In mathematics, a rod group is a three-dimensional
line group whose
point group
In geometry, a point group is a group (mathematics), mathematical group of symmetry operations (isometry, isometries in a Euclidean space) that have a Fixed point (mathematics), fixed point in common. The Origin (mathematics), coordinate origin o ...
is one of the axial
crystallographic point group
In crystallography, a crystallographic point group is a three-dimensional point group whose symmetry operations are compatible with a three-dimensional crystallographic lattice. According to the crystallographic restriction it may only contain o ...
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 (an infinite array of discrete points). Space groups (symmetry groups ...
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 group (mathematics), mathematical group of symmetry operations (isometry, isometries in a Euclidean space) that have a Fixed point (mathematics), fixed point in common. The Origin (mathematics), coordinate origin o ...
*
Crystallographic point group
In crystallography, a crystallographic point group is a three-dimensional point group whose symmetry operations are compatible with a three-dimensional crystallographic lattice. According to the crystallographic restriction it may only contain o ...
*
Space group
In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that ...
*
Line group
*
Frieze group
In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. The term is derived from friezes in architecture and decorative arts, where such repeating patterns are often used. Frieze patterns can be classif ...
*
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 , 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 in ...
, location=Berlin, New York , edition=5th , isbn=978-1-4020-0715-6 , doi= 10.1107/97809553602060000105 , year=2002 , volume=E, url-access=subscription
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