4 
3 
2 
2 
6 
4 
6 
2 

(b) Rotation or rotoinversion axis: All symmetry operations (except 1 and ) have a characteristic axis (the rotation or rotoinversion axis). The direction u of this axis is invariant under the symmetry operation: The + sign is for rotations, the − sign for rotoinversions.
In the case of a kfold rotation, the direction u can be calculated by the equationwhere is an arbitrary direction. The direction Y(W)v is invariant under the symmetry operation W as the multiplication with W just permutes the terms of Y. If the application of equation (1.2.2.14) results in , then the direction v is perpendicular to u and another direction v has to be selected. In the case of a rotoinversion W, the direction Y(−W)v gives the direction of the rotoinversion axis. For , .
(c) Sense of rotation (for rotations or rotoinversions with ): The sense of rotation is determined by the sign of the determinant of the matrix Z, given by = , where u is the vector of equation (1.2.2.14) and x is a nonparallel vector of u, e.g. one of the basis vectors.

Examples are given later.
(2) Analysis of the translation column w:
(a) If W is the matrix of a rotation of order k or of a reflection (), then , and one determines the intrinsic translation part (or screw part or glide part) of the symmetry operation, also called the intrinsic translation component of the symmetry operation, by orThe vector with the column of coefficients is called the screw or glide vector. This vector is invariant under the symmetry operation: . Indeed, multiplication with W permutes only the terms on the right side of equation (1.2.2.16). Thus, the screw vector of a screw rotation is parallel to the screw axis. The glide vector of a glide reflection is left invariant for the same reason. It is parallel to the glide plane because (−W + I)(I + W) = O.
If t = o holds, then (W, w) describes a rotation or reflection. For , (W, w) describes a screw rotation or glide reflection. One forms the socalled reduced operation by subtracting the intrinsic translation part from (W, w): The column is called the location part (or the location component of the translation part) of the symmetry operation because it determines the position of the rotation or screw–rotation axis or of the reflection or glide–reflection plane in space.
(b) The set of fixed points of a symmetry operation is obtained by solving the equation Equation (1.2.2.18) has a unique solution for all rotoinversions (including , excluding ). There is a onedimensional set of solutions for rotations (the rotation axis) and a twodimensional set of solutions for reflections (the mirror plane). For translations, screw rotations and glide reflections, there are no solutions: there are no fixed points. However, a solution is found for the reduced operation, i.e. after subtraction of the intrinsic translation part, cf. equation (1.2.2.17)


(Note that the reduced operation of a translation is the identity, whose set of fixed points is the whole space.)
The formulae of this section enable the user to find the geometric contents of any symmetry operation. In practice, the geometric meanings for all symmetry operations which are listed in the General position blocks of the spacegroup tables of Part 2 can be found in the corresponding Symmetry operations blocks of the spacegroup tables. The explanation of the symbols for the symmetry operations is found in Sections 1.4.2
and 2.1.3.9
.
The procedure for the geometric interpretation of the matrix–column pairs (W, w) of the symmetry operations is illustrated by three examples of the space group , No. 230 (cf. the spacegroup tables of Chapter 2.3
).
Examples
(1) Consider the symmetry operation [symmetry operation (15) of the General position block of the space group ]. Its matrix–column pair is given by
Type of operation: the values of det(W) = 1 and tr(W) = 1 show that the symmetry operation is a fourfold rotation.
The direction of rotation axis u: The application of equation (1.2.2.14) with the matrixyields the direction u = [001] of the fourfold rotation axis.
Sense of rotation: The negative sense of rotation follows from det(Z) = −1, where the matrix(here, is taken as a vector nonparallel to u).
Screw component: The intrinsic translation part (screw component) of the symmetry operation is calculated from
Location of the symmetry operation: The location of the fourfold screw rotation is given by the fixed points of the reduced symmetry operation . The set of fixed points is obtained from the equation
Following the conventions for the designation of symmetry operations adopted in this volume (cf. Section 1.4.2
and 2.1.3.9
), the symbol of the symmetry operation , , is given by .
(2) The symmetry operation with corresponds to the entry No. (30) of the General position block of the space group , No. 230.
Type of operation: the values of det(W) = −1 and tr(W) = 0 show that symmetry operation is a threefold rotoinversion.
The direction of rotoinversion axis u:yields the direction from .
Sense of rotation: The positive sense of rotation follows from the positive sign of the determinant of the matrix Z, det(Z) = 1, where the matrix(here, is taken as a vector nonparallel to u).
Location of the symmetry operation: The solution , of the fixedpoint equation of the rotoinversiongives the coordinates of the inversion centre on the rotoinversion axis. An obvious description of a line along the direction and passing through the point is given by the parametric expression . The choice of the free parameter results in the description of the rotoinversion axis found in the Symmetry operation block. The convention adopted in this volume to have zero constant at the z coordinate of the description of the axis determines the specific choice of the free parameter.
The geometric characteristics of the symmetry operation are reflected in its symbol .
(3) The matrix–column pair (W, w) of the symmetry operation (37) of the General position block of the space group is given by
Type of operation: The values of the determinant det(W) = −1 and the trace tr(W) = +1 indicate that the symmetry operation is a reflection.
Normal u of the reflection plane: The orientation of the reflection plane in space is determined by its normal u, which is directed along . The direction of u follows from the matrix equationwhere is arbitrary.
Glide component: The glide component , determined from the equationindicates that the symmetry operation is a dglide reflection. As expected, the translation vectorcorresponds to a centring translation.
Location of the symmetry operation: The location of the dglide plane follows from the set of fixed points of the reduced symmetry operationThus, the set of fixed points (the dglide plane) can be described as .
The symbol of the symmetry operation , found in the Symmetry operations block of the spacegroup table of in Chapter 2.3
, comprises the essential geometric characteristics of the symmetry operation, i.e. its type, glide component and location. It is worth repeating that according to the conventions adopted in the spacegroup tables of Part 2, the mirror planes are specified by their sets of fixed points and not by the normals to the planes (cf. Section 1.4.2
for more details).

The specification of the symmetry operations by their types, screw or glide components and locations is sufficient to determine the corresponding matrix–column pairs (W, w). The general idea is to determine the image points of some points X under the symmetry operation by applying geometrical considerations. The 12 unknown coefficients of (W, w) (nine coefficients and three coefficients ) can then be calculated as solutions of 12 inhomogeneous linear equations obtained from the system of equations (1.2.2.1) written for four pairs (point image point), provided the points X are linearly independent. In fact, because of the special form of the matrix–column pairs, in many cases it is possible to reduce and simplify considerably the calculations necessary for the determination of (W, w): the determination of the image points of the origin O and of the three `coordinate points' , and under the symmetry operation is sufficient for the determination of its matrix–column pair.
Example
What is the pair (W, w) for a glide reflection with the plane through the origin, the normal of the glide plane parallel to c, and with the glide vector ?
(a) Image of the origin O: The origin is left invariant by the reflection part of the mapping; it is shifted by the glide part to 1/2, 1/2, 0 which are the coordinates of . Therefore, .
(b) Images of the coordinate points. Neither of the points A and B are affected by the reflection part, but A is then shifted to 3/2, 1/2, 0 and B to 1/2, 3/2, 0. This results in the equations , , for A and , , for B.
One obtains , , and . Point : 0, 0, 1 is reflected to and then shifted to .
This means , , = or , .
(c) The matrix–column pair is thuswhich can be represented by the coordinate triplet [cf. Section 1.2.2.1.1 for the shorthand notation of (W, w)].

The problem of the determination of (W, w) discussed above is simplified if it is reduced to the special case of the derivation of matrix–column pairs of spacegroup symmetry operations (General position block) from their symbols (Symmetry operations block) found in the spacegroup tables of Part 2 of this volume. The main simplification comes from the fact that for all symmetry operations of space groups, the rotation parts W referring to conventional coordinate systems are known and listed in Tables 1.2.2.1 and 1.2.2.2. In this way, given the symbol of the symmetry operation and using the tabulated data, one can write down directly the corresponding rotation part W.
The translation part w of the symmetry operation has two components: . The intrinsic translation part (or screw or glide component) is given explicitly in the symmetry operation symbol. The location part of w is derived from the equationsHere, are the coordinates of an arbitrary fixed point of the symmetry operation.
In the 1970s, when the International Union of Crystallography (IUCr) planned a new series of International Tables for Crystallography to replace the series International Tables for Xray Crystallography (1952), there was some confusion about the use of the term symmetry element. Crystallographers and mineralogists had used this term for rotation and rotoinversion axes and reflection planes, in particular for the description of the morphology of crystals, for a long time, although there had been no strict definition of `symmetry element'. With the impact of mathematical group theory in crystallography the term element was introduced with another meaning, in which an element is a member of a set, in particular as a group element of a group. In crystallography these group elements, however, were the symmetry operations of the symmetry groups, not the crystallographic symmetry elements. Therefore, the IUCr Commission on Crystallographic Nomenclature appointed an Adhoc Committee on the Nomenclature of Symmetry with P. M. de Wolff as Chairman to propose definitions for terms of crystallographic symmetry and for several classifications of crystallographic space groups and point groups.
In the reports of the Adhoc Committee, de Wolff et al. (1989) and (1992) with Addenda, Flack et al. (2000), the results were published. To define the term symmetry element for any symmetry operation was more complicated than had been envisaged previously, in particular for unusual screw and glide components.
According to the proposals of the Committee the following procedure has been adopted (cf. also Table 1.2.3.1):
(1) No symmetry element is defined for the identity and the (lattice) translations.
Name of symmetry element  Geometric element  Defining operation (d.o.)  Operations in element set 
Mirror plane 
Plane p 
Reflection through p 
D.o. and its coplanar equivalents^{†} 
Glide plane 
Plane p 
Glide reflection through p; 2v (not v) a latticetranslation vector 
D.o. and its coplanar equivalents^{†} 
Rotation axis 
Line l 
Rotation around l, angle 2π/N, N = 2, 3, 4 or 6 
1st … (N − 1)th powers of d.o. and their coaxial equivalents^{‡} 
Screw axis 
Line l 
Screw rotation around l, angle 2π/N, u = j/N times shortest lattice translation along l, righthand screw, N = 2, 3, 4 or 6, j = 1, …, (N − 1) 
1st … (N − 1)th powers of d.o. and their coaxial equivalents^{‡} 
Rotoinversion axis 
Line l and point P on l 
Rotoinversion: rotation around l, angle 2π/N, followed by inversion through P, N = 3, 4 or 6 
D.o. and its inverse 
Centre 
Point P 
Inversion through P 
D.o. only 
^{†}That is, all glide reflections through the same reflection plane, with glide vectors v differing from that of the d.o. (taken to be zero for reflections) by a latticetranslation vector. The glide planes a, b, c, n, d and e are distinguished ( cf. Table 2.1.2.1
).
^{‡}That is, all rotations and screw rotations around the same axis l, with the same angle and sense of rotation and the same screw vector u (zero for rotation) up to a latticetranslation vector.

(2) For any symmetry operation of point groups and space groups with the exception of the rotoinversions , and , the geometric element is defined as the set of fixed points (the second column of Table 1.2.3.1) of the reduced operation, cf. equation (1.2.2.17). For reflections and glide reflections this is a plane; for rotations and screw rotations it is a line, for the inversion it is a point. For the rotoinversions , and the geometric element is a line with a point (the inversion centre) on this line.
(3) The element set (cf. the last column of Table 1.2.3.1) is defined as a set of operations that share the same geometric element. The element set can consist of symmetry operations of the same type (such as the powers of a rotation) or of different types, e.g. by a reflection and a glide reflection through the same plane. The defining operation (d.o.) may be any symmetry operation from the element set that suffices to identify the symmetry element. In most cases, the `simplest' symmetry operation from the element set is chosen as the d.o. (cf. the third column of Table 1.2.3.1). For reflections and glide reflections the element set includes the defining operation and all glide reflections through the same reflection plane but with glide vectors differing by a latticetranslation vector, i.e. the socalled coplanar equivalents. For rotations and screw rotations of angle 2π/k the element set is the defining operation, its 1st … (k − 1)th powers and all rotations and screw rotations with screw vectors differing from that of the defining operation by a latticetranslation vector, known as coaxial equivalents. For a rotoinversion the element set includes the defining operation and its inverse.
(4) The combination of the geometric element and its element set is indicated by the name symmetry element. The names of the symmetry elements (first column of Table 1.2.3.1) are combinations of the name of the defining operation attached to the name of the corresponding geometric element. Names of symmetry elements are mirror plane, glide plane, rotation axis, screw axis, rotoinversion axis and centre.^{2} This allows such statements as this point lies on a rotation axis or these operations belong to a glide plane.

Examples
(1) Glide and mirror planes. The element set of a glide plane with a glide vector v consists of infinitely many different glide reflections with glide vectors that are obtained from v by adding any latticetranslation vector parallel to the glide plane, including centring translations of centred cells.
(a) It is important to note that if among the infinitely many glide reflections of the element set of the same plane there exists one operation with zero glide vector, then this operation is taken as the defining operation (d.o). Consider, for example, the symmetry operation , , of (63) [General position block]. This is an nglide reflection through the plane . However, the corresponding symmetry element is a mirror plane, as among the glide reflections of the element set of the plane one finds the reflection [symmetry operation (6) of the General position block].
(b) The symmetry operation is a glide reflection. Its geometric element is the plane . Its symmetry element is a glide plane in space group Pmmn (59) because there is no lattice translation by which the glide vector can be changed to o. If, however, the same mapping is a symmetry operation of space group Cmmm (65), then its symmetry element is a reflection plane because the glide vector with components can be cancelled through a translation , which is a lattice translation in a C lattice. Evidently, the correct specification of the symmetry element is possible only with respect to a specific translation lattice.
(c) Similarly, in Cmme (67) with an aglide reflection , the bglide reflection also occurs. The geometric element is the plane and the symmetry element is an eglide plane.
In fact, all vectors , integers, are glide vectors of glide reflections through the (001) plane of a space group with a Ccentred lattice. Among them one finds a glide reflection b with a glide vector related to by the centring translation; an aglide reflection and a bglide reflection share the same plane as a geometric element. Their symmetry element is thus an eglide plane.
(d) In general, the eglide planes are symmetry elements characterized by the existence of two glide reflections through the same plane with perpendicular glide vectors and with the additional requirement that at least one glide vector is along a crystal axis (de Wolff et al., 1992). The eglide designation of glide planes occurs only when a centred cell represents the choice of basis (cf. Table 2.1.2.2
). The `double' eglide planes are indicated by special graphical symbols on the symmetryelement diagrams of the space groups (cf. Tables 2.1.2.3
and 2.1.2.4
). For example, consider the space group (108). The symmetry operations [General position (0, 0, 0) block] and [General position (1/2, 1/2, 1/2) block] are glide reflections through the same plane, and their glide vectors and are related by the centring translation. The corresponding symmetry element is an eglide plane and it is easily recognized on the symmetryelement diagram of shown in Chapter 2.3
.

(2) Screw and rotation axes. The element set of a screw axis is formed by a screw rotation of angle with a screw vector u, its (N − 1) powers and all its coaxial equivalents, i.e. screw rotations around the same axis, with the same angle and sense of rotation, with screw vectors obtained by adding a latticetranslation vector parallel to u.
(3) Special case. In point groups , and space groups (175), (191) and (192) the geometric elements of the defining operations and are the same. To make the element sets unique, the geometric elements should not be given just by a line and a point on it, but should be labelled by these operations. Then the element sets and thus the symmetry element are unique (Flack et al., 2000).
