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Geometric element

geometric element is an element in space (plane, line, point, or a combination of these) about which asymmetry operation is performed. Geometric elements are classified on the basis of the dimensionality N of the space on which they act, the  upper limit  on the dimensionality of the symmetry element being N-1.

One-dimensional space

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

Two-dimensional space

In this space, two types of geometric elements exist: zero and one-dimensional:

  • rotations points
  • reflection lines (mirror lines)

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

Three-dimensional space

In this space, three types of geometric elements exist: zero, one- and two-dimensional:

  • inversion centers
  • rotations 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.


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. & Abrahams, S. C. (1989). 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. Acta Cryst., A 45, 494−499.