In mathematics, a layer group is a three-dimensional extension of a
wallpaper group
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
, with reflections in the third dimension. It is a
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 uncha ...
with a two-dimensional lattice, meaning that it is symmetric over repeats in the two lattice directions. The symmetry group at each lattice point is an 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 u ...
with the main axis being perpendicular to the lattice plane.
Table of the 80 layer 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 poin ...
or lattice type, and by their point groups:
See also
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Point group
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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 u ...
*
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 uncha ...
*
Rod 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 ...
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Wallpaper group
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
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
Bilbao Crystallographic Server under "Subperiodic Groups: Layer, Rod and Frieze Groups"
Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvinby Frank Farris. He constructs layer groups from wallpaper groups using negating isometries.
Euclidean symmetries
Discrete groups