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<title>Line group</title>

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Line group

</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">A line group is a mathematical way of describing <a href="Symmetry" title="Symmetry">symmetries</a> 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 <a href="Isometry" title="Isometry">isometries</a> or symmetry transformations.

One constructs a line group by taking a <a href="Point_group" title="Point group">point group</a> 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 <a href="Space_group" title="Space group">space group</a>. 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 <a href="Unit_cell" title="Unit cell">unit cell</a> segment.

<div class="mw-heading mw-heading2"><h2 id="One-dimensional">One-dimensional</h2></div>

There are 2 <a href="One-dimensional_symmetry_group" title="One-dimensional symmetry group">one-dimensional line groups</a>. They are the infinite limits of the discrete <a href="Point_groups_in_two_dimensions" title="Point groups in two dimensions">two-dimensional point groups</a> Cn and Dn:

<th colspan="4">Notations

</th>

<th rowspan="2">Description

</th>

<th rowspan="2">Example

</th></tr>

<tr>

</th>

<th><a href="Orbifold_notation" title="Orbifold notation">Orbifold</a>

</th>

<th><a href="Coxeter_notation" title="Coxeter notation">Coxeter</a>

</th>

<th>P.G.

</th></tr>

<tr>

<th>C∞

</th>

<td>Translations. Abstract group Z, the integers under addition

</td>

<td>... --> --> --> --> ...

</td></tr>

<tr>

<th>D∞

</th>

<td>Reflections. Abstract group Dih∞, the <a href="Infinite_dihedral_group" title="Infinite dihedral group">infinite dihedral group</a>

</td>

<td>... --> <-- --> <-- ...

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Two-dimensional">Two-dimensional</h2></div>

There are 7 <a href="Frieze_group" title="Frieze group">frieze groups</a>, which involve reflections along the line, reflections perpendicular to the line, and 180° rotations in the two dimensions.

<caption>7 frieze group notations and diagram

</caption>

</th>

<th><a href="Orbifold_notation" title="Orbifold notation">Orbifold</a>

</th>

<th><a href="Schoenflies_notation" title="Schoenflies notation">Schönflies</a>

</th>

<th><a href="John_Horton_Conway" title="John Horton Conway">Conway</a>

</th>

<th><a href="Coxeter_notation" title="Coxeter notation">Coxeter</a>

</th>

<th>Fundamental domain

</th></tr>

<td>

</td></tr>

<td>

</td></tr>

<td>

</td></tr>

<td>

</td></tr>

<td>

</td></tr>

<td>

</td></tr>

<td>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Three-dimensional">Three-dimensional</h2></div>

There are 13 infinite families of three-dimensional line groups,<a href="#cite_note-1">[1]</a> derived from the 7 infinite families of axial <a href="Point_groups_in_three_dimensions" title="Point groups in three dimensions">three-dimensional point groups</a>. 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 <a href="Hermann-Mauguin_notation" class="mw-redirect" title="Hermann-Mauguin notation">Hermann-Mauguin notation</a>, and for the point groups, <a href="Sch%C3%B6nflies_notation" class="mw-redirect" title="Schönflies notation">Schönflies notation</a>. There appears to be no comparable notation for the line groups. These groups can also be interpreted as patterns of <a href="Wallpaper_group" title="Wallpaper group">wallpaper groups</a><a href="#cite_note-2">[2]</a> 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:

<th colspan="5">Point group

</th>

<th colspan="7">Line group

</th></tr>

<tr>

<th colspan="2">H-M

</th>

<th rowspan="2">Schönf.

</th>

</th>

</th>

<th colspan="2">H-M

</th>

<th rowspan="2">Offset type

</th>

<th colspan="3">Wallpaper

</th>

<th rowspan="2">Coxeter [∞h,2,pv]

</th></tr>

<tr>

<th>Odd n

</th>

<th>Odd n

</th>

</th>

<th><a href="Orbifold_notation" title="Orbifold notation">Orbifold</a>

</th>

<th>Diagram

</th></tr>

<td colspan="2">n

</td>

<td>[n]+

</td>

<td colspan="2">Pnq

</td>

<td>Helical: q

</td>

<td>p1

</td>

<td>

</td>

<td>[∞+,2,n+]

</td></tr>

<td>2n

</td>

<td>n

</td>

<td>[2+,2n+]

</td>

<td>P2n

</td>

<td>Pn

</td>

<td>None

</td>

<td>p11g, pg(h)

</td>

<td>

</td>

<td>[(∞,2)+,2n+]

</td></tr>

<td>n/m

</td>

<td>2n

</td>

<td>[2,n+]

</td>

<td>Pn/m

</td>

<td>P2n

</td>

<td>None

</td>

<td>p11m, pm(h)

</td>

<td>

</td>

<td>[∞+,2,n]

</td></tr>

<td colspan="2">2n/m

</td>

<td>[2,2n+]

</td>

<td colspan="2">P2nn/m

</td>

<td>Zigzag

</td>

<td>c11m, cm(h)

</td>

<td>

</td>

<td>[∞+,2+,2n]

</td></tr>

<td rowspan="2">nmm

</td>

<td rowspan="2">nm

</td>

<td rowspan="2">[n]

</td>

<td>Pnmm

</td>

<td>Pnm

</td>

<td>None

</td>

<td>p1m1, pm(v)

</td>

<td>

</td>

<td>[∞,2,n+]

</td></tr>

<td>Pncc

</td>

<td>Pnc

</td>

<td>None

</td>

<td>p1g1, pg(v)

</td>

<td>

</td>

<td>[∞+,(2,n)+]

</td></tr>

<td colspan="2">2nmm

</td>

<td>[2n]

</td>

<td colspan="2">P2nnmc

</td>

<td>Zigzag

</td>

<td>c1m1, cm(v)

</td>

<td>

</td>

<td>[∞,2+,2n+]

</td></tr>

<td>n22

</td>

<td>n2

</td>

<td>[2,n]+

</td>

<td>Pnq22

</td>

<td>Pnq2

</td>

<td>Helical: q

</td>

<td>p2

</td>

<td>

</td>

<td>[∞,2,n]+

</td></tr>

<td rowspan="2">2n2m

</td>

<td rowspan="2">nm

</td>

<td rowspan="2">[2+,2n]

</td>

<td>P2n2m

</td>

<td>Pnm

</td>

<td>None

</td>

<td>p2gm, pmg(v)

</td>

<td>

</td>

<td>[(∞,2)+,2n]

</td></tr>

<td>P2n2c

</td>

<td>Pnc

</td>

<td>None

</td>

<td>p2gg, pgg

</td>

<td>

</td>

<td>[+(∞,(2),2n)+]

</td></tr>

<td rowspan="2">n/mmm

</td>

<td rowspan="2">2n2m

</td>

<td rowspan="2">[2,n]

</td>

<td>Pn/mmm

</td>

<td>P2n2m

</td>

<td>None

</td>

<td>p2mm, pmm

</td>

<td>

</td>

<td>[∞,2,n]

</td></tr>

<td>Pn/mcc

</td>

<td>P2n2c

</td>

<td>None

</td>

<td>p2mg, pmg(h)

</td>

<td>

</td>

<td>[∞,(2,n)+]

</td></tr>

<td colspan="2">2n/mmm

</td>

<td>[2,2n]

</td>

<td colspan="2">P2nn/mcm

</td>

<td>Zigzag

</td>

<td>c2mm, cmm

</td>

<td>

</td>

<td>[∞,2+,2n]

</td></tr></tbody></table>

The offset types are:

<ul><li>None. Offsets along the axis include no offsets around it to within repeats of the unit cell around the axis.</li>

<li>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.</li>

<li>Zigzag offset. Helical offset of 1/2 relative to the unit cell around the axis.</li></ul>

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 <a href="Crystallographic_point_group" title="Crystallographic point group">crystallographic point group</a>, a symmetry of some three-dimensional lattice, then the resulting line group is called a <a href="Rod_group" title="Rod group">rod group</a>. There are 75 rod groups.

<ul><li>The <a href="Coxeter_notation" title="Coxeter notation">Coxeter notation</a> is based on the rectangular wallpaper groups, with the vertical axis wrapped into a cylinder of symmetry order n or 2n.</li></ul>

Going to the <a href="Continuum_limit" title="Continuum limit">continuum limit</a>, 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.

<div class="mw-heading mw-heading3"><h3 id="Helical_symmetry">Helical symmetry</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Screw_axis" title="Screw axis">Screw axis</a></div>

The groups Cn(q) and Dn(q) express the symmetries of helical objects. Cn(q) is for n helices oriented in the same direction, while Dn(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.

<a href="Nucleic_acid" title="Nucleic acid">Nucleic acids</a>, <a href="DNA" title="DNA">DNA</a> and <a href="RNA" title="RNA">RNA</a>, 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).

<ul><li><a href="Point_group" title="Point group">Point group</a></li>

<li><a href="Space_group" title="Space group">Space group</a></li>

<li><a href="One-dimensional_symmetry_group" title="One-dimensional symmetry group">One-dimensional symmetry group</a></li>

<li><a href="Frieze_group" title="Frieze group">Frieze group</a></li>

<li><a href="Rod_group" title="Rod group">Rod group</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFDamnjanovicMilosevic2010" class="citation cs2">Damnjanovic, Milan; Milosevic, Ivanka (2010), <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-3-642-11172-3_2">"Line Groups Structure"</a>, Line Groups in Physics, Lecture Notes in Physics, vol. 801, Springer, pp. 7–27, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-11172-3_2">10.1007/978-3-642-11172-3_2</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-11171-6</bdi></cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFRassat1996" class="citation cs2">Rassat, André (1996), "Symmetry in Spheroalcanes, Fullerenes, Tubules, and Other Column-Like Aggregates", in Tsoucaris, Georges; Atwood, J.L; Lipkowski, Janusz (eds.), Crystallography of Supramolecular Compounds, NATO Science Series C: (closed), vol. 480, Springer, pp. 181–201, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7923-4051-5</bdi></cite> (books.google.com <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=bnDIQ3uEPbMC&pg=PA181">[1]</a>)

</li>

</ol></div></div></div><div><div class="zim-footer">

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