Difference between revisions of "Fixed-point-free space group"
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| − | + | <font color="blue">Groupes d'espace sans points fixes, groupes d'espace sans torsion, groupes de Bieberbach</font> (''Fr''). <font color="red">Fixpunktfreie Raumgruppe, Bieberbachgruppe</font> (''Ge''). <font color="black">Gruppi spaziali senza punti fissi, gruppi spaziali senza torsione, gruppi di Bieberbach</font> (''It''). <font color="purple">不動点を持たない空間群、捩れのない空間群、ビーベルバッハ群</font> (''Ja''). <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. | |
| − | ==Fixed-point-free space groups in E<sup> | + | ==Fixed-point-free space groups in ''E''<sup>2</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>. | + | Only two fixed-point-free space groups exist in ''E''<sup>2</sup>: ''p''1 and ''pg''. |
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| + | ==Fixed-point-free space groups in ''E''<sup>3</sup>== | ||
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| + | 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]] | ||
Latest revision as of 14:04, 13 November 2017
Groupes d'espace sans points fixes, groupes d'espace sans torsion, groupes de Bieberbach (Fr). Fixpunktfreie Raumgruppe, Bieberbachgruppe (Ge). Gruppi spaziali senza punti fissi, gruppi spaziali senza torsione, gruppi di Bieberbach (It). 不動点を持たない空間群、捩れのない空間群、ビーベルバッハ群 (Ja). 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 E2
Only two fixed-point-free space groups exist in E2: p1 and pg.
Fixed-point-free space groups in E3
Thirteen fixed-point-free space groups exist in E3: P1, P21, Pc, Cc, P212121, Pca21, Pna21, P41, P43, P31, P32, P61, P65.
See also
- Crystallographic orbit
- Point configuration
- Wyckoff position
- Chapter 1.4.4.2 of International Tables for Crystallography, Volume A, 6th edition