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A line group is a mathematical way of describing
symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, and they may involve those dimensions in its
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
or symmetry transformations. One constructs a line group by taking a
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 ...
in the full dimensions of the space, and then adding translations or offsets along the line to each of the point group's elements, in the fashion of constructing a
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 ...
. These offsets include the repeats, and a fraction of the repeat, one fraction for each element. For convenience, the fractions are scaled to the size of the repeat; they are thus within the line's
unit cell In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector In mathematics, a unit vector i ...
segment.


One-dimensional

There are 2 one-dimensional line groups. They are the infinite limits of the discrete two-dimensional point groups C''n'' and D''n'':


Two-dimensional

There are 7 frieze groups, which involve reflections along the line, reflections perpendicular to the line, and 180° rotations in the two dimensions.


Three-dimensional

There are 13 infinite families of three-dimensional line groups, derived from the 7 infinite families of axial three-dimensional point groups. As with space groups in general, line groups with the same point group can have different patterns of offsets. Each of the families is based on a group of rotations around the axis with order ''n''. The groups are listed in Hermann-Mauguin notation, and for the point groups,
Schönflies notation The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe ...
. There appears to be no comparable notation for the line groups. These groups can also be interpreted as patterns of
wallpaper group A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry, symmetries in the pattern. Such patterns occur frequently in architecture a ...
s (books.google.co

wrapped around a cylinder ''n'' times and infinitely repeating along the cylinder's axis, much like the three-dimensional point groups and the frieze groups. A table of these groups: The offset types are: * None. Offsets along the axis include no offsets around it to within repeats of the unit cell around the axis. * Helical offset with helicity ''q''. For a unit offset along the axis, there is an offset of q around it. A point that has repeated offsets will trace out a helix. * Zigzag offset. Helical offset of 1/2 relative to the unit cell around the axis. Note that the wallpaper groups pm, pg, cm, and pmg appear twice. Each appearance has a different orientation relative to the line-group axis; reflection parallel (h) or perpendicular (v). The other groups have no such orientation: p1, p2, pmm, pgg, cmm. If the point group is constrained to be a
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 ...
, a symmetry of some three-dimensional lattice, then the resulting line group is called a rod group. There are 75 rod groups. * The
Coxeter notation In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, ...
is based on the rectangular wallpaper groups, with the vertical axis wrapped into a cylinder of symmetry order ''n'' or ''2n''. Going to the
continuum limit In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model characterizes its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approximate real-world pr ...
, with ''n'' to ∞, the possible point groups become C, C∞h, C∞v, D, and D∞h, and the line groups have the appropriate possible offsets, with the exception of zigzag.


Helical symmetry

The groups C''n''(''q'') and D''n''(''q'') express the symmetries of helical objects. C''n''(''q'') is for ''n'' helices oriented in the same direction, while D''n''(''q'') is for ''n'' unoriented helices and ''2n'' helices with alternating orientations. Reversing the sign of ''q'' creates a mirror image, reversing the helices' chirality or handedness.
Nucleic acid Nucleic acids are large biomolecules that are crucial in all cells and viruses. They are composed of nucleotides, which are the monomer components: a pentose, 5-carbon sugar, a phosphate group and a nitrogenous base. The two main classes of nuclei ...
s,
DNA Deoxyribonucleic acid (; DNA) is a polymer composed of two polynucleotide chains that coil around each other to form a double helix. The polymer carries genetic instructions for the development, functioning, growth and reproduction of al ...
and
RNA Ribonucleic acid (RNA) is a polymeric molecule that is essential for most biological functions, either by performing the function itself (non-coding RNA) or by forming a template for the production of proteins (messenger RNA). RNA and deoxyrib ...
, are well known for their helical symmetry. Nucleic acids have a well-defined direction, giving single strands C1(''q''). Double strands have opposite directions and are on opposite sides of the helix axis, giving them D1(''q'').


See also

*
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 ...
*
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 ...
* One-dimensional symmetry group * Frieze group * Rod group


References

{{reflist Euclidean symmetries Discrete groups