Line group

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.

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:

Notations				Description	Example
Intl	Orbifold	Coxeter	P.G.	Description	Example
p1	∞∞	[∞]⁺	C_∞	Translations. Abstract group Z, the integers under addition	... --> --> --> --> ...
p1m	*∞∞	[∞]	D_∞	Reflections. Abstract group Dih_∞, the infinite dihedral group	... --> <-- --> <-- ...

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.

7 frieze group notations and diagram
IUC	Orbifold	Schönflies	Conway	Coxeter
p1	∞∞	C_∞	C_∞	[∞,1]⁺
p1m1	*∞∞	C_∞v	CD_2∞	[∞,1]
p11g	∞x	S_2∞	CC_2∞	[∞⁺,2⁺]
p11m	∞*	C_∞h	±C_∞	[∞⁺,2]
p2	22∞	D_∞	D_2∞	[∞,2]⁺
p2mg	2*∞	D_∞d	DD_4∞	[∞,2⁺]
p2mm	*22∞	D_∞h	±D_2∞	[∞,2]

Three-dimensional

There are 13 infinite families of three-dimensional line groups,[1] 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[2] 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
H-M		Schönf.	Orb.	Cox.	H-M		Offset type	Wallpaper			Coxeter [∞_h,2,p_v]
Even n	Odd n	Schönf.	Orb.	Cox.	Even n	Odd n	Offset type	IUC	Orbifold	Diagram	Coxeter [∞_h,2,p_v]
n		C_n	nn	[n]⁺	Pn_q		Helical: q	p1	o		[∞⁺,2,n⁺]
2n	n	S_2n	n×	[2⁺,2n⁺]	P2n	Pn	None	p11g, pg(h)	××		[(∞,2)⁺,2n⁺]
n/m	2n	C_nh	n*	[2,n⁺]	Pn/m	P2n	None	p11m, pm(h)	**		[∞⁺,2,n]
2n/m		C_2nh	(2n)*	[2,2n⁺]	P2n_n/m		Zigzag	c11m, cm(h)	*×		[∞⁺,2⁺,2n]
nmm	nm	C_nv	*nn	[n]	Pnmm	Pnm	None	p1m1, pm(v)	**		[∞,2,n⁺]
nmm	nm	C_nv	*nn	[n]	Pncc	Pnc	None	p1g1, pg(v)	××		[∞⁺,(2,n)⁺]
2nmm		C_2nv	*(2n)(2n)	[2n]	P2n_nmc		Zigzag	c1m1, cm(v)	*×		[∞,2⁺,2n⁺]
n22	n2	D_n	n22	[2,n]⁺	Pn_q22	Pn_q2	Helical: q	p2	2222		[∞,2,n]⁺
2n2m	nm	D_nd	2*n	[2⁺,2n]	P2n2m	Pnm	None	p2gm, pmg(v)	22*		[(∞,2)⁺,2n]
2n2m	nm	D_nd	2*n	[2⁺,2n]	P2n2c	Pnc	None	p2gg, pgg	22×		[⁺(∞,(2),2n)⁺]
n/mmm	2n2m	D_nh	*n22	[2,n]	Pn/mmm	P2n2m	None	p2mm, pmm	*2222		[∞,2,n]
n/mmm	2n2m	D_nh	*n22	[2,n]	Pn/mcc	P2n2c	None	p2mg, pmg(h)	22*		[∞,(2,n)⁺]
2n/mmm		D_2nh	*(2n)22	[2,2n]	P2n_n/mcm		Zigzag	c2mm, cmm	2*22		[∞,2⁺,2n]

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, 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 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.

Helical symmetry

The Boerdijk–Coxeter helix, a chain of regular tetrahedra, shows helical symmetry without an integer number of turns to repeat an original orientation.

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 acids, DNA and RNA, 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).

References

Damnjanovic, Milan; Milosevic, Ivanka (2010), "Line Groups Structure", Line Groups in Physics, Lecture Notes in Physics, vol. 801, Springer, pp. 7–27, doi:10.1007/978-3-642-11172-3_2, 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 (eds.), Crystallography of Supramolecular Compounds, NATO Science Series C: (closed), vol. 480, Springer, pp. 181–201, ISBN 978-0-7923-4051-5 (books.google.com )

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[1] Damnjanovic, Milan; Milosevic, Ivanka (2010), "Line Groups Structure", Line Groups in Physics, Lecture Notes in Physics, vol. 801, Springer, pp. 7–27, doi:10.1007/978-3-642-11172-3_2, ISBN 978-3-642-11171-6

[2] 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, ISBN 978-0-7923-4051-5 (books.google.com )