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)	*x		[∞⁺,2⁺,2n]
nmm	nm	C_nv	*nn	[n]	Pnmm	Pnm	None	p1m1, pm(v)	**		[∞,2,n⁺]
nmm	nm	C_nv	*nn	[n]	Pncc	Pnc	Planar reflection	p1g1, pg(v)	xx		[∞⁺,(2,n)⁺]
2nmm		C_2nv	*(2n)(2n)	[2n]	P2n_nmc		Zigzag	c1m1, cm(v)	*x		[∞,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(h)	22*		[(∞,2)⁺,2n]
2n2m	nm	D_nd	2*n	[2⁺,2n]	P2n2c	Pnc	Planar reflection	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	Planar reflection	p2mg, pmg(v)	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:

No offset.
Helical offset with helicity q. For C_n(q) and D_n(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. D_n(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.

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 |q| helices oriented in the same direction, while D_n(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 C_n(1). Double strands have opposite directions and are on opposite sides of the helix axis, giving them D_n(1).

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References

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 (eds.), Crystallography of Supramolecular Compounds, NATO Science Series C: (closed), 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" (PDF), Line Groups in Physics: Theory and Applications to Nanotubes and Polymers (Lecture Notes in Physics), Springer, 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), 480, Springer, pp. 181–201, ISBN 978-0-7923-4051-5 (books.google.com )