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 ever ...
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 ...
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 ever ...
*
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 ...
*
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 unch ...
*
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 i ...
, 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
Bilbao Crystallographic Server is an open access website offering online crystallographic database and programs aimed at analyzing, calculating and visualizing problems of structural and mathematical crystallography, solid state physics and struct ...
Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin
Euclidean symmetries
Discrete groups