Talk:Frieze group

Cleanup
Major tidy-up planned here. The current version is vague and incorrect in several places. Also the list of frieze groups is in a different order to the diagram. The diagram is not explained (the group is not the pattern) nor is it particularly well-chosen, as the friezes come over as noise. I will work on it sporadically over the next hour. --AndrewKepert 06:35, 3 Feb 2004 (UTC)

Okay, getting there. Still some work to do, but it will be dark soon and I haven't yet put lights on my bike for the year. --AndrewKepert 08:15, 3 Feb 2004 (UTC)

I have finished the overhaul - further tweaking needed. The old image is below: --AndrewKepert 06:50, 9 Feb 2004 (UTC)


 * The image below gives seven sample patterns on the strip whose symmetry groups are the seven frieze groups. However, the order does not correspond with the order in the list above.  (TO FIX)


 * [[image:frieze.png|Image of frieze groups]]

Comments on recent edits
Just for any other observers' benefits, here is a posting I just made to User talk:Patrick:

I noticed that you have contributed a large sequence of edits on Frieze group in the last couple of weeks. Unfortunately your edits have introduced a number of factual errors and problems with the language. I also feel you have introduced too much unnecessary detail, some of which belongs in, or is duplicated in, other articles. Natch. this is all IMHO so take it with 0.0647g of NaCl if you prefer.
 * Below you mention only one factual error, which I corrected.--Patrick 20:16, 1 August 2005 (UTC)

I will be brief, and use the 1/4/2005 edit of Linas as my point of reference. The side-by-side comparison is here:. My apologies if not all the edits I refer to are yours.
 * The opening two sentences are far less clear than the original. In the first, you have made an attempt to refer to the equivalence classes of groups, but the strict mathematical "class of groups of transformations of pairs of numbers" is four levels of abstraction deep (well, five if you consider numbers an abstraction of quantity).  Too deep for a general reference work such as WP.  IMO the original captures the essence of the groups, and if it is necessary to add the extra level of abstraction, this can be done via a "strictly speaking" comment after the definition, possibly linked to Isomorphism class or whatever.  The second sentence is currently incorrect - a group is not an isometry.  (Ironically, you have equated an object at one level of the tower of abstraction with one at a different level.)
 * I moved the formal part a little down and corrected the error. The levels of abstraction are inherent in the subject, I found that leaving one level out was confusing, e.g. a symmetry group of type 2 is only a subgroup of one of type 7 if for the latter we take a smaller translation distance; also combining reflection and rotation is different depending on the positions of the centers, so we can not simply take the origin at the symmetry center.--Patrick 13:10, 1 August 2005 (UTC)


 * The introduction of cartesian coordinates is not necessary for the definition of Frieze group, or even the characterisation of the 7 classes. Coords should not be introduced unannounced.  However, they are useful in analysing the groups in later sections.
 * I did not change that.--Patrick 13:18, 1 August 2005 (UTC)


 * You have substituted the &deg; (&amp;deg;) sequence for that symbol in whatever code table your computer and browser likes. It is better to use the character entity.  See
 * It is at the bottom of the edit page, so apparently a Wikimedia standard.--Patrick 13:18, 1 August 2005 (UTC)


 * The listing of finite symmetries of a strip is unnecessary detail, amplifying an off-hand remark in the original. Maybe trim back and refer to Point group.
 * It could be moved there, but since this is specifically about a strip, it is also very much related to the subject of this article.--Patrick 13:24, 1 August 2005 (UTC)


 * I am not sure of your intent in the "1D" sections and "Mathematics of ..." sections. I am guessing that you are attempting to demonstrate that these are the only groups by first analysing the action of the group on the long axis and then adding "thickness".  However this should be made explicit - the article is not about isometries of R.
 * I moved the 1D section to a separate article.--Patrick 13:32, 1 August 2005 (UTC)

Finally, a request: "Show preview" and "Comment"! Or at least a brief description of what you are doing and what you plan to do in the talk page. The history page for your edits doesn't help me see what you have done. I don't have time to work on the article just now, and it seems that your are still progressing with your edits. I may visit again in a week or two.

All the best, Andrew Kepert 10:12, 1 August 2005 (UTC)


 * Hi, I just want to tentatively add my voice to AndrewKepert's comments. Although I'm not familiar with the history of the Frieze group article, I can see there is currently a problem with target audience. At the very least, the introduction to Frieze group should be something along the lines of the introduction Wallpaper group, aimed at a general audience, with all the mathematical terminology separated off in a "formal development" section or something similar. Dmharvey Talk 11:02, 1 August 2005 (UTC)


 * Good idea, I changed the intro.--Patrick 13:32, 1 August 2005 (UTC)


 * (following Patrick's intro changes as a result of Andrew's constructive criticism)... I think that's a definite improvement.
 * Sometime I want to add some real-life photos of these groups occurring in real life, like for what's happening in Wallpaper group. Some time in the next few weeks I'll get around to it hopefully. Having a pretty one in the introduction would be especially illuminating. Dmharvey Talk 16:10, 1 August 2005 (UTC)

numbering of groups
This numbering by 1, 2, ... 7, is this standard notation? If not, doesn't it add an extra layer of confusion where unnecessary? Dmharvey Talk 10:00, 10 August 2005 (UTC)

couple of photos
I'm trying to collect some photos to work into this article...

group 7


Dmharvey 03:07, 12 March 2006 (UTC)

Abstract structure of 7th frieze group
The article says that the 7th frieze group "is isomorphic to a semidirect product of Z × C2 with C2". Isn't it also isomorphic to C2 × (the semidirect product of Z by C2)? And isn't this a rather more helpful way of describing its structure? Maproom (talk) 13:44, 6 July 2008 (UTC)


 * Every semidirect product of Z &times; C2 and C2 has a copy of C2 as a direct factor, and a copy of Z semi C2 as a direct complement, so I think you are right. In fact, (the semidirect product of Z by C2) is called the infinite dihedral group so the name would definitely be nicer. JackSchmidt (talk) 16:25, 6 July 2008 (UTC)

Infinite dihedral group
Is there some reason to avoid mentioning the infinite dihedral group? It is just group #4, like, literally, this is how dihedral groups are defined. This also simplifies the exposition of group #7. JackSchmidt (talk) 14:00, 10 July 2008 (UTC)


 * It is algebraically isomorphic (I added that), but, the way infinite dihedral groups are defined here, not as group of isometries.--Patrick (talk) 21:24, 10 July 2008 (UTC)

Alternative Notations
I think that we ought to mention alternative notations for the frieze groups.

Fundamentals of Crystallographic Symmetry by Paolo G. Radaelli discusses the frieze groups starting at internal page number 19. He uses Hermann–Mauguin notation for the groups.

The International Union of Crystallography has published International Tables for Crystallography, where Volume E discusses the frieze groups, but it is behind a paywall. SpringerLink: The 7 frieze groups reproduces that part, and it is also behind a paywall. However, its Fulltext Preview gives them numbers and uses the Hermann-Mauguin symbols for them. I'll give the IUCr's numbering, the H-M symbol, and the number in the article:

1: p1 (Article #1): b b b b b 2: p211 (Article #5): b b b b b q q q q q 3: p1m1 (Article #4): bdbdbdbdbd 4: p11m (Article #3): b b b b b p p p p p 5: p11g (Article #2): b b b b b p p p p p 6: p2mm (Article #7): bdbdbdbdbd pqpqpqpqpq 7: p2mg (Article #6): bdbdbdbdbd qpqpqpqpqp

Lpetrich (talk) 20:29, 16 June 2011 (UTC)


 * Good idea. (The foot prints in the article are also useful examples too!) I've not seen many sources for the Frieze groups to identify them. I created table at Template:Frieze group notations that can be shared between other articles as well. Only conflict from your notation above, p1 is listed as the pure two directional translational Wallpaper group, so I listed this version as p111'', unsure if that's reasonable, but it seems a good compromise for now. I really don't believe wallpaper and frieze groups ought to have the same symbol! Tom Ruen (talk) 21:12, 19 June 2011 (UTC)


 * Here's a source that names them, including p1, but I wonder why identical to a wallpaper group?


 * Looking further, looks like a number of overlaps - p1, p1m1, p211, p2mg, and p2mm ALL exist as both Frieze groups and Wallpaper groups. So if that's true, it means ambiguity on which group is implied based on context?! Tom Ruen (talk) 21:29, 19 June 2011 (UTC)

One would so so by context. The elements in these groups have this action on a vector x that makes it x':

x' = R.x + D

where R is a rotation / reflection matrix and D is a translation vector. The R's form point groups. D has the form D(grid) + D(subgrid). Each R value is associated with a D(subgrid) value. The Hermann-Mauguin notation describes the R / D(subgrid) combinations, so that's why the frieze and the wallpaper notations overlap. To illustrate, I'll give the R and D values for each sort of element of a frieze group:

Grid unit vector: {1,0} R/R matrix, subgrid vector, and their action:
 * 1) {{1,0},{0,1}}, {0,0}, x0' = x0 + n, x1' = x1, translation only
 * 2) {{1,0},{0,-1}}, {0,0}, x0' = x0 + n, x1' = -x1, reflection across repeat direction
 * 3) {{1,0},{0,-1}}, {1/2,0}, x0' = x0 + n, x1' = -x1 + 1/2, glide reflection across repeat direction
 * 4) {{-1,0},{0,1}}, {0,0}, x0' = - x0 + n, x1' = x1, reflection along repeat direction
 * 5) {{-1,0},{0,-1}}, {0,0}, x0' = - x0 + n, x1' = -x1, 180d rotation
 * 6) {{-1,0},{0,-1}}, {1/2,0}, x0' = - x0 + n, x1' = -x1 + 1/2, 180d rotation with glide

Membership:
 * p1: 1
 * 1) p11g: 1,3
 * 2) p11m: 1,2
 * 3) p1m1: 1,4
 * 4) p211: 1,5
 * 5) p2mg: 1,3,4,6
 * 6) p2mm: 1,2,4,5

The overlap in notation extends to three-dimensional groups. Bilbao Crystallographic Server lists "rod groups", "layer groups", and 3D space groups, all with overlapping notations, and all with notations that overlap those of the wallpaper and frieze groups. This is likely for the same reason: same or similar R + D(subgrid) sets.


 * Rod groups: like frieze groups, but with two dimensions around a one-dimensional grid
 * Layer groups: like frieze groups, but with one dimension around a two-dimensional grid

(lpetrich forgot to sign earlier)


 * Thanks for the information, lots more to learn! I've barely touched looking at the 3-space groups. Is there anything else that should be added now? I did find one source that said p2 instead of p211. It looks like there's long and short names (like p2mg vs mg), so if there's difference names that are equivalent names that should be listed? Tom Ruen (talk) 23:47, 20 June 2011 (UTC)

The wallpaper groups are listed with both full and abbreviated Hermann-Mauguin symbols; we could do likewise with the frieze groups.

BTW, I've discovered general rod groups. The ones at Bilbao and in some of the literature have crystallographic constraints; their point groups can only be ones for grids. I suspect that I've duplicated something in the professional literature, but I haven't been able to find out where.

Lpetrich (talk) 11:48, 22 June 2011 (UTC)

Extension to three dimensions: line, rod, and layer groups
I've come across generalizations of the frieze groups to three dimensions.

Line groups

A three-dimensional line group is a combination of an axial point group and repeats along the axis, a one-dimensional lattice. From the 7 infinite families of axial point groups, one finds 13 infinite families of line groups.

Line Groups Structure, the second chapter of Amazon.com: Line Groups in Physics: Theory and Applications to Nanotubes and Polymers (Lecture Notes in Physics) (9783642111716): Milan Damnjanovic, Ivanka Milosevic: Books

Rod groups

The 75 rod groups are the line groups whose point groups are crystallographic ones. Source: Bilbao Crystallographic Server: Subperiodic Groups: Layer, Rod and Frieze Groups Retrieval Tools

Layer groups

The 80 layer groups are subgroups of space groups; they are combinations of the crystallographic point groups with a two-dimensional lattice. Also at Bilbao.

Should I create separate articles for them or additional sections of the frieze-group article? I'd list all the rod and layer groups in them and try to describe the families of line groups.

Lpetrich (talk) 15:15, 23 June 2011 (UTC)


 * I suggest, separate articles, with links from frieze group, wallpaper group, and point group. Maproom (talk) 15:28, 23 June 2011 (UTC)


 * Cool stuff! Definitely separate article(s) is/are needed. Would this Coxeter_helix "polyhedron" (helix chain of tetrahedra) have a line symmetry group? Tom Ruen (talk) 19:54, 23 June 2011 (UTC)
 * Coxeter helix.png

Yes indeed, the group that I've called Cn,m -- the cyclic point group with the offsets making a helix with m turns over the the entire cycle. I've had to resort to the expedient of inventing a Schönflies-inspired notation, because my sources do not use a clear, Schönflies-style notation. — Preceding unsigned comment added by Lpetrich (talk • contribs) 00:01, 24 June 2011 (UTC)

First illustration should be replaced
The illustration at the top of the article, entitled "Example Frieze group patterns," should be replaced. It purports to show portions of two friezes. The one at the bottom is OK: it's recognizably a portion of a p2mg frieze. But what in the world is going on at the top? If this is a portion of a frieze, then it consists at most a single fundamental domain, i.e., it could just as well be replaced by anything at all! It certainly doesn't illustrate the concept of a frieze in any helpful way. Ishboyfay (talk) 21:26, 25 September 2013 (UTC)


 * I agree. The upper part of the image shows something which might be a single repeating unit, but is probably only part of one. It does not help the reader to understand frieze groups, indeed it hinders. Maproom (talk) 22:40, 25 September 2013 (UTC)


 * I looked at and this one stood out as simple, and two clear examples, p2, p2mg I think: Tom Ruen (talk) 23:00, 25 September 2013 (UTC)
 * Meander alagrek.svg


 * Now in use -- thanks! Ishboyfay (talk) 20:53, 30 September 2013 (UTC)

Notation & terminology
I'm having trouble correlating the IUC notation in the 3 different tables where it's used. The first large table (with purple example images) uses the term p1m1, whereas the second table (really a list called "Summarized") uses p2m1. The third table (oblique/rectangular lattice) goes back to using p1m1. Note the differences.

Here is the other instance. The 1st table uses the notation p2, the 2nd table uses p2, and the 3rd table uses p211.

I am not at all familiar with IUC notation. Is p2m1 synonymous with p1m1? Likewise, is p2 synonymous with p211? If not, then there's a typo. If so, I still think the tables should be consistent with each other. --Officiallyover (talk) 22:37, 20 February 2014 (UTC)


 * On a quick look, its clear they are indexed 1-7 inconsistently, and yes p211=p2. Tom Ruen (talk) 06:54, 21 February 2014 (UTC)

numbering of groups
The text refers to the groups by number. I have made this consistent with the ordering in the table, but it would be nice if the table itself were numbered for easy reference. I'm not sure how to edit the table. Is the markup for the table located on some other page? Will Orrick (talk) 21:29, 10 December 2015 (UTC)


 * I've now realized that the table is a template that has been transcluded in the article. It appears that, at some point, the order of the groups in the table changed, but the article did not keep pace.  Since the table is a template, it might be problematic to add numbering or make other changes that would affect other articles that transclude the template. Will Orrick (talk) 04:00, 11 December 2015 (UTC)


 * Thanks for your effort. The table is here Template:Frieze group notations, and without any indexing, but it could be added. It is also used here Orbifold_notation. It seems better to avoid indexing in the text, but if we want to include it, I guess we should look at references and see if they index, and if they are consistent. For now I moved the short indexed list before the larger table to help. Tom Ruen (talk) 06:32, 11 December 2015 (UTC)


 * I agree that it would be better to avoid indexing, if a suitable substitute could be found. I have added descriptive text in places, where before there were bare lists of indices.  One further question: why is G included as a symmetry of group 7, but not of group 3?  It seems to me that it should either be included in both, or omitted from both.  I have a preference for the latter, since, if my understanding is correct, the G in these cases is the composition of T and H. Will Orrick (talk) 09:24, 12 December 2015 (UTC)


 * Regarding the listing of G in groups 3 and 7, some additional thinking has brought me around to the opposite point of view, namely that G ought to be listed for both. It seems better to list all of the symmetries for each group, even when some are compositions of others.  For some examples: T is listed in group 2, even though it is the composition of two Gs; R is listed in group 7, even thought it is a composition of H and V.  Moreover, while it is true that the G in groups 3 and 7 is the composition of T and H, it is equally true that T is the composition of G and H in these groups.  The discussion of possible minimal sets of generators is a separate one that is already done very well in the article.  It seems that the removal of G from group 3 was only done in 2014.  I'm going to go ahead and add G back, both in the template and in the summary table.  I will include a remark that it arises as the composition of T and H. Will Orrick (talk) 13:39, 13 December 2015 (UTC)

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