In the spacegroup tables of Chapter 2.3
, for each space group there are at least two diagrams displaying the symmetry (there are more diagrams for space groups of low symmetry). The symmetryelement diagram displays the location and orientation of the symmetry elements of the space group. The generalposition diagrams show the arrangement of a set of symmetryequivalent points of the general position. Because of the periodicity of the arrangements, the presentation of the contents of one unit cell is sufficient. Both types of diagrams are orthogonal projections of the spacegroup unit cell onto the plane of projection along a basis vector of the conventional crystallographic coordinate system. The symmetry elements of triclinic, monoclinic and orthorhombic groups are shown in three different projections along the basis vectors. The thin lines outlining the projection are the traces of the side planes of the unit cell.
Detailed explanations of the diagrams of space groups are found in Section 2.1.3.6
. In this section, after a very brief introduction to the diagrams, we will focus mainly on certain important but very often overlooked features of the diagrams.
Symmetryelement diagram
The graphical symbols of the symmetry elements used in the diagrams are explained in Section 2.1.2
. The heights along the projection direction above the plane of the diagram are indicated for rotation or screw axes and mirror or glide planes parallel to the projection plane, for rotoinversion axes and inversion centres. The heights (if different from zero) are given as fractions of the shortest translation vector along the projection direction. In Fig. 1.4.2.6 (left) the symmetry elements of (unique axis b, cell choice 1) are represented graphically in a projection of the unit cell along the monoclinic axis b. The directions of the basis vectors c and a can be read directly from the figure. The origin (upper left corner of the unit cell) lies on a centre of inversion indicated by a small open circle. The black lenticular symbols with tails represent the twofold screw axes parallel to b. The cglide plane at height along b is shown as a bent arrow with the arrowhead pointing along c.

Figure 1.4.2.6
 top  pdf  Symmetryelement diagram (left) and generalposition diagram (right) for the space group , No. 14 (unique axis b, cell choice 1).

The crystallographic symmetry operations are visualized geometrically by the related symmetry elements. Whereas the symmetry element of a symmetry operation is uniquely defined, more than one symmetry operation may belong to the same symmetry element (cf. Section 1.2.3
). The following examples illustrate some important features of the diagrams related to the fact that the symmetryelement symbols that are displayed visualize all symmetry operations that belong to the element sets of the symmetry elements.
Examples
(1) Visualization of the twofold screw rotations of (Fig. 1.4.2.6). The second coset of the decomposition of with respect to its translation subgroup shown in Table 1.4.2.6 is formed by the infinite set of twofold screw rotations represented by the coordinate triplets , (where are integers). To analyse how these symmetry operations are visualized, it is convenient to consider two special cases:
(i) , i.e. = ; these operations correspond to twofold screw rotations around the infinitely many screw axes parallel to the line , i.e. around the lines . The symbols of the symmetry elements (i.e. of the twofold screw axes) located in the unit cell at ; ; ; (and the translationally equivalent and ) are shown in the symmetryelement diagram (Fig. 1.4.2.6);
(ii) , i.e. = ; these symmetry operations correspond to screw rotations around the line with screw components , i.e. with a screw component to which all lattice translations parallel to the screw axis are added. These operations, infinite in number, share the same geometric element, i.e. they form the element set of the same symmetry element, and geometrically they are represented just by one graphical symbol on the symmetryelement diagrams located exactly at .
(iii) The rest of the symmetry operations in the coset, i.e. those with the translation parts , are combinations of the two special cases above.

(2) Inversion centres of (Fig. 1.4.2.6). The element set of an inversion centre consists of only one symmetry operation, viz. the inversion through the point located at the centre. In other words, to each inversion centre displayed on a symmetryelement diagram there corresponds one symmetry operation of inversion. The infinitely many inversions = = of are located at points . Apart from translational equivalence, there are eight centres located in the unit cell: four at y = 0, namely at 0, 0, 0; ; ; and four at height of b. It is important to note that only inversion centres at y = 0 are indicated on the diagram.
A similar rule is applied to all pairs of symmetry elements of the same type (such as e.g. twofold rotation axes, planes etc.) whose heights differ by of the shortest lattice direction along the projection direction. For example, the cglide plane symbol in Fig. 1.4.2.6 with the fraction next to it represents not only the cglide plane located at height but also the one at height .
(3) Glide reflections visualized by mirror planes. As discussed in Section 1.2.3
, the element set of a mirror or glide plane consists of a defining operation and all its coplanar equivalents (cf. Table 1.2.3.1
). The corresponding symmetry element is a mirror plane if among the infinite set of the coplanar glide reflections there is one with zero glide vector. Thus, the symmetry element is a mirror plane and the graphical symbol for a mirror plane is used for its representation on the symmetryelement diagrams of the space groups. For example, the mirror plane shown on the symmetryelement diagram of Fmm2 (42), cf. Fig. 1.4.2.3, represents all glide reflections of the element set of the defining operation [symmetry operation (4) of the generalposition set, cf. Fig. 1.4.2.2], including the nglide reflection [entry (4) of the generalposition set]. In a similar way, the graphical symbols of the mirror planes also represent the nglide reflections [entry (3) of the generalposition set] of Fmm2.

Generalposition diagram
The graphical presentations of the spacegroup symmetries provided by the generalposition diagrams consist of a set of generalposition points which are symmetry equivalent under the symmetry operations of the space group. Starting with a point in the upper left corner of the unit cell, indicated by an open circle with a sign `+', all the displayed points inside and near the unit cell are images of the starting point under some symmetry operation of the space group. Because of the onetoone correspondence between the image points and the symmetry operations, the number of generalposition points in the unit cell (excluding the points that are equivalent by integer translations) equals the multiplicity of the general position. The coordinates of the points in the projection plane can be read directly from the diagram. For all systems except cubic, only one parameter is necessary to describe the height along the projection direction. For example, if the height of the starting point above the projection plane is indicated by a `+' sign, then signs `+', `−' or their combinations with fractions (e.g. , etc.) are used to specify the heights of the image points. A circle divided by a vertical line represents two points with different coordinates along the projection direction but identical coordinates in the projection plane. A comma `,' in the circle indicates an image point obtained by a symmetry operation of the second kind [i.e. with , cf. Section 1.2.2
].
Notes:
(1) The close relation between the symmetryelement and the generalposition diagrams is obvious. For example, the points shown on the generalposition diagram are images of a generalposition point under the action of the spacegroup symmetry operations displayed by the corresponding symmetry elements on the symmetryelement diagram. With some practice each of the diagrams can be generated from the other. In a number of texts, the two diagrams are considered as completely equivalent descriptions of the same space group. This statement is true for most of the space groups. However, there are a number of space groups for which the point configuration displayed on the generalposition diagram has higher symmetry than the generating space group (Suescun & Nespolo, 2012; Müller, 2012). For example, consider the diagrams of the space group P2, No. 3 (unique axis b, cell choice 1) shown in Fig. 1.4.2.7. It is easy to recognise that, apart from the twofold rotations, the point configuration shown in the generalposition diagram is symmetric with respect to a reflection through a plane containing the generalposition points, and as a result the space group of the generalposition configuration is of P2/m type, and not of P2. There are a number of space groups for which the generalposition diagram displays higher spacegroup symmetry, for example: P1, , P4mm, P6 etc. The analysis of the eigensymmetry groups of the generalposition orbits results in a systematic procedure for the determination of such space groups: the generalposition diagrams do not reflect the spacegroup symmetry correctly if the generalposition orbits are noncharacteristic, i.e. their eigensymmetry groups are supergroups of the space groups. (An introduction to terms like eigensymmetry groups, characteristic and noncharacteristic orbits, and further discussion of space groups with noncharacteristic generalposition orbits are given in Section 1.4.4.4.)

Figure 1.4.2.7
 top  pdf  Symmetryelement diagram (left) and generalposition diagram (right) for the space group P2, No. 3 (unique axis b, cell choice 1).

(2) The graphical presentation of the generalposition points of cubic groups is more difficult: three different parameters are required to specify the height of the points along the projection direction. To make the presentation clearer, the generalposition points are grouped around points of higher site symmetry and represented in the form of polyhedra. For most of the space groups the initial general point is taken as 0.048, 0.12, 0.089, and the polyhedra are centred at 0, 0, 0 (and its equivalent points). Additional generalposition diagrams are shown for space groups with special sites different from that have sitesymmetry groups of equal or higher order. Consider, for example, the two generalposition diagrams of the space group (214) shown in Fig. 1.4.2.8. The polyhedra of the lefthand diagram are centred at special points of highest sitesymmetry, namely, at and its equivalent points in the unit cell. The sitesymmetry groups are of the type 32 leading to polyhedra in the form of twisted trigonal antiprisms (cf. Table 3.2.3.2
). The polyhedra (sphenoids) of the righthand diagram are attached to the origin and its equivalent points in the unit cell, sitesymmetry group of the type 3. The fractions attached to the polyhedra indicate the heights of the highsymmetry points along the projection direction (cf. Section 2.1.3.6
for further explanations of the diagrams).
