# Fixed-point-free space group

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 Revision as of 15:03, 10 April 2017 (view source)m (→See also: 6th edition of ITA)← Older edit Revision as of 11:02, 15 May 2017 (view source)m (Style edits to align with printed edition)Newer edit → Line 1: Line 1: - [[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. - ==Fixed-point-free space groups in E2== + ==Fixed-point-free space groups in ''E''2== - Only two ﬁxed-point-free space groups exist in E2: ''p''1 and ''pg''. + Only two ﬁxed-point-free space groups exist in ''E''2: ''p''1 and ''pg''. - ==Fixed-point-free space groups in E3== + ==Fixed-point-free space groups in ''E''3== - Thirteen ﬁxed-point-free space groups exist in E3: ''P''1, ''P''21, ''Pc'', ''Cc'', ''P''212121, ''Pca''21, ''Pna''21, ''P''41, ''P''43, ''P''31, ''P''32, ''P''61, ''P''65. + Thirteen ﬁxed-point-free space groups exist in ''E''3: ''P''1, ''P''21, ''Pc'', ''Cc'', ''P''212121, ''Pca''21, ''Pna''21, ''P''41, ''P''43, ''P''31, ''P''32, ''P''61, ''P''65. == See also == == See also == - *[[crystallographic orbit]] + *[[Crystallographic orbit]] - *[[point configuration]] + *[[Point configuration]] *[[Wyckoff position]] *[[Wyckoff position]] - * Section 1.4.4.2 of the ''International Tables of Crystallography'', Volume A, 6th edition + * 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 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.