# Geometric element

(Difference between revisions)
 Revision as of 14:39, 13 November 2017 (view source) (Tidied translations and added German and Spanish (U. Mueller))← Older edit Latest revision as of 15:06, 30 November 2018 (view source) (Text moved to symmetry element, where it actually belongs.) Line 1: Line 1: Élément géométrique (''Fr''). Geometrische Element (''Ge''). Elemento geometrico (''It''). 幾何的要素 (''Ja''). Elemento geométrico (''Sp''). 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. 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== + 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. - 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== ==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.