Difference between revisions of "Geometric element"
From Online Dictionary of Crystallography
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<font color="blue">Élément géométrique</font> (''Fr''); <font color="black">Elemento geometrico</font> (''It''); <font color="purple">幾何的要素</font> (''Ja''). | <font color="blue">Élément géométrique</font> (''Fr''); <font color="black">Elemento geometrico</font> (''It''); <font color="purple">幾何的要素</font> (''Ja''). | ||
| − | A '''geometric element''' is an element in space (plane, line, point, or a combination of these) about which a [[symmetry 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. | + | A '''geometric element''' is an element in space (plane, line, point, or a combination of these) about which a [[symmetry 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== | ==One-dimensional space== | ||
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==Two-dimensional space== | ==Two-dimensional space== | ||
In this space, two types of geometric elements exist: zero and one-dimensional: | In this space, two types of geometric elements exist: zero and one-dimensional: | ||
| − | *''' | + | *'''rotation points''' |
*'''reflection lines''' (mirror lines) | *'''reflection lines''' (mirror lines) | ||
The inversion centre (point) does not exist in spaces of even number of dimensions. | The inversion centre (point) does not exist in spaces of even number of dimensions. | ||
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In this space, three types of geometric elements exist: zero, one- and two-dimensional: | In this space, three types of geometric elements exist: zero, one- and two-dimensional: | ||
*'''inversion centres''' | *'''inversion centres''' | ||
| − | *''' | + | *'''rotation axes''' |
*'''reflection planes''' (mirror planes) | *'''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. | 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. | ||
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*Section 1.2.3 of ''International Tables for Crystallography, Volume A'', 6<sup>th</sup> edition | *Section 1.2.3 of ''International Tables for Crystallography, Volume A'', 6<sup>th</sup> edition | ||
| − | == | + | ==Reference== |
| − | 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. | + | 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.'' A'''45''', 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'' |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 10:02, 12 May 2017
Élément géométrique (Fr); Elemento geometrico (It); 幾何的要素 (Ja).
A geometric element is an element in space (plane, line, point, or a combination of these) about which a symmetry 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.
Contents
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:
- 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: zero, one- and two-dimensional:
- 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
- Symmetry element
- Section 1.2.3 of International Tables for Crystallography, Volume A, 6th edition
Reference
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