International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 6773
Section 1.4.5. Sections and projections of space groups
B. Souvignier^{d}

In crystallography, twodimensional sections and projections of crystal structures play an important role, e.g. in structure determination by Fourier and Patterson methods or in the treatment of twin boundaries and domain walls. Planar sections of threedimensional scattering density functions are used for finding approximate locations of atoms in a crystal structure. They are indispensable for the location of Patterson peaks corresponding to vectors between equivalent atoms in different asymmetric units (the Harker vectors).
A twodimensional section of a crystal pattern takes out a slice of a crystal pattern. In the mathematical idealization, this slice is regarded as a twodimensional plane, allowing one, however, to distinguish its upper and lower side. Depending on how the slice is oriented with respect to the crystal lattice, the slice will be invariant by translations of the crystal pattern along zero, one or two linearly independent directions. A section resulting in a slice with twodimensional translational symmetry is called a rational section and is by far the most important case for crystallography.
Because the slice is regarded as a twosided plane, the symmetries of the full crystal pattern that leave the slice invariant fall into two types:
Therefore, the symmetries of twodimensional rational sections are described by layer groups, i.e. subgroups of space groups with a twodimensional translation lattice. Layer groups are subperiodic groups and for their elaborate discussion we refer to Chapter 1.7 and IT E (2010).
Analogous to twodimensional sections of a crystal pattern, one can also consider the penetration of crystal patterns by a straight line, which is the idealization of a onedimensional section taking out a rod of the crystal pattern. If the penetration line is along the direction of a translational symmetry of the crystal pattern, the rod has onedimensional translational symmetry and its group of symmetries is a rod group, i.e. a subgroup of a space group with a onedimensional translation lattice. Rod groups are also subperiodic groups, cf. IT E for their detailed treatment and listing.
A projection along a direction d into a plane maps a point of a crystal pattern to the intersection of the plane with the line along d through the point. If the projection direction is not along a rational lattice direction, the projection of the crystal pattern will contain points with arbitrarily small distances and additional restrictions are required to obtain a discrete pattern (e.g. the cutandproject method used in the context of quasicrystals). We avoid any such complication by assuming that d is along a rational lattice direction. Furthermore, one is usually only interested in orthogonal projections in which the projection direction is perpendicular to the projection plane. This has the effect that spheres in threedimensional space are mapped to circles in the projection plane.
Although it is also possible to regard the projection plane as a twosided plane by taking into account from which side of the plane a point is projected into it, this is usually not done. Therefore, the symmetries of projections are described by ordinary plane groups.
Sections and projections are related by the projection–slice theorem (Bracewell, 2003) of Fourier theory: A section in reciprocal space containing the origin (the socalled zero layer) corresponds to a projection in direct space and vice versa. The projection direction in the one space is normal to the slice in the other space. This correspondence is illustrated schematically in Fig. 1.4.5.1. The top part shows a rectangular lattice with b/a = 2 and a slice along the line defined by 2x + y = 0. Normalizing a = 1, the distance between two neighbouring lattice points in the slice is . If the pattern is restricted to this slice, the points of the corresponding diffraction pattern in reciprocal space must have distance and this is precisely obtained by projecting the lattice points of the reciprocal lattice onto the slice.
The different, but related, viewpoints of sections and projections can be stated in a simple way as follows: For a section perpendicular to the c axis, only those points of a crystal pattern are considered which have z coordinate equal to a fixed value or in a small interval around . For a projection along the c axis, all points of the crystal pattern are considered, but their z coordinate is simply ignored. This means that all points of the crystal pattern that differ only by their z coordinate are regarded as the same point.
For a space group and a point X in the threedimensional point space , the sitesymmetry group of X is the subgroup of operations of that fix X. Analogously, one can also look at the subgroup of operations fixing a onedimensional line or a twodimensional plane. If the line is along a rational direction, it will be fixed at least by the translations of along that direction. However, it may also be fixed by a symmetry operation that reverses the direction of the line. The resulting subgroup of that fixes the line is a rod group.
Similarly, a plane having a normal vector along a rational direction is fixed by translations of corresponding to a twodimensional lattice. Again, the plane may also be fixed by additional symmetry operations, e.g. by a twofold rotation around an axis lying in the plane, by a rotation around an axis normal to the plane or by a reflection in the plane.
Definition
A rational planar section of a crystal pattern is the intersection of the crystal pattern with a plane containing two linearly independent translation vectors of the crystal pattern. The intersecting plane is called the section plane.
A rational linear section of a crystal pattern is the intersection of the crystal pattern with a line containing a translation vector of the crystal pattern. The intersecting line is called the penetration line.
A planar section is determined by a vector d which is perpendicular to the section plane and a continuous parameter s, called the height, which gives the position of the plane on the line along d.
A linear section is specified by a vector d parallel to the penetration line and a point in a plane perpendicular to d giving the intersection of the line with that plane.
Definition
From now on we will only consider rational sections and omit this attribute. Moreover, we will concentrate on the case of planar sections, since this is by far the most relevant case for crystallographic applications. The treatment of onedimensional sections is analogous, but in general much easier.
Let d be a vector perpendicular to the section plane. In most cases, d is chosen as the shortest lattice vector perpendicular to the section plane. However, in the triclinic and monoclinic crystal family this may not be possible, since the translations of the crystal pattern may not contain a vector perpendicular to the section plane. In that case, we assume that d captures the periodicity of the crystal pattern perpendicular to the section plane. This is achieved by choosing d as the shortest nonzero projection of a lattice vector to the line through the origin which is perpendicular to the section plane. Because of the periodicity of the crystal pattern along d, it is enough to consider heights s with , since for an integer m the sectional layer groups at heights s and s + m are conjugate subgroups of . This is a consequence of the orbit–stabilizer theorem in Section 1.1.7 , applied to the group acting on the planes in . The layer at height s is mapped to the layer at height s + m by the translation through md. Thus, the two layers lie in the same orbit under . According to the orbit–stabilizer theorem, the corresponding stabilizers, being just the layer groups at heights s and s + m, are then conjugate by the translation through md.
Since we assume a rational section, the sectional layer group will always contain translations along two independent directions , which, we assume, form a crystallographic basis for the lattice of translations fixing the section plane. The points in the section plane at height s are then given by . In order to determine whether the sectional layer group contains additional symmetry operations which are not translations, the following simple remark is crucial:
Let be an operation of a sectional layer group. Then the rotational part of maps d either to +d or to −d. In the former case, is sidepreserving, in the latter case it is sidereversing. Moreover, since the section plane remains fixed under , the vectors and are mapped to linear combinations of and by the rotational part of . Therefore, with respect to the (usually nonconventional) basis , , d of threedimensional space and some choice of origin, the operation has an augmented matrix of the formHere, . Moreover, if , i.e. is sidepreserving, then is necessarily zero, since otherwise the plane is shifted along d. On the other hand, if , i.e. is sidereversing, then a plane situated at height s along d is only fixed if .
From these considerations it is straightforward to determine the conditions under which a spacegroup operation belongs to a certain sectional layer group (excluding translations):
The sidepreserving operations will belong to the sectional layer groups for all planes perpendicular to d, independent of the height s:
Sidereversing operations will only occur in the sectional layer groups for planes at special heights along d:
Note that, because of the periodicity along d, a sidereversing operation that occurs at height s gives rise to a sidereversing operation of the same type occurring at height : if is a sidereversing symmetry operation fixing a layer at height s, then maps a point in the layer at height with coordinates (with respect to the layeradapted basis ) to a point with coordinates and hence the composition of with the translation by d maps to , i.e. it fixes the layer at height . This shows that the composition with the translation by d provides a onetoone correspondence between the sidereversing symmetry operations in the layer group at height s with those at height .
If a section allows any sidereversing symmetry at all, then the sidepreserving symmetries of the section form a subgroup of index 2 in the sectional layer group. Since the sidepreserving symmetries exist independently of the height parameter s, the full sectional layer group is always generated by the sidepreserving subgroup and either none or a single sidereversing symmetry.
Summarizing, one can conclude that for a given space group the interesting sections are those for which the perpendicular vector d is parallel or perpendicular to a symmetry direction of the group, e.g. an axis of a rotation or rotoinversion or the normal vector of a reflection or glide reflection.
Example
Consider the space group of type (31). In its standard setting, the cosets of relative to the translation subgroup are represented by the operations given in Table 1.4.5.1.

Since this is an orthorhombic group, it is natural to consider sections along the coordinate axes. The spacegroup diagrams displayed in Fig. 1.4.5.2, which show the orthogonal projections of the symmetry elements along these directions, are very helpful.
As we have seen, a section of a crystal pattern is determined by a vector d and a height s along this vector. Choosing two vectors and perpendicular to d, the points of the section plane at height s are precisely given by the vectors . In contrast to that, a projection of a crystal pattern along d is obtained by mapping an arbitrary point to the point of the plane spanned by and , thereby ignoring the coordinate along the d direction.
Definition
In a projection of a crystal pattern along the projection direction d, a point X of the crystal pattern is mapped to the intersection of the line through X along d with a fixed plane perpendicular to d.
One may think of the projection plane as the plane perpendicular to d and containing the origin, but every plane perpendicular to d will give the same result.
Let be the line along d. If a symmetry operation of a space group maps to a line parallel to , then maps every plane perpendicular to d again to a plane perpendicular to d. This means that points that are projected to a single point (i.e. points on a line parallel to ) are mapped by to points that are again projected to a single point and thus the operation gives rise to a symmetry of the projection of the crystal pattern. Conversely, an operation that maps to a line that is inclined to does not result in a symmetry of the projection, since the points on are projected to a single point, whereas the image points under are projected to a line. In summary, the operations of that map to a line parallel to give rise to symmetries of the projection forming a plane group, sometimes called a wallpaper group.
Let be the subgroup of consisting of those mapping the line to a line parallel to , then is called the scanning group along d. The scanning group can be read off a coset decomposition relative to the translation subgroup of . Since translations map lines to parallel lines, one only has to check whether a coset representative maps to a line parallel to . This is precisely the case if the linear part of maps d to d or to −d. Therefore, is the union of those cosets relative to for which the linear part of maps d to d or to −d.
If the operations of a space group are written as augmented matrices with respect to a (usually nonconventional) basis , , d such that and are perpendicular to d, then an operation of the scanning group is of the formwith (just as for planar sections). Then the action of on the projection along d is obtained by ignoring the z coordinate, i.e. by cutting out the upper 2 × 2 block of the linear part and the first two components of the translation part. This gives rise to the planegroup operation
The mapping that assigns to each operation of the scanning group its action on the projection is in fact a homomorphism from to a plane group and the kernel of this homomorphism are the operations of the formi.e. translations along d and reflections with normal vector parallel to d.
Definition
The symmetry group of the projection along the projection direction d is the plane group of actions on the projection of those operations of that map the line along d to a line parallel to .
This group is isomorphic to the quotient group of the scanning group along d by the group of translations along d and reflections with normal vector parallel to d.
Example
We consider again the space group of type (31) for which the augmented matrices of the coset representatives with respect to the translation subgroup (in the standard setting) are given by
Since the linear parts of all four matrices are diagonal matrices, the scanning group for projections along the coordinate axes is always the full group .
For the projection along the direction [100], one has to cut out the lower 2 × 2 part of the linear parts and the second and third component of the translation part, thus choosing as a basis for the projection plane. This gives as matrices for the projected operationsin which the third and fourth operations are clearly redundant and which is thus a plane group of type p1g1 (plane group No. 4 with short symbol pg).
The projection along the direction [010] gives for the basis , of the projection plane (thus picking out the first and third rows and columns) the matriceswhere the second matrix is the product of the third and fourth. The third operation is a centring translation, the fourth a reflection, thus the resulting plane group is of type c1m1 (plane group No. 5 with short symbol cm).
Finally, the projection along the direction [001] results for the basis of the projection plane in the matriceswhere again the second matrix is the product of two others. The third operation is a glide reflection and the fourth is a reflection, thus the corresponding plane group is of type p2mg (plane group No. 7). Note that in order to obtain the plane group p2mg in its standard setting, the origin has to be shifted to (with respect to the plane basis ).
As for the sectional layer groups, the typical projection directions considered are symmetry directions of the space group , i.e. directions along rotation or screw axes or normal to reflection or glide planes. In order to relate the coordinate system of the plane group to that of the space group, not only the basis vectors perpendicular to the projection direction d have to be given, but also the origin for the plane group. This is done by specifying a line parallel to the projection direction which is projected to the origin of the plane group in its conventional setting. The spacegroup tables list the plane groups for the projections along symmetry directions of the group in the block `Symmetry of special projections'.
It is not hard to determine the corresponding types of planegroup operations for the different types of spacegroup operations, as is shown by the following list of simple rules:

The relationship between the symmetry operations in threedimensional space and the corresponding symmetry operations of a projection as listed above can be seen directly in the diagrams of the corresponding groups. In Fig. 1.4.5.4, the top diagram shows the orthogonal projection of the symmetryelement diagram of along the [001] direction and the bottom diagram shows the diagram for the plane group p2mg, which is precisely the symmetry group of the projection of along [001]. Firstly, one sees immediately that in order to match the two diagrams, the origin in the projection plane has to be shifted to (as already noted in the example above). Secondly, keeping in mind that the projection direction d is perpendicular to the drawing plane, one sees the correspondence between the twofold screw rotations in with the twofold rotations in p2mg [rule (iii)], the correspondence between the reflections with normal vector perpendicular to d in and the reflections in p2mg [rule (vii)] and the correspondence between the diagonal glide reflections in (indicated by the dotdash lines) and the glide reflections in p2mg {rule (vii); note that the diagonal glide vector has a component perpendicular to the projection direction [001]}.
Example
Let be a space group of type (117), then the interesting projection directions (i.e. symmetry directions) are [100], [010], [001], [110] and . However, the directions [100] and [010] are symmetryrelated by the fourfold rotoinversion and thus result in the same projection. The same holds for the directions [110] and . The three remaining directions are genuinely different and the projections along these directions will be discussed in detail below. The corresponding information given in the spacegroup tables under the heading `Symmetry of special projections' is reproduced in Fig. 1.4.5.5 for .

Orthogonal projection along [001] of the symmetryelement diagram for (31) (top) and the diagram for plane group p2mg (7) (bottom). 
Coset representatives of relative to its translation subgroup can be extracted from the generalpositions block in the spacegroup tables of and are given in Table 1.4.5.2.

Note that for directions different from those considered above, additional nontrivial plane groups may be obtained. For example, for the projection direction , the scanning group consists of the cosets of and . The operation acts as a glide reflection and the resulting plane group is of type c1m1 (plane group No. 5).
References
International Tables for Crystallography (2010). Vol. E, Subperiodic Groups, edited by V. Kopský & D. B. Litvin, 2nd ed. Chichester: John Wiley. [Abbreviated as IT E.]Bracewell, R. N. (2003). Fourier Analysis and Imaging. New York: Springer Science+Business Media.
Vainshtein, B. K. (1994). Fundamentals of Crystals. Symmetry, and Methods of Structural Crystallography. Berlin, Heidelberg, New York: Springer.