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

### From Online Dictionary of Crystallography

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− | [[Space group]]s with no special [[Wyckoff position]]s (''i | + | [[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. |

− | ==Fixed-point-free space groups in E<sup>2</sup>== | + | ==Fixed-point-free space groups in ''E''<sup>2</sup>== |

− | Only two ﬁxed-point-free space groups exist in E<sup>2</sup>: ''p''1 and ''pg''. | + | Only two ﬁxed-point-free space groups exist in ''E''<sup>2</sup>: ''p''1 and ''pg''. |

− | ==Fixed-point-free space groups in E<sup>3</sup>== | + | ==Fixed-point-free space groups in ''E''<sup>3</sup>== |

− | Thirteen ﬁxed-point-free space groups exist in E<sup>3</sup>: ''P''1, ''P''2<sub>1</sub>, ''Pc'', ''Cc'', ''P''2<sub>1</sub>2<sub>1</sub>2<sub>1</sub>, ''Pca''2<sub>1</sub>, ''Pna''2<sub>1</sub>, ''P''4<sub>1</sub>, ''P''4<sub>3</sub>, ''P''3<sub>1</sub>, ''P''3<sub>2</sub>, ''P''6<sub>1</sub>, ''P''6<sub>5</sub>. | + | Thirteen ﬁxed-point-free space groups exist in ''E''<sup>3</sup>: ''P''1, ''P''2<sub>1</sub>, ''Pc'', ''Cc'', ''P''2<sub>1</sub>2<sub>1</sub>2<sub>1</sub>, ''Pca''2<sub>1</sub>, ''Pna''2<sub>1</sub>, ''P''4<sub>1</sub>, ''P''4<sub>3</sub>, ''P''3<sub>1</sub>, ''P''3<sub>2</sub>, ''P''6<sub>1</sub>, ''P''6<sub>5</sub>. |

== See also == | == See also == | ||

− | *[[ | + | *[[Crystallographic orbit]] |

− | *[[ | + | *[[Point configuration]] |

*[[Wyckoff position]] | *[[Wyckoff position]] | ||

− | * | + | * Chapter 1.4.4.2 of ''International Tables for Crystallography'', Volume A, 6th edition |

[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] |

## Revision as of 11:02, 15 May 2017

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