# Difference between revisions of "Fixed-point-free space group"

### From Online Dictionary of Crystallography

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+ | <font color="red">Fixpunktfreie Raumgruppe, Bieberbachgruppe</font> (''Ge''). <font color="green">Grupo espacial sin puntos fijos</font> (''Sp''). | ||

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[[Space group]]s with no special [[Wyckoff position]]s (''i.e''. with no special [[crystallographic orbit]]s) are called '''ﬁxed-point-free space groups''' or '''torsion-free space groups''' or '''Bieberbach groups'''. In ﬁxed-point-free space groups every element other than the identity has infinite order. | [[Space group]]s with no special [[Wyckoff position]]s (''i.e''. with no special [[crystallographic orbit]]s) are called '''ﬁxed-point-free space groups''' or '''torsion-free space groups''' or '''Bieberbach groups'''. In ﬁxed-point-free space groups every element other than the identity has infinite order. | ||

## Revision as of 13:33, 13 November 2017

Fixpunktfreie Raumgruppe, Bieberbachgruppe (*Ge*). Grupo espacial sin puntos fijos (*Sp*).

Space groups with no special Wyckoff positions (*i.e*. with no special crystallographic orbits) are called **ﬁxed-point-free space groups** or **torsion-free space groups** or **Bieberbach groups**. In ﬁxed-point-free space groups every element other than the identity has infinite order.

## Fixed-point-free space groups in *E*^{2}

Only two ﬁxed-point-free space groups exist in *E*^{2}: *p*1 and *pg*.

## Fixed-point-free space groups in *E*^{3}

Thirteen ﬁxed-point-free space groups exist in *E*^{3}: *P*1, *P*2_{1}, *Pc*, *Cc*, *P*2_{1}2_{1}2_{1}, *Pca*2_{1}, *Pna*2_{1}, *P*4_{1}, *P*4_{3}, *P*3_{1}, *P*3_{2}, *P*6_{1}, *P*6_{5}.

## See also

- Crystallographic orbit
- Point configuration
- Wyckoff position
- Chapter 1.4.4.2 of
*International Tables for Crystallography*,*Volume A*, 6th edition