In mathematics, a rod group is a three-dimensional
line group whose
point group
In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...
is one of the axial
crystallographic point group
In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal un ...
s. This constraint means that the point group must be the symmetry of some three-dimensional lattice.
Table of the 75 rod groups, organized by
crystal system
In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their poi ...
or lattice type, and by their point groups:
The double entries are for orientation variants of a group relative to the perpendicular-directions lattice.
Among these groups, there are 8 enantiomorphic pairs.
See also
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Point group
In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...
*
Crystallographic point group
In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal un ...
*
Space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it ...
*
Line group
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Frieze group
In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetrie ...
*
Layer group
References
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* {{Citation , editor1-last=Kopsky , editor1-first=V. , editor2-last=Litvin , editor2-first=D.B. , title=International Tables for Crystallography, Volume E: Subperiodic groups , series=International Tables for Crystallography , url=http://it.iucr.org/E/ , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , edition=5th , isbn=978-1-4020-0715-6 , doi= 10.1107/97809553602060000105 , year=2002 , volume=E
External links
"Subperiodic Groups: Layer, Rod and Frieze Groups"on
Bilbao Crystallographic Server
Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin
Euclidean symmetries
Discrete groups