Symmetry element

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عنصر التماثل (Ar). Élément de symétrie (Fr). Symmetrieelement (Ge). Elemento di simmetria (It). 対称要素 (Ja). Elemento de simetría (Sp).

A symmetry element (of a given crystal structure or object) is defined as a concept with a double meaning, namely the combination of a geometric element with the set of symmetry operations having this geometric element in common (termed its element set). Together with the identity and the translations, for which a geometric element is not defined, the element sets cover all symmetry operations.


One-dimensional space

The only symmetry element that exists in this space is the reflection point (mirror point).

Two-dimensional space

In this space, two types of symmetry elements exist:

  • rotation points
  • reflection lines (mirror lines)

The inversion centre (point) does not exist in spaces of even number of dimensions.

Three-dimensional space

In this space, three types of geometric elements exist:

  • inversion centres
  • rotation axes
  • reflection planes (mirror planes)

For roto-inversion operations, the geometric element is a combination of a line, about which the rotation is performed, and a point (inversion point) with respect to which the inversion is performed.

See also


Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Senechal, M., Shoemaker, D. P., Wondratschek, H., Hahn, Th., Wilson, A. J. C. and Abrahams, S. C. (1989). Acta Cryst. A45, 494−499. Definition of symmetry elements in space groups and point groups. Report of the International Union of Crystallography Ad-hoc Committee on the Nomenclature of Symmetry