Talk:Line group

Coxeter Notation?
The 3D groups are still a bit confusing to me. Looking at the Coxeter notation, I'm thinking there's 4 reflectional cases in 3-space, by Coxeter-Dynkin diagram:
 * 1) Two parallel mirrors:, [1,1,&infin;]
 * 2) Two parallel mirrors and one orthogonal mirror:, [1,&infin;,2]
 * 3) Two parallel mirrors and two mutually-orthogonal mirrors:, [&infin;,2,2]
 * 4) Two parallel mirrors and two non-orthogonal mirrors, with angle &pi;/n:, [&infin;,2,n]

I would expect all the groups to be derived from these, something like:
 * [1,1,&infin;], [1,1,&infin;]+
 * [1,&infin;,2], [1,&infin;+,2], [1,&infin;,2], [1,&infin;,2+], [1,&infin;+,2+]
 * 1) [&infin;,2,2], [&infin;,2+,2], [&infin;,2+,2+], [&infin;,(2,2)+], [&infin;+,2,2], [&infin;+,2+,2], [&infin;+,2+,2+], [&infin;+,(2,2)+], [(&infin;,2)+,2], [(&infin;,2)+,2+], [&infin;,2,2]+
 * 2) [&infin;,2,n], [&infin;,2+,2n], [&infin;,2+,2n+], [&infin;,(2,n)+], [&infin;+,2,2n], [&infin;+,2+,2n], [&infin;+,2+,2n+], [&infin;+,(2,n)+], [(&infin;,2)+,2n], [(&infin;,2)+,2n+], [(&infin;,n)+,2], [(&infin;,n)+,2+], [&infin;,2,n]+

Oh, that's just a quick symbolic enumeration, might not be complete and could be large duplicated groups, and excluding arbitrary glide-rotation angles. I'll check with someone! Tom Ruen (talk) 01:53, 24 June 2011 (UTC)

By all means do so. If you can find another reference for line groups as a result of your search, that would be welcome. The reference I found, that second book chapter, was almost unintelligible, and I had to refer to my independent derivation of the line-group families to make sense out of it.

I'm also half-thinking of adding sections to the various point and space group articles on how to construct these groups.

Lpetrich (talk) 12:33, 24 June 2011 (UTC)

I think we can see some of them as a subset of the wallpaper groups generated by two sets of parallel mirrors, [&infin;,2,&infin;], and then moving one of the set of parallel mirrors to angle &pi;/n, as [&infin;,2,n]. So these wallpaper groups, instead of being infinite one direction, it wraps into a cylinder and sector fundamental domains in 3-space, with pattern period n. Tom Ruen (talk) 21:00, 24 June 2011 (UTC)

Hey, these below are 13 families, and I notice p1 and p2 allow non-orthogonal angles, so that degree of freedom that could perhaps account for the helix symmetries? I think I'd call these "cylindrical groups" more than line groups since they define symmetries on the surface of a cylinder. They're almost the same as wallpaper groups, but all the mirrors have to be perpendicular or parallel to the cylinder axis. Tom Ruen (talk) 21:15, 24 June 2011 (UTC)

Cylinder wallpaper groups
I added the 13 wallpaper group names (above) to the table, doing my best to read the descriptions for a proper association, but it was guesswork and needs to be checked. The first columns "point groups" are still a bit confusing! Tom Ruen (talk) 03:18, 26 June 2011 (UTC)

I'm glad if you have a source that associates wallpaper groups, but it needs to express which direction associated with the line direction. Also, the groups don't make sense to me, specifically the "offsets" as "none" with the p1m1 wallpaper group which has mirrors in one direction, and not the other, so one of these case offsets ought to be reflective. Tom Ruen (talk) 05:38, 26 June 2011 (UTC)

(TomRuen's inclusion)

I made this change because I worked out mathematically what wallpaper groups correspond to what 3D line groups. It's much like the calculation of what frieze groups correspond to what 3D point groups -- you work out the action of the group in cylindrical coordinates. Despite this calculation being straightforward, I was unable to find any references in the literature to anyone having done it. So I think that we might want to retract this connection as possible Original Research.

Lpetrich (talk) 06:27, 26 June 2011 (UTC)


 * Definitely the entire table seems questionable without sources for confirmation. But still I don't know why your tabvle has "none" on "offset type" if p1m1 has parallel mirrors? One of them ought to be offset by mirrors! Something is fishy. Tom Ruen (talk) 06:43, 26 June 2011 (UTC)

I indeed have a source: Table 2.2 in Line Groups Structure

It can be difficult to follow, so I can help you out if you wish, explaining the notation.

Here is how all these groups work: they take a vector x and make a vector x' by doing x' = R.x + D The R matrices are the elements of the associated point group, while the D vectors are the translations / offsets. The D values for the identity element define the lattice. The other D's are related to it by D(R) = D(identity) + D(R,sublattice) where each R has a unique D(R,sublattice) value.

For line groups, D is one-dimensional, that is, it's always a multiple of some vector D0. For convenience, we can take the lattice repeat distance as 1.

Here is what the offset types are:
 * None: D(R,sublattice) = 0 for all R -- R can be a pure rotation or a reflection
 * Helical: D(rotation by k out of n, sublattice) = q*(k/n) mod 1
 * Zigzag: like helical, but for even n and q = n/2. As one rotates, D(sublattice) alternates between 0 and 1/2
 * Planar reflection: D(pure rotation, sublattice) = 0, D(reflection, sublattice) = 1/2

Should I also give the explicit forms of the point-group matrices? Lpetrich (talk) 08:17, 26 June 2011 (UTC)

Jackpot!

I found a reference for a line-group-wallpaper-group connection. It's in the book "Crystallography of Supramolecular Compounds", available online at books.google.com. The important stuff is in pages 191-197. The constructions are in pages 191 and 194, and a table of the results is in page 195. Andr&eacute; Rassat identifies 17 groups, but 2 sets of 3 groups each are sets of special cases of 2 of the line groups. I was able to verify that Rassat had listed all the line groups, even if he did not call them "line groups". Furthermore, he listed their associated wallpaper groups, and his results agree with my calculations. Thus, I conclude that identifying wallpaper groups with 3D line groups is not Original Research, and is thus therefore legitimate for Wikipedia.

Lpetrich (talk) 14:49, 26 June 2011 (UTC)


 * Thanks for trying to explain. I accept I was mis-interpreting the meaning of "offset type", and still don't really follow the meaning in relation to the visual wallpaper diagrams. The CoSC reference table has a column (b) "ring-approach" which mixes a Sch&ouml;nflies notation and an "offset" operation: T, $$T_{2}$$, $$T^{p}_{q}$$, V, $$H_{2}$$. Maybe that's similar to your offsetting? ALSO it has 3 types of p1,p2 giving a total of 17 groups. Tom Ruen (talk) 19:20, 27 June 2011 (UTC)

Lpetrich, you should look at the table at and check your "point groups", mainly Sch&ouml;nflies, whcih doesn't match the "ring approach" column codes in all cases. Tom Ruen (talk) 20:50, 27 June 2011 (UTC)

I checked, and I found the matches. I also reverted the Cnv case for planar reflection. Here are the matrices for the 3D point groups:

T(sp,sx,a) = ((cos(a), -sp*sin(a), 0), (sin(a), sp*cos(a), 0), (0, 0, sx))

k is an integer that defines the lattice
 * 1) C(n): T(1,1,2pi*k/n)
 * 2) S(2n): T(1,1,2pi*k/n), T(1,-1,2pi*(k+1/2)/n)
 * 3) C(n,h): T(1,1,2pi*k/n), T(1,-1,2pi*k/n)
 * 4) C(n,v): T(1,1,2pi*k/n), T(-1,1,2pi*k/n)
 * 5) D(n): T(1,1,2pi*k/n), T(-1,-1,2pi*k/n)
 * 6) D(n,d): T(1,1,2pi*k/n), T(1,-1,2pi*(k+1/2)/n), T(-1,1,2pi*k/n), T(-1,-1,pi*(k+1/2)/n)
 * 7) D(n,h): T(1,1,2pi*k/n), T(1,-1,2pi*k/n), T(-1,1,2pi*k/n), T(-1,-1,2pi*k/n)

In cylindrical coordinates, T(sp,sx,a).(phi,z) = (sp*phi + a, sx*z)

Here, w = 2pi/n
 * 1) C(n): (phi + w*k, z)
 * 2) S(2n): (phi + w*k, z), (phi + w*(k+1/2), -z)
 * 3) C(n,h): (phi + w*k, z), (phi + w*k, -z)
 * 4) C(n,v): (phi + w*k, z), (-phi + w*k, z)
 * 5) D(n): (phi + w*k, z), (-phi + w*k, -z)
 * 6) D(n,d): (phi + w*k, z), (phi + w*(k+1/2), -z), (-phi + w*k, z), (-phi + w*(k+1/2), -z)
 * 7) D(n,h): (phi + w*k, z), (phi + w*k, -z), (-phi + w*k, z), (-phi + w*k, -z)

Distilled into a multiplier of phi, a multiplier of z, and a sub-grid offset in phi:
 * 1) C(n): (1,1,0)
 * 2) S(2n): (1,1,0), (1,-1,1/2)
 * 3) C(n,h): (1,1,0), (1,-1,0)
 * 4) C(n,v): (1,1,0), (-1,1,0)
 * 5) D(n): (1,1,0), (-1,-1,0)
 * 6) D(n,d): (1,1,0), (1,-1,1/2), (-1,1,0), (-1,-1,1/2)
 * 7) D(n,h): (1,1,0), (1,-1,0), (-1,1,0), (-1,-1,0)

It's easy to derive the point-group / frieze-group connection from that.

Now the line group. In cylindrical coordinates, T(sp,sx,a).(phi,z) + D(h) = (sp*phi + a, sx*z + h)

The offsets. Adding an integer to z is assumed here.
 * 1) C(n)(q): T(1,1,2pi*k/n): q*k/n
 * 2) S(2n): T(1,1,2pi*k/n): 0, T(1,-1,pi*(k+1/2)/n): 0
 * 3) C(n,h)(None): T(1,1,2pi*k/n): 0, T(1,-1,2pi*k/n): 0
 * 4) C(n,h)(Zigzag): T(1,1,2pi*k/n): k/2, T(1,-1,2pi*k/n): k/2
 * 5) C(n,v)(None): T(1,1,2pi*k/n): 0, T(-1,1,2pi*k/n): 0
 * 6) C(n,v)(Pln.Rfl.): T(1,1,2pi*k/n): 0, T(-1,1,2pi*k/n): 1/2
 * 7) D(n): T(1,1,2pi*k/n): q*k/n, T(-1,1,2pi*k/n): q*k/n
 * 8) D(n,d)(None): T(1,1,2pi*k/n): 0, T(1,-1,pi*(k+1/2)/n): 0, T(1,1,2pi*k/n): 0, T(1,-1,pi*(k+1/2)/n): 0
 * 9) D(n,d)(Pln.Rfl.): T(1,1,2pi*k/n): 0, T(1,-1,pi*(k+1/2)/n): 0, T(1,1,2pi*k/n): 1/2, T(1,-1,pi*(k+1/2)/n): 1/2
 * 10) D(n,h)(None): T(1,1,2pi*k/n): 0, T(1,-1,pi*k/n): 0, T(1,1,2pi*k/n): 0, T(1,-1,pi*k/n): 0
 * 11) D(n,h)(Pln.Rfl.): T(1,1,2pi*k/n): 0, T(1,-1,pi*k/n): 0, T(1,1,2pi*k/n): 1/2, T(1,-1,pi*k/n): 1/2
 * 12) D(n,h)(Zigzag): T(1,1,2pi*k/n): k/2, T(1,-1,pi*k/n): k/2, T(1,1,2pi*k/n): k/2, T(1,-1,pi*k/n): k/2

k and m are integers that define the lattice
 * 1) C(n)(q): (phi + w*k, z + m + k*q/n)
 * 2) S(2n): (phi + w*k, z + m), (phi + w*(k+1/2), -z + m)
 * 3) C(n,h)(None): (phi + w*k, z + m), (phi + w*k, -z + m)
 * 4) C(n,h)(Zigzag): (phi + w*k, z + m + k/2), (phi + w*k, -z + m + k/2)
 * 5) C(n,v)(None): (phi + w*k, z + m), (-phi + w*k, z + m)
 * 6) C(n,v)(Pln.Rfl.): (phi + w*k, z + m), (-phi + w*k, z + m + 1/2)
 * 7) D(n)(q): (phi + w*k, z + m + k*q/n), (-phi + w*k, -z + m + k*q/n)
 * 8) D(n,d)(None): (phi + w*k, z + m), (phi + w*(k+1/2), -z + m), (-phi + w*k, z + m), (-phi + w*(k+1/2), -z + m)
 * 9) D(n,d)(Pln.Rfl.): (phi + w*k, z + m), (phi + w*(k+1/2), -z + m), (-phi + w*k, z + m + 1/2), (-phi + w*(k+1/2), -z + m + 1/2)
 * 10) D(n,h)(None): (phi + w*k, z + m), (phi + w*k, -z + m), (-phi + w*k, z + m), (-phi + w*k, -z + m)
 * 11) D(n,h)(Pln.Rfl.): (phi + w*k, z + m), (phi + w*k, -z + m), (-phi + w*k, z + m + 1/2), (-phi + w*k, -z + m + 1/2)
 * 12) D(n,h)(Zigzag): (phi + w*k, z + m + k/2), (phi + w*k, -z + m + k/2), (-phi + w*k, z + m + k/2), (-phi + w*k, -z + m + k/2)

For the zigzag ones, we can redefine w as w/2, and split k into even and odd cases: 2k and 2k+1. One gets

(phi + w*k, z + m), (phi + w*(k+1/2), z + m + 1/2)

Distilled into a multiplier of phi, a multiplier of z, and sub-grid offsets in phi and z:
 * 1) C(n)(q): (1,1,0,0) (lattice may be oblique)
 * 2) S(2n): (1,1,0,0), (1,-1,1/2,0)
 * 3) C(n,h)(None): (1,1,0,0), (1,-1,0,0)
 * 4) C(n,h)(Zigzag): (1,1,0,0), (1,1,1/2,1/2), (1,-1,0,0), (1,-1,1/2,1/2)
 * 5) C(n,v)(None): (1,1,0,0), (-1,1,0,0)
 * 6) C(n,v)(Pln.Rfl.): (1,1,0,0), (-1,1,0,1/2)
 * 7) D(n)(q): (1,1,0,0), (-1,-1,0,0) (lattice may be oblique)
 * 8) D(n,d)(None): (1,1,0,0), (1,-1,1/2,0), (-1,1,0,0), (-1,-1,1/2,0)
 * 9) D(n,d)(Pln.Rfl.): (1,1,0,0), (1,-1,1/2,0), (-1,1,0,1/2), (-1,-1,1/2,1/2)
 * 10) D(n,h)(None): (1,1,0,0), (1,-1,0,0), (-1,1,0,0), (-1,-1,0,0)
 * 11) D(n,h)(Pln.Rfl.): (1,1,0,0), (1,-1,0,0), (-1,1,0,1/2), (-1,-1,0,1/2)
 * 12) D(n,h)(Zigzag): (1,1,0,0), (1,1,1/2,1/2), (1,-1,0,0), (1,-1,1/2,1/2), (-1,1,0,0), (-1,1,1/2,1/2), (-1,-1,0,0), (-1,-1,1/2,1/2)

etc.

Shall I give the mathematical forms of the wallpaper-group matrices and offsets?

Rassat's article splits C(n) and D(n) into three special cases each. q = 0, q = n/2 for even n, and q nonzero. I see no reason to follow that split, because my other main source doesn't make it.

I still think that going into a lot of detail about the wallpaper groups is unnecessary. I think that the Hermann-Mauguin symbols should be enough. If we need any diagrams, it should be something like in "Line Groups Structure"

BTW, that group goes into detail about phi and z actions.

Lpetrich (talk) 00:06, 28 June 2011 (UTC)


 * I'll see if I can understand more of your descriptions above. For now the wallpaper correspondence and diagrams are my only doorway to understanding since the Intl notations (and your offset types) are yet cryptic to me. I removed the triple-diagrams for p1/p2. I moved the Coxeter notation after the wallpaper notation, since Coxeter's notation is only the 2D wallpaper as the limit as n->&infin;. Tom Ruen (talk) 02:32, 28 June 2011 (UTC)

The international / Hermann-Mauguin notation is often used for space groups. In fact, every rod group is a member of some line-group family for some suitable n -- it's easiest to see when the H-M symbols match.

A great help for me has been the computer-algebra software package Mathematica. It can crunch through all the vector and matrix calculations needed for testing various hypotheses, like whether a purported group is closed or whether one group is a rotated version of another. If Mathematica or Maple are beyond your budget, I suggest trying Maxima or Sage -- they are freely downloadable.

Lpetrich (talk) 03:34, 28 June 2011 (UTC)


 * Thanks for the suggestions. Lots of good stuff out there I'm sure, but I don't need to do too much computations now. Mostly just help in reading different sources and understanding the notations. My interest in the Coxeter notation, and these line groups specifically, is because they represent duocylindrical groups of 4-polytopes of interest, like duoprism, and grand antiprism. So these groups in general are the point groups [n,2,m] in 4-space, and good to see their relation to the wallpaper groups. I'm also trying to follow Conway's notation for the 3-sphere point groups, and hoping I can get a correspondence table with Coxeter's notation. There's always more! Tom Ruen (talk) 04:37, 28 June 2011 (UTC)


 * p.s. If the international / Hermann-Mauguin notation supports non-crystalographic groups (orders besides 1,2,3,4,6), are there symbols for Icosahedral groups? (532, *532) Tom Ruen (talk) 04:48, 28 June 2011 (UTC)

Indeed there are: Crystallography of quasicrystals: applications to icosahedral symmetry mentions for pure rotation, 532, and for rotation + reflection $\overline{2n}$$\overline{n}$2/m.

Three-dimensional point groups (chap 2 in book) goes into a lot of detail about them and their Hermann-Mauguin symbols.

C(n) has n, for n-fold rotation axes.

D(n) has n22 for n even, n2 for n odd. The 2's are for 180&deg; rotation in the perpendicular plane. For regular polygons and n even, there are two families: vertex-vertex and edge-edge, while for n odd, there is only one family: vertex-edge. Thus the two 2's for even and one 2 for odd.

The symbol $\overline{2n}$ means generation by a n-fold rotation followed by inversion of all three coordinates. This is equivalent to rotation by (n+2)/(2n) of a circle, with an axial-direction inversion. Book page 31 has a nice illustration. Separate + and - for 3 and 4, overlapping for 6. This also explains S(n) for odd n: it's C(n,h).
 * n = 2k+1 -- (2k+3)/(2(2k+1)) generated by 1/(2(2k+1)). The +axial,-axial will not overlap. Thus, S(2(2k+1)) is $\overline{n}$
 * n = 2(2k+1) -- (m+1)/(2m+1) generated by 1/(2k+1). The +axial,-axial will overlap. Thus C(2k+1,h) is $\overline{2n}$
 * n = 4k -- (2k+1)/(4k), generated by 1/(4k). The +axial,-axial will not overlap. Thus, S(4k) is $\overline{2n}$

Turning to mirroring, there are two 3D notations:
 * nm for n-fold rotations combined with mirroring perpendicular to the axis (mirror plane contains axis)
 * n/m for n-fold rotations combined with mirroring along the axis (mirror plane does not contain axis). (2k+1)/m is the same as $\overline{2n}$, and is written that way. That's why C(n,h) is written n/m for n even and $\overline{n}$ for n odd.

C(n,v) is nmm for n even and nm odd, for the reason that D(n) is n22 for n even, n2 for n odd.

D(n,h) is n/mmm when n is even; it is short for n/m 2/m 2/m (n along axis, 2 in perp plane, another 2 in perp plane)

D(n,h) is $\overline{2n}$2m when n is odd; it is short for n/m 2 2/m, (n along axis, 180&deg; rotation in perp plane, another 2 in perp plane)

Analogously, D(n,d) is $\overline{n}$2m when n is even, and $\overline{2n}$m when n is odd.

Now the Platonic-solid groups and their H-M notations.

Rotations of a tetrahedron are about twofold and threefold axes, one family each. Edges, vertices/faces. Thus, the rotational tetrahedral group T's H-M symbol is 23.

Rotations of a cube or octahedron are about twofold, threefold, and fourfold axes, one family each. Edges, vertices/faces, faces/vertices. Thus, the rotational octahedral group O's H-M symbol is 432.

Rotations of a dodecahedron or icosahedron are about twofold, threefold, and fivefold axes, one family each. Edges, vertices/faces, faces/vertices. Thus, the rotational icosahedral group I's H-M symbol is 532 or 235.

The full isometry group of rotations and reflections a tetrahedron has no inversions, for obvious reasons. It is Td, with H-M symbol $\overline{n}$ 3 2/m, or $\overline{2n}$3m -- a pair of opposite edges flipped, face flipped, single edge flipped.

A tetrahedral group with inversions is the pyritohedral group, and is distinct from Td. It is Th, with H-M symbol 2/m, or m -- edge rotation inverted, face/vertex rotation inverted

An octahedral group's rotations and reflections is the original with inversions. It is Oh, with H-M symbol 4/m 2/m, or mm -- face/vertex rotation inverted, vertex/face rotation inverted, and edge rotation inverted.

An icosahedral group's rotations and reflections is the original with inversions. It is Ih, with H-M symbol  2/m, or m also or m -- face/vertex rotation inverted, vertex/face rotation inverted, and edge rotation inverted.

So yes, there are H-M symbols for the icosahedral groups, along with axial non-crystallographic axial point groups.

Lpetrich (talk) 20:27, 30 June 2011 (UTC)


 * How do you write D5h? $\overline{2n}$2m --> $\overline{2n}$2m or $\overline{5}$.2m?! Tom Ruen (talk) 20:58, 30 June 2011 (UTC)

I think that one would use punctuation to disambiguate such cases, like your $\overline{3}$.2m

BTW, the link to that Andr&eacute; Rassat paper is [http://books.google.com/books?id=bnDIQ3uEPbMC&printsec=frontcover#v=onepage&q&f=false Crystallography of supramolecular ... - Google Books] and I'll give for reference his wallpaper-group identifications: My preference for mentioning them would be to stick to standard notations as much as possible, unless one's willing to make a note of one's departure from it. For wallpaper groups, that would be what's in Wallpaper group. I used AR's names in this list, he used p2mm, p2mg, p2gg, c2mm, instead of more typical pmm, pmg, pgg, cmm, which would be more appropriate for the article.
 * p1: C(n, helicity q) - he has 3 special cases
 * p2: D(n, helicity q) - he has 3 special cases
 * pm:
 * pm(h): C(n,h, none)
 * pm(v): C(n,v, none)
 * pg:
 * pg(h): S(2n, none)
 * pg(v): C(n,v, pln.ref. offset)
 * cm:
 * cm(h): C(2n,h, zigzag)
 * cm(v): C(2n,v, zigzag)
 * p2mm: D(n,h)
 * p2mg:
 * p2mg(h): D(n,d, none)
 * p2mg(v): D(n,h, pln.ref. offset)
 * p2gg: D(n,d, pln.ref. offset)
 * c2mm: D(n,h, zigzag)

I'd also prefer internally linking to Hermann-Mauguin notation, since that page has some description of it, as opposed to the "IUC" page.


 * My concern on the HM notation label is that there appears to be two distinct notations, one for 2D ({p,c}{1,2,3,4,6}{1,m,g}2), one for 3D (crystalographic). So one reference I had called the wallpaper groups IUC, so I thought that would be more clear, currently as a redirect IUC notation, which might equally be linked into a section of Hermann-Mauguin notation if such a section existed. Tom Ruen (talk) 22:57, 30 June 2011 (UTC)

Frieze groups
Hi, I noticed that the part of this article dealing with Frieze groups contains a table that seems to be the same as the one in the article about Frieze groups. I think it'd be better if it were removed from here while keeping only a see-that-article link to Frieze groups. Regards, --Tomaxer (talk) 17:11, 30 June 2011 (UTC)


 * I made a smaller summary table. Tom Ruen (talk) 20:58, 30 June 2011 (UTC)


 * Thanks for the effort. --Tomaxer (talk) 18:59, 1 July 2011 (UTC)

D(n,d) for n odd
I had to revert D2nd to Dnd because Dnd is reasonable for n odd. In crystallographic point group, one can find the groups D1d and D3d, which are both counterexamples to even-only. — Preceding unsigned comment added by Lpetrich (talk • contribs) 23:07, 7 July 2011 (UTC)

Just because Dnd exists for odd n does't mean it makes sense in the 3D line group. I can't evaluate the crystallographic point groups, I'm only going by the Coxeter notation, which requires 2n due to the simple or glide reflections in the related wallpaper group(s). An odd n would create a double-covering over the wallpaper cylinder to repeat the pattern. -TomRuen 75.146.178.58 (talk) 01:08, 8 July 2011 (UTC)


 * Okay, I see Dnd=[2+,2n], so the 2n is contained implicitly. 75.146.178.58 (talk) 01:27, 8 July 2011 (UTC)