A line group is a mathematical way of describing symmetries 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 or symmetry transformations.
One constructs a line group by taking a point group 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. 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 segment.
|p1||∞∞||[∞]+||C∞||Translations. Abstract group Z, the integers under addition||... --> --> --> --> ...|
|p1m||*∞∞||[∞]||D∞||Reflections. Abstract group Dih∞, the infinite dihedral group||... --> <-- --> <-- ...|
There are 7 frieze groups, which involve reflections along the line, reflections perpendicular to the line, and 180° rotations in the two dimensions.
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. There appears to be no comparable notation for the line groups. These groups can also be interpreted as patterns of wallpaper groups 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:
|Point group||Line group|
|Even n||Odd n||Even n||Odd n||IUC||Orbifold||Diagram|
|Pncc||Pnc||Planar reflection||p1g1, pg(v)||xx||[∞+,(2,n)+]|
|P2n2c||Pnc||Planar reflection||p2gg, pgg||22×||[+(∞,(2),2n)+]|
|Pn/mcc||P2n2c||Planar reflection||p2mg, pmg(v)||22*||[∞,(2,n)+]|
The offset types are:
- No offset.
- Helical offset with helicity q. For Cn(q) and Dn(q), axial rotation k out of n has an offset (q/n)k mod 1. A particle subjected to the rotations in sequence will thus trace out a helix. Dn(q) includes 180° rotations on axes in the perpendicular plane; those axes have the same helical pattern of offsets relative to their directions.
- Zigzag offset. Helical offset for helicity q = n for total number 2n. Axial rotation k out of 2n has 1/2 if odd, 0 if even, and likewise for the other elements.
- Planar-reflection offset. Every element that is a reflection along a direction in the perpendicular plane has an offset of 1/2. This is analogous to what happens in frieze groups p11g and p2mg.
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.
- The Coxeter notation 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, 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.
The groups Cn(q) and Dn(q) express the symmetries of helical objects. Cn(q) is for |q| helices oriented in the same direction, while Dn(q) is for |q| unoriented helices and 2|q|, helices with alternating orientations. Reversing the sign of q creates a mirror image, reversing the helices' chirality or handedness. The helices may have their own internal repeat lengths; n becomes the number of turns necessary to produce an integer number of internal repeats. But if the helix's coiling and internal repeating are incommensurable (ratio not a rational number), then n is effectively ∞.
Nucleic acids, DNA and RNA, are well known for their helical symmetry. Nucleic acids have a well-defined direction, giving single strands Cn(1). Double strands have opposite directions and are on opposite sides of the helix axis, giving them Dn(1).
- Damnjanovic, Milan; Milosevic, Ivanka (2010), "Line Groups Structure" (PDF), Line Groups in Physics: Theory and Applications to Nanotubes and Polymers (Lecture Notes in Physics), Springer, ISBN 978-3-642-11171-6
- Rassat, André (1996), "Symmetry in Spheroalcanes, Fullerenes, Tubules, and Other Column-Like Aggregates", in Tsoucaris, Georges; Atwood, J.L; Lipkowski, Janusz, Crystallography of Supramolecular Compounds, NATO Science Series C: (closed), 480, Springer, pp. 181–201, ISBN 978-0-7923-4051-5 (books.google.com )