International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 1.4, pp. 7-11
https://doi.org/10.1107/97809553602060000503

## Chapter 1.4. Graphical symbols for symmetry elements in one, two and three dimensions

Th. Hahna*

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

This chapter lists the graphical symbols for symmetry elements used throughout this volume. The lists are accompanied by notes and cross-references to recent IUCr nomenclature reports.

### 1.4.1. Symmetry planes normal to the plane of projection (three dimensions) and symmetry lines in the plane of the figure (two dimensions)

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Symmetry plane or symmetry lineGraphical symbolGlide vector in units of lattice translation vectors parallel and normal to the projection planePrinted symbol
None m
Axial' glide plane lattice vector normal to projection plane a, b or c
Double' glide plane (in centred cells only) e
Diagonal' glide plane n
Diamond' glide plane (pair of planes; in centred cells only) along line parallel to projection plane, combined with normal to projection plane (arrow indicates direction parallel to the projection plane for which the normal component is positive) d
For further explanations of the double' glide plane e see Note (iv) below and Note (x) in Section 1.3.2 .
Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance and . The second power of a glide reflection d is a centring vector.

### 1.4.2. Symmetry planes parallel to the plane of projection

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Symmetry planeGraphical symbolGlide vector in units of lattice translation vectors parallel to the projection planePrinted symbol
Reflection plane, mirror plane None m
Axial' glide plane lattice vector in the direction of the arrow a, b or c
Double' glide plane (in centred cells only) e
Diagonal' glide plane n
Diamond' glide plane§ (pair of planes; in centred cells only) in the direction of the arrow; the glide vector is always half of a centring vector, i.e. one quarter of a diagonal of the conventional face-centred cell d
The symbols are given at the upper left corner of the space-group diagrams. A fraction h attached to a symbol indicates two symmetry planes with heights' h and above the plane of projection; e.g. stands for and . No fraction means and (cf. Section 2.2.6 ).
For further explanations of the double' glide plane e see Note (iv) below and Note (x) in Section 1.3.2 .
§Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance and . The second power of a glide reflection d is a centring vector.

### 1.4.3. Symmetry planes inclined to the plane of projection (in cubic space groups of classes and only)

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Symmetry planeGraphical symbol for planes normal toGlide vector in units of lattice translation vectors for planes normal toPrinted symbol
[011] and [101] and [011] and [101] and
Reflection plane, mirror plane None None m
Axial' glide plane lattice vector along [100] a or b
Axial' glide plane lattice vector along or along [011]
Double' glide plane [in space groups (217) and (229) only] Two glide vectors: along [100] and along or along [011] Two glide vectors: along [010] and along or along [101] e
Diagonal' glide plane One glide vector: along or along [111]§ One glide vector: along or along [111]§ n
Diamond' glide plane†† (pair of planes; in centred cells only) along or along [111] d
along or along
The symbols represent orthographic projections. In the cubic space-group diagrams, complete orthographic projections of the symmetry elements around high-symmetry points, such as ; ; , are given as inserts'.
For further explanations of the double' glide plane e see Note (iv) below and Note (x) in Section 1.3.2 .
§In the space groups (216), (225) and (227), the shortest lattice translation vectors in the glide directions are or and or , respectively.
The glide vector is half of a centring vector, i.e. one quarter of the diagonal of the conventional body-centred cell in space groups (220) and (230).
††Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance and . The second power of a glide reflection d is a centring vector.

### 1.4.4. Notes on graphical symbols of symmetry planes

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 (i) The graphical symbols and their explanations (columns 2 and 3) are independent of the projection direction and the labelling of the basis vectors. They are, therefore, applicable to any projection diagram of a space group. The printed symbols of glide planes (column 4), however, may change with a change of the basis vectors, as shown by the following example. In the rhombohedral space groups (161) and (167), the dotted line refers to a c glide when described with hexagonal axes' and projected along [001]; for a description with rhombohedral axes' and projection along [111], the same dotted glide plane would be called n. The dash-dotted n glide in the hexagonal description becomes an a, b or c glide in the rhombohedral description; cf. the first footnote in Section 1.3.1. (ii) The graphical symbols for glide planes in column 2 are not only used for the glide planes defined in Chapter 1.3 , but also for the further glide planes g which are mentioned in Section 1.3.2 (Note x ) and listed in Table 4.3.2.1 ; they are explained in Sections 2.2.9 and 11.1.2 . (iii) In monoclinic space groups, the parallel' glide vector of a glide plane may be along a lattice translation vector which is inclined to the projection plane. (iv) In 1992, the International Union of Crystallography introduced the double' glide plane e and the graphical symbol ..--..-- for e glide planes oriented normal' and inclined' to the plane of projection (de Wolff et al., 1992); for details of e glide planes see Chapter 1.3 . Note that the graphical symbol for e glide planes oriented parallel' to the projection plane has already been used in IT (1935) and IT (1952).

### 1.4.5. Symmetry axes normal to the plane of projection and symmetry points in the plane of the figure

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Symmetry axis or symmetry pointGraphical symbolScrew vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axisPrinted symbol (partial elements in parentheses)
Identity None None 1
None 2
Twofold screw axis: 2 sub 1'
None 3
Threefold screw axis: 3 sub 1'
Threefold screw axis: 3 sub 2'
None 4 (2)
Fourfold screw axis: 4 sub 1'
Fourfold screw axis: 4 sub 2'
Fourfold screw axis: 4 sub 3'
None 6 (3,2)
Sixfold screw axis: 6 sub 1'
Sixfold screw axis: 6 sub 2'
Sixfold screw axis: 6 sub 3'
Sixfold screw axis: 6 sub 4'
Sixfold screw axis: 6 sub 5'
None
Inversion axis: 3 bar' None
Inversion axis: 4 bar' None
Inversion axis: 6 bar' None
Twofold rotation axis with centre of symmetry None
Twofold screw axis with centre of symmetry
Fourfold rotation axis with centre of symmetry None
4 sub 2' screw axis with centre of symmetry
Sixfold rotation axis with centre of symmetry None
6 sub 3' screw axis with centre of symmetry

Notes on the heights' h of symmetry points , , and :

 (1) Centres of symmetry and , as well as inversion points and on and axes parallel to [001], occur in pairs at heights' h and . In the space-group diagrams, only one fraction h is given, e.g. stands for and . No fraction means and . In cubic space groups, however, because of their complexity, both fractions are given for vertical axes, including and . (2) Symmetries and contain vertical and axes; their and inversion points coincide with the centres of symmetry. This is not indicated in the space-group diagrams. (3) Symmetries and also contain vertical and axes, but their and inversion points alternate with the centres of symmetry; i.e. points at h and interleave with or points at and . In the tetragonal and hexagonal space-group diagrams, only one fraction for and one for or is given. In the cubic diagrams, all four fractions are listed for ; e.g. (No. 223): : ; : .

### 1.4.6. Symmetry axes parallel to the plane of projection

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Symmetry axisGraphical symbolScrew vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axisPrinted symbol (partial elements in parentheses)
Twofold rotation axis None 2
Twofold screw axis: 2 sub 1'
Fourfold rotation axis None 4 (2)
Fourfold screw axis: 4 sub 1'
Fourfold screw axis: 4 sub 2'
Fourfold screw axis: 4 sub 3'
Inversion axis: 4 bar' None
Inversion point on 4 bar'-axis point
The symbols for horizontal symmetry axes are given outside the unit cell of the space-group diagrams. Twofold axes always occur in pairs, at heights' h and above the plane of projection; here, a fraction h attached to such a symbol indicates two axes with heights h and . No fraction stands for and . The rule of pairwise occurrence, however, is not valid for the horizontal fourfold axes in cubic space groups; here, all heights are given, including and . This applies also to the horizontal axes and the inversion points located on these axes.

### 1.4.7. Symmetry axes inclined to the plane of projection (in cubic space groups only)

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Symmetry axisGraphical symbolScrew vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axisPrinted symbol (partial elements in parentheses)
Twofold rotation axis None 2
Twofold screw axis: 2 sub 1'
Threefold rotation axis None 3
Threefold screw axis: 3 sub 1'
Threefold screw axis: 3 sub 2'
Inversion axis: 3 bar' None
The dots mark the intersection points of axes with the plane at . In some cases, the intersection points are obscured by symbols of symmetry elements with height ; examples: (203), origin choice 2; (222), origin choice 2; (223); (229); (230).

### References

Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). I. Band, edited by C. Hermann. Berlin: Borntraeger. [Reprint with corrections: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]
International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]
Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Hahn, Th., Senechal, M., Shoemaker, D. P., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1992). Symbols for symmetry elements and symmetry operations. Final Report of the International Union of Crystallography Ad-hoc Committee on the Nomenclature of Symmetry. Acta Cryst. A48, 727–732.