Geometric element

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<font color="blue">Élément géométrique</font> (''Fr''). <font color="red">Geometrische Element</font> (''Ge''). <font color="black">Elemento geometrico</font> (''It''). <font color="purple">幾何的要素</font> (''Ja''). <font color="green">Elemento geométrico</font> (''Sp'').
<font color="blue">Élément géométrique</font> (''Fr''). <font color="red">Geometrische Element</font> (''Ge''). <font color="black">Elemento geometrico</font> (''It''). <font color="purple">幾何的要素</font> (''Ja''). <font color="green">Elemento geométrico</font> (''Sp'').
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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.
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==One-dimensional space==
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In one-dimensional spaces ''N''-1 = 0 and the only geometric element is a point. In two-dimensional spaces ''N''-1 = 1 and we have points and lines. In three-dimensional spaces ''N''-1 = 2 and we have points, lines and planes. In four-dimensional spaces ''N''-1 = 3 and we have points, lines, planes and hyperplanes.
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The only geometric element that exists in this space is the '''reflection point''' (mirror point).
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==Two-dimensional space==
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In this space, two types of geometric elements exist: zero and one-dimensional:
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*'''rotation points'''
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*'''reflection lines''' (mirror lines)
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The inversion centre (point) does not exist in spaces of even number of dimensions.
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==Three-dimensional space==
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In this space, three types of geometric elements exist: zero, one- and two-dimensional:
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*'''inversion centres'''
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*'''rotation axes'''
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*'''reflection planes''' (mirror planes)
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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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==See also==
==See also==

Latest revision as of 15:06, 30 November 2018

Élément géométrique (Fr). Geometrische Element (Ge). Elemento geometrico (It). 幾何的要素 (Ja). Elemento geométrico (Sp).


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.

In one-dimensional spaces N-1 = 0 and the only geometric element is a point. In two-dimensional spaces N-1 = 1 and we have points and lines. In three-dimensional spaces N-1 = 2 and we have points, lines and planes. In four-dimensional spaces N-1 = 3 and we have points, lines, planes and hyperplanes.

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