Projections of crystal structures are used by crystallographers in special cases. Use of socalled `twodimensional data' (zerolayer intensities) results in the projection of a crystal structure along the normal to the reciprocallattice net. A detailed treatment of projections of space groups, including basic definitions and illustrative examples, is given in Section 1.4.5.3
.
Even though the projection of a finite object along any direction may be useful, the projection of a periodic object such as a crystal structure is only sensible along a rational lattice direction (lattice row). Projection along a nonrational direction results in a constant density in at least one direction.
Data listed in the spacegroup tables. Under the heading Symmetry of special projections, the following data are listed for three projections of each space group; no projection data are given for the plane groups.
(i) The projection direction. All projections are orthogonal, i.e. the projection is made onto a plane normal to the projection direction. This ensures that spherical atoms appear as circles in the projection. For each space group, three projections are listed. If a lattice has three kinds of symmetry directions, the three projection directions correspond to the primary, secondary and tertiary symmetry directions of the lattice (cf. Table 2.1.3.1). If a lattice contains fewer than three kinds of symmetry directions, as in the triclinic, monoclinic and rhombohedral cases, the additional projection direction(s) are taken along coordinate axes, i.e. lattice rows lacking symmetry.
The directions for which projection data are listed are as follows:
(ii) The Hermann–Mauguin symbol of the plane group resulting from the projection of the space group. If necessary, the symbols are given in oriented form; for example, plane group pm is expressed either as p1m1 or as p11m (cf. Section 1.4.1.5
for explanations of Hermann–Mauguin symbols of plane groups).
(iii) Relations between the basis vectors a′, b′ of the plane group and the basis vectors a, b, c of the space group. Each set of basis vectors refers to the conventional coordinate system of the plane group or space group, as employed in Chapters 2.2
and 2.3
. The basis vectors of the twodimensional cell are always called a′ and b′ irrespective of which two of the basis vectors a, b, c of the threedimensional cell are projected to form the plane cell. All relations between the basis vectors of the two cells are expressed as vector equations, i.e. a′ and b′ are given as linear combinations of a, b and c. For the triclinic or monoclinic space groups, basis vectors a, b or c inclined to the plane of projection are replaced by the projected vectors .
For primitive threedimensional cells, the metrical relations between the lattice parameters of the space group and the plane group are collected in Table 2.1.3.9. The additional relations for centred cells can be derived easily from the table.
(iv) Location of the origin of the plane group with respect to the unit cell of the space group. The same description is used as for the location of symmetry elements (cf. Section 2.1.3.9).

Example
`Origin at x, 0, 0' or `Origin at '.
Projections of centred cells (lattices). For centred lattices, two different cases may occur:
(i) The projection direction is parallel to a latticecentring vector. In this case, the projected plane cell is primitive for the centring types A, B, C, I and R. For Fcentred lattices, the multiplicity is reduced from 4 to 2 because ccentred plane cells result from projections along face diagonals of threedimensional F cells.
(ii) The projection direction is not parallel to a latticecentring vector (general projection direction). In this case, the plane cell has the same multiplicity as the threedimensional cell. Usually, however, this centred plane cell is unconventional and a transformation is required to obtain the conventional plane cell. This transformation has been carried out for the projection data in this volume.
Examples
(1) Projection along [] of a cubic Icentred cell leads to an unconventional quadratic ccentred plane cell. A simple cell transformation leads to the conventional quadratic p cell.
(2) Projection along [] of an orthorhombic Icentred cell leads to a rectangular ccentred plane cell, which is conventional.
(3) Projection along [] of an Rcentred cell (both in obverse and reverse setting) results in a triple hexagonal plane cell h (the twodimensional analogue of the H cell, cf. Table 2.1.1.2). A simple cell transformation leads to the conventional hexagonal p cell.


Projections of symmetry elements. A symmetry element of a space group does not project as a symmetry element unless its orientation bears a special relation to the projection direction; all translation components of a symmetry operation along the projection direction vanish, whereas those perpendicular to the projection direction (i.e. parallel to the plane of projection) may be retained. This is summarized in Table 2.1.3.10 for the various crystallographic symmetry elements. From this table the following conclusions can be drawn:
(i) nfold rotation axes and nfold screw axes, as well as rotoinversion axes , parallel to the projection direction project as nfold rotation points; a axis projects as a sixfold, a axis as a threefold rotation point. For the latter, a doubling of the projected electron density occurs owing to the mirror plane normal to the projection direction .
(ii) nfold rotation axes and nfold screw axes normal to the projection direction (i.e. parallel to the plane of projection) do not project as symmetry elements if n is odd. If n is even, all rotation and rotoinversion axes project as mirror lines: the same applies to the screw axes and because they contain an axis 2. Screw axes , , , , and project as glide lines because they contain .
(iii) Reflection planes normal to the projection direction do not project as symmetry elements but lead to a doubling of the projected electron density owing to overlap of atoms. Projection of a glide plane results in an additional translation; the new translation vector is equal to the glide vector of the glide plane. Thus, a reduction of the translation period in that particular direction takes place.
(iv) Reflection planes parallel to the projection direction project as reflection lines. Glide planes project as glide lines or as reflection lines, depending upon whether the glide vector has or does not have a component parallel to the projection plane.
(v) Centres of symmetry, as well as axes in arbitrary orientation, project as twofold rotation points.

A detailed discussion of the correspondence between the symmetry elements and their projections is given in Section 1.4.5.3
.
Further details about the geometry of projections can be found in publications by Buerger (1965) and Biedl (1966).