Talk:Wallpaper group

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Contents 1 Re: Guide to recognising wallpaper groups 2 Re: Illustrating plane isometries 3 regarding the recent makeover of this page 4 request for photographs of wallpaper groups 5 other stuff to do 6 Discrimination table 7 orbifold notation examples 8 Diagrams on Commons 9 Shapes and colours? 10 Orientation of diagram vs. that of computer-generated images 11 Fidgeting 12 Enumeration proof stub filled 13 Crystallographic notation 14 Problem with italics in headers 15 Splitting the page 16 Illustrations 17 Article split up? 18 Roger Penrose tiles? 19 miscellaneous 20 Formal definition 21 Images showing a rhombus 22 A note on p1

[edit] Re: Guide to recognising wallpaper groups

At least for me, the biggest difficulty is recognizing which part of the pattern is the tile. After that, finding out the group is much easier. So maybe we should add a paragraph on how to identify the tile, ideally with example(s). Trapolator 04:03, 6 Jun 2005 (UTC)

[edit] Re: Illustrating plane isometries

A simple, easily remembered, illustration of plane isometries requires no images, merely a geometric sans-serif font.

• [d  ] Identity
• [dp] Rotation
• [dd] Translation
• [db] Reflection
• [dq] Glide reflection

I used this in the Isometries as reflections section of Euclidean plane isometry. KSmrq 14:00, 8 Jun 2005 (UTC)

[edit] regarding the recent makeover of this page

I have completely redone the wallpaper group page. I borrowed some pics and links but most of the text is new. Also there is LOTS of art.

I expanded the section on euclidean plane isometry-ies and moved it into its own article. It has some material in common with Coordinate rotations and reflections. I believe these articles could be profitably merged.

Things to do include:

• stuff on the "to do" list later on this discussion page
• still need to finish labelling the example patterns of Wallpaper group; I've only done the first few. I will finish this off in the next few days. I don't suppose anyone else will have much luck with it; I happen to have the book "Grammar of Ornament" here with me.
• there should be a discussion of lattices somewhere.
• the "stub" bits need to be filled out, especially relationship between informal and formal approaches. I've noticed the text is not very explicit on that point as it stands.
• needs pretty pictures illustrating the various types of euclidean isometries.
• needs REFERENCES

Dmharvey Talk 17:02, 4 Jun 2005 (UTC)

[edit] request for photographs of wallpaper groups

I hereby initiate a project to collect photographs illustrating the 17 wallpaper groups. (See examples to the side).

These patterns are used in all kinds of artistic situations, especially in architecture (bricks, tilings, pavings, etc) and in decorative art.

I am looking specifically for photographs. The article already has some "diagrammatic" representations, but I think the article could be made far more appealing if we show examples that people are familiar with in everyday life, and examples with artistic/aesthetic merit.

Also, in the next month or so, I intend to edit this article so it becomes more accessible to the non-mathematically inclined. It is an excellent example of a mathematical article that could have wide appeal.

Please deposit links to the images on this discussion page. When there are sufficiently many, I will put them in the article proper.

When you add an image, please try to identify which of the 17 patterns it corresponds to (see article), and include it in the filename. Please try to match the filename conventions I have used above. Also, give some indication in the comment field of the source of the image (e.g. "pattern on the oval office ceiling"). We might as well keep reasonably high-res versions available (the examples I have given are approx 1MB jpegs).

Make sure your photo includes a few "cells", so that the repetitive nature of the pattern can be easily seen. If possible, try to rotate the image into a sensible orientation, and make sure the brightness/saturation etc is reasonable. (I can do this myself if need be.)

Note that it is virtually impossible to get an exact representation. For example, in the p4g photo shown here, some of the tiles are slightly orange-coloured, in a manner not strictly matching the p4g description. But it's pretty close, and gets the general idea across.

If you think you can improve on one of the images already present, please go ahead! Call it for example "Wallpaper_group-cmm-2.jpg", etc.

I have made a small beginning with some of the easy ones, from my bathroom and garage (see thumbs).

Thank you so much. Dmharvey 21:32, 31 May 2005 (UTC)

[edit] other stuff to do

• historical links to follow up at http://www.mi.sanu.ac.yu/vismath/ana/ana6.htm, would be nice to include some of this
• need a page on the crystallographic restriction theorem, i.e. why is it that the only allowable rotations are of order 2, 3, 4, or 6. Also cover higher dimensional cases involving totient function.
• would be nice eventually to have a (possibly informal) discussion on how to prove that there are exactly 17 groups
• need an informal discussion of when two groups are considered isomorphic. After all it may be difficult for someone without group theory background to recognise why two completely different looking patterns are actually "the same", or why two similar looking patterns are actually "different".
• I think there is some kind of concept of geometric isomorphism which is a priori different from isomorphism as abstract groups; although if I recall correctly it turns out that they are the same concept; i.e. all the 17 groups are non-isomorphic even abstractly.
• would be nice if this article was more in sync with the frieze groups article.

[edit] Discrimination table

Although the text said the following table explained how to decide which group to assign to a pattern, a list was used. The list was compact in the source, but seemed awkward on the page. To try to make the decision tree easier to read, I have created a version as a table with nested tables. In an abundance of caution, should others violently disagree with the wisdom of this, I have left the previous list version in the source, commented out. Admittedly, just shuffling the deck chairs. :-) KSmrq 02:17, 11 Jun 2005 (UTC)

Thanks KSmrq, I think your table is a vast improvement in legibility. I might come back soon and change some of the labels/terminology in the table, to more closely match that used in the discussion earlier in the article. Dmharvey Talk 11:22, 23 Jun 2005 (UTC)

The most uncomfortable terminology is "rotocenter"; but alternatives sprawled out of control, so I stuck with it. Substitute "center of rotation" and you'll see what I mean. Incidentally, I found it much harder to format a table like this in Wikipedia syntax than I would in XHTML/CSS2. Admittedly, part of that was climbing the learning curve. KSmrq 05:43, 26 Jun 2005 (UTC)

Hey what happened to the table? It used to have grid lines, where did they go? Looked much better before. Dmharvey Talk 28 June 2005 16:15 (UTC)

Grid lines show up for me. Could be your browser or the new WikiMedia software. The lines do assist readability. 68.63.244.30 28 June 2005 17:23 (UTC)

[edit] orbifold notation examples

Hi KSmrq, are you sure the "cmm" example for the orbifold notation is correct? I find that after following your instructions for the first three symbols 2*2, I am already forced to have the entire cmm group, so it is not true that the last 2 represents an independent rotation. Dmharvey Talk 28 June 2005 18:07 (UTC)

Also could you please fix up the text to clarify exactly how the terms dihedral and cyclic are used in this context. Dmharvey Talk 28 June 2005 18:07 (UTC)

Sorry, I didn't notice your questions until now. I have added links for dihedral and cyclic, which may help a little.

As for cmm and 2*22, look at the fundamental region. (Tess is handy for this.) On its border we see three rotation centers, none of which is a symmetry image of the others. (I've modified the text a little to clarify.) One of them lies on a glide reflection axis intersection but not any mirror axis. The other two lie on intersecting 45° mirror axes.

Using the idea of a fractional Euler characteristic we can count the contribution of each notated feature. The first "2" counts as 1/2; the "*" counts as 1; and each following "2" counts as 1/4. The sum is exactly 2, as is required for a wallpaper orbifold. In fact, I've been contemplating filling the "proof of 17" stub with this idea, but am a little worried about the background required. The rule is that "n" before a "*" or "x" counts as (n−1)/n; after, half that. Both "*" and "x" themselves count as 1, and "o" counts as 2. So 4*2 (p4g) adds up as 3/4 + 1 + 1/4. Similarly, 3*3 (p31m) yields 2/3 + 1 + 1/3. KSmrq 10:51, 2005 July 25 (UTC)

[edit] Diagrams on Commons

I'm currently working on a german version of this article. For this, I created a new set of diagrams. As those images are on Wikimedia Commons, you could include them here as well. The diagrams are somewhat different than the ones currently used.

Main Advantages in my opinion are:

• Different equivalence classes of symmetry elements are colored (and rotated) differently
• Higher resolution, available as SVG as well
• F-shaped tile mark

I often use a smaller cell, which might not be rectangular in some cases, and which might also cut elemental cells in half. I do this because I think smaller cells are more intuitive, but if there is a reason why the diagrams as currently listed are better, I'd perhaps change my diagrams as well. What do you think?

If someone is interested in the XML and XSLT I used to create those images, I'm willing to release them under GPL, so feel free to ask. But beware, It's a crude job in some places, so don't expect too much.

Your diagrams are a huge improvement. I would definitely like them included in the english version, in fact to replace the current diagrams. Actually, I have a bigger job in mind: I think the "illustrations from architecture and art" need a separate page, with plenty of cross linking back and forth. Something like Wallpaper group (examples). Besides, I have a whole heap of further such examples sitting on my computer, waiting to be cleaned up. Unfortunately I don't have time right now to work on this, but please feel free to go ahead and start doing it, if you feel so inclined. Dmharvey Talk 19:59, 22 July 2005 (UTC)

Thank's for the praise. I will gladly exchange the existing images for my own, but as wikimedia servers are somewhat instable at the moment, I will do so probably next week. I was thinking about an extra examples page as well. As examples would need very little language specific text and would thus apply to all languages, perhaps such an example page would best be located on Commons as well. The images should be moved there in any case, I believe. I think I can do this as soon as I get access to the Commons bulk File upload service. -- Martin von Gagern 22:58, 22 July 2005 (UTC)

I agree that the example images should be on Commons. (I apologise for not uploading them there myself to begin with. I will do so from now on.) However, I think there should be a separate example pages on each wikipedia. The article should be part of the encyclopaedia(s) after all. Dmharvey Talk 13:30, 23 July 2005 (UTC)

I think it is sometimes helpful to have both versions, showing sometimes a different orientation, a different choice of fundamental domain, etc.--Patrick 22:46, 25 July 2005 (UTC)

Yes, the new pictures are an improvement; well done.

Coincidentally, I've recently discovered Tess is a wonderful tool for making symmetry examples, and wanted to make a new set of small images, based on "d" rather than "F". (I use "d" already on the Euclidean plane isometry page, for a few reasons.) The idea was to have a visible mnemonic; I cannot look at either Hermann-Mauguin or Conway notation and immediately picture the symmetry, and I'm sure many readers will have the same problem and thus appreciate a visual hint now and then. Especially, I thought it would be nice to accompany the classification table, so we can see all the patterns together.

An idea that just occurs to me is to trim my hints to the fundamental regions. That works nicely with Conway notation, keeps them small, and perhaps avoids duplication. I'm not sure the new pictures will work at the tiny size (inline?!) I'm contemplating. It might be nice to depict an orbifold instead of a fundamental region, but that's a graphics challenge. Meanwhile, using different colors for non-isomorphic features (centers, axes) may suffice.

One subtlety is that some of the groups allow axes of different lengths, and possibly at arbitrary angles; I'd like that depicted.

The examples themselves look clean to me, but the (old?) key looks like a crude scan, not SVG quality. Also, it doesn't quite visually match the images. It may be a temporary server problem, but p4g, p4m, and p6m are missing for me. KSmrq 11:50, 2005 July 25 (UTC)

The legend images are old, I think they have been removed some day, but for now, I think they are better than having just text describing the symbols.--Patrick 22:37, 25 July 2005 (UTC)

Re missing images: I fixed that, it was due to the use of group names with an extra m at the end.--Patrick 22:37, 25 July 2005 (UTC)

The files need to be uploaded to Commons with the correct name, and use of the alternate template removed. Otherwise consistency of the page becomes hard to maintain. I found this a problem when I tried to give all the images a descriptive tag, as web standards require. For the heck of it, I used the Conway notation, and found some minor issues: As I understand it, x should always come last, and should be lower case. KSmrq 06:10, 2005 July 26 (UTC)

I believe I read my notation with the additional m in some book as well, but I would not bet on this. I uploaded the files as SVG and used your naming scheme for this. So once the images can be migrated to use SVG, names would match notation. But before this can happen, MediaWiki bugs #5109 and #5110 need to be fixed. Otherwise, the proportions of the images are incorrect and the lines of glide reflection not dashed. I also fixed an error in the cm image pointed out by Patrick. If things turn out well, I might even get around to do SVG versions of the legend images. -- Martin von Gagern 01:40, 27 February 2006 (UTC)

[edit] Shapes and colours?

Someone added the following:

Sometimes two categorizations are meaningful, one based on shapes and one also on colors.

No further explanation is given. I don't understand what is meant. Please expand or I will remove it. Thanks Dmharvey Talk 11:19, 24 July 2005 (UTC)

E.g. a red p and a black q can together form a symmetric image, but only if we ignore color.--Patrick 23:42, 24 July 2005 (UTC)

Similarly, ignoring colors the image shown is p4, otherwise p2, I think.--Patrick 23:47, 24 July 2005 (UTC)

[edit] Orientation of diagram vs. that of computer-generated images

It is a little confusing that the orientation of the diagram is sometimes different from that of the computer-generated image.--Patrick 10:49, 25 July 2005 (UTC)

The computer generated images are pretty much the only things left from the previous incarnation of this page. I don't mind if they disappear, especially if they get replaced by something matching the diagrams. Dmharvey Talk 18:11, 25 July 2005 (UTC)

[edit] Fidgeting

When I filled in the section on orbifold notation, I used the spelling "center". The rest of the article uses "centre", so I changed to that everywhere. Also, the word "rotocenter" was used in several places, and I changed that to "rotation centre", for both clarity and consistency. In the classification table, I changed my original "rotocenter" to "rot. centre"; it's not quite so pretty, but may be easier to understand — and it still fits.

The constant use of quotation marks around group names like "p3m1" seemed distracting in an article that uses them so often, so I switched to either italics or boldface, depending on context.

I promise, to atone for this silly fidgeting I will fill in another section stub. :-) KSmrq 11:37, 2005 July 26 (UTC)

[edit] Enumeration proof stub filled

Done! I'm not thrilled with my effort, but I sketched the pretty orbifold approach to enumerating the groups. My hunch is that a really satisfying, accessible, and complete proof through any route would be too long. Instead, I have inserted a provocative handwave. Is it ideal? Doubtful. Is it better than nothing? I hope so.

One benefit is that anyone can enumerate the groups, and can do a sanity check on orbifold notation. For example, is 2*2 a wallpaper group? No, the sum is 1/2+1+1/4, which is too small. Is 444 a wallpaper group? No, the sum is 3/4+3/4+3/4, which is too large. I know of no easy sanity check on crystallographic notation.

However, this topological approach offers no geometric insight. Most of the pleasure of wallpaper groups is in the geometry, and we often use them as a stepping stone to the full 3D crystallographic space groups. Sadly, I haven't found a way to condense a geometric approach to any acceptable length.

Caveat: I have not myself fully absorbed the orbifold ideas, thus I may not present them as they deserve.

But I do like filling the stub. :-) KSmrq 10:22, 2005 July 27 (UTC)

[edit] Crystallographic notation

I would greatly appreciate another set of eyes on the description I've written of crystallographic (Hermann-Mauguin) notation. Have I told any lies? Is there a better way to explain it? What pictures, if any, should be added?

Anyway, the section is finally filled. (That's two in two days!) Enjoy. KSmrq 20:32, 2005 July 27 (UTC)

It says "Here are all the names" and then 11 groups are listed.--Patrick 00:01, 29 July 2005 (UTC)

Thanks. I had copied the table from another site, not realizing it was incomplete. (Tess uses these pairs.) I have already fixed that problem, but it raised a question perhaps you understand. The group p2 is listed as short for p211, but p1, p3, p4, and p6 are different. Why is p6 not short for p611; or should it be? KSmrq 11:39, 2005 July 29 (UTC)

Thanks. I don't know.--Patrick 19:54, 29 July 2005 (UTC)

I see you found something to say about the distinction between p and c cells. Sadly, after reading it several times in context I still found it more confusing than helpful. The phrase you replaced does trouble me as well, being little more than a handwave. Honestly, we may never be able to say enough in a single short sentence. For now, your new material is commented out.

Ideally, any explanation should make sense in 3D as well, since this is crystallographic notation. (In 3D we have more cell types and the idea of closest-packing structures.) Part of the problem is that no rhombus is visible in the diagram(s) for cmm. Another is that the notation discussion considerably precedes the diagrams. Actually [sic], I have little confidence I could look at a pattern or its distilled cell and be able to apply the sentence, which bodes ill for our readers.

I vaguely remember that years back I sorted out something plausible and coherent for 3D, but I've not taken the time to try to resurrect it. One way to look at it is to imagine taking that rhombus as a p cell, then trying to notate the other symmetries; the whole 2-axis notation system falls apart. The central idea is not that the primitive cell has equal sides (a rhombus), but that the symmetries don't align with the cell sides. We'd have a problem empirically confirming a rhombus (from measurement error in lengths and angles); less so the symmetries. KSmrq 13:39, 2005 August 2 (UTC)

I think the figures of cm and cmm should show a rhombus, as I indicated on commons:Image talk:Wallpaper group diagram cmm.png; that is the smallest possible cell that is repeated by translation, while, as usual, having the translation vectors as its sides.--Patrick 22:33, 2 August 2005 (UTC)

The cm figure has a rhombus that is visible, though subtle. The cmm figure has no visible rhombus. On close inspection I can imagine that one is indicated by a darker shade of gray interspersed with the dots of the glide axes and hidden behind the cell boundaries, but that's rather close to a "polar bear in a snowstorm".

I've introduced an extended description of the cell distinction, split out as a separate paragraph. KSmrq 04:16, 2005 August 3 (UTC)

I don't know what you are trying to say: either that the groups have little to do with a rhombus, or that you agree that the figures should (more clearly) show a rhombus.--Patrick 07:14, 3 August 2005 (UTC)

Ah. First, the captions for cm and cmm refer to a rhombus, which is essentially invisible for cmm. If any text, whether caption or otherwise, is to refer to the rhombus, it should be visible. I believe a diamond shape will work for cmm as it did for cm, which might suffice.

The caption provisionally compensates for what is missing in the diagram. I have clarified it further. Surely you don't mean that important info missing from an image should also be missing from the text?--Patrick 20:32, 3 August 2005 (UTC)

Yes, the new caption helps. And, yes, I do mean the text should not refer to an invisible feature. If you feel that the information is important, and are not swayed by the reservations I expressed, then I expect you will want to amend the image and then augment the text. Remember, the previous attempt left me scratching my head in puzzlement, and I'm someone helping edit the article, not someone who knows nothing about wallpaper groups trying to read it. Are you incredulous, or confused, or something else with regard to my comments? Because I feel like I'm just repeating myself, which probably isn't that helpful. KSmrq 23:07, 2005 August 3 (UTC)

Second, the discussion of notation no longer refers to a rhombus, for reasons I gave above. If the cmm figure is modified, we then have the option to discuss that aspect of c cells. I'm all for visual references, so long as the language is clear and helpful. Unfortunately, the clause you introduced seemed (to me) to be confusing for discrimination and silent for explanation. For all I know, mine may seem the same for you! KSmrq 17:58, 2005 August 3 (UTC)

Another thing: "translation vector" seems clearer and more common than "translation axis" ("axis" is used for reflection and for 3D rotation, and for a line in the center). Also, there are two, but without explanation one is referred to as the "main". That seems odd.--Patrick 06:13, 4 August 2005 (UTC)

Thanks; good catch. Crystallography has conventions for choosing these axes, and choices for the cell origin as well. Axis, not vector, is the correct term in this context, so I suppose more explanation is required. It's a little frustrating, because we encounter the notation section at a point in the article where we have not yet (if ever) introduced the background needed for a proper discussion.

Anyway, briefly, we typically use the minimal translation vectors of a primitive cell as oblique axes, giving a non-orthonormal basis in which every lattice point has integer coordinates. In crystallography, positions of atoms making up a crystal are given with respect to this coordinate system. Here is an example of a calcite description:

  Calcite
  Graf D L
  American Mineralogist 46 (1961) 1283-1316
  Crystallographic tables for the rhombohedral carbonates
  
  4.9900 4.9900 17.0615 90 90 120 R-3c
  
  atom     x y   z
  Ca       0 0   0
  C        0 0 .25
  O    .2578 0 .25

The first three numbers give the lengths of the axes; the next three, the angles; then comes the space group, which in this case is based on a rhombohedral cell. (Of course, the rhombohedral symmetry partially constrains the axis lengths and completely constrains the angles.) The calcium atom is taken as the cell origin, the carbon is at a special fraction (1/4) of the z axis, and the oxygen is at a less special position with respect to the symmetry. Thus the action of the symmetry group yields the chemical formula CaCO₃, with more copies of the oxygen atom.

Now the challenge is, how to say just enough to explain wallpaper group notation without dragging in all of crystallography! KSmrq 17:38, 2005 August 4 (UTC)

I have reverted your (Patrick's) edit back to my original clause, "they permit the same method of symmetry description in the other cases", to convey the correct meaning. Your edit did not make it clearer, it changed it to something completely different. My meaning, apparently misunderstood, was that by using c we can say cm and mean that the mirror is perpendicular to the first cell axis, just as pm does, and similarly for cmm. Would you prefer that I say, "in the remaining two cases"? Or perhaps you can suggest a better wording, now that you know (I hope!) what I'm trying to say. KSmrq 07:00, 2005 August 6 (UTC)

Hmm; Conway notation is looking more and more appealing! I have attempted to clarify centred cells again, along with the axes. Still left wanting an explanation is "primary" axis. Sigh. KSmrq 08:36, 2005 August 6 (UTC)

Patrick, the description of primary axis choice is a definite improvement over nothing (which is what I had said); thanks. A problem or two remains. Your phrasing is

"if there is a mirror perpendicular to a translation axis we choose that axis as the main one"

This gives us no guidance for the groups pg (p1g1) and pgg (p2gg); and for p4g (p4gm) and p31m, it's confusing. I think the underlying cause of these difficulties is not our clumsy explanations, but the way the notation really works, which is a bit backwards. We know the groups and their symmetries, and use a notation that distinguishes the groups and that depends on a proper choice of axes. The axes in each case are chosen to make the notation work. Still, we're making progress. KSmrq 19:47, 2005 August 6 (UTC)

All this discussion really points to the inadequacy of the crystallographic notation. There is nothing canonical about it. Given a new group, you would not be able to guess accurately the name used for the last 100 years. The orbifold notation really does uniquely define a symmetry, in a way that can never be mistaken and can be calculated; it generalizes to all two-dimensional symmetries, of the plane, sphere, and hyperbolic plane, and the frieze groups. Jan 25 2006.

[edit] Problem with italics in headers

I noticed that when italics are used in a section header, after section editing one ends up at the beginning of the page instead of at the section. I find this very inconvenient. Therefore I suggest that we use normal text in section headers.--Patrick 21:56, 28 July 2005 (UTC)

Fascinating; that is an inconvenient bug. I'll follow your suggestion and change all the section headers to normal, as the context there sets off the group names anyway. (If Dmharvey doesn't object.) I'd still like to see consistent bold group names in the text. Do you disagree with that convention or just don't want to be bothered with doing it? KSmrq 12:01, 2005 July 29 (UTC)

Thanks for changing the headers. I do not find bolding very important because fortunately the codes are not normal words, except cm, but even that is not used once in the meaning of centimetre in the article. However, for uniformity I'll try to conform.--Patrick 19:50, 29 July 2005 (UTC)

[edit] Splitting the page

Hello KSmrq and Patrick, you've both been doing a lot of great stuff to this page. I wish I had time to edit now but I don't. I have been keeping an eye on progress though.

My two cents is: The article is now way too long.

My proposal to remedy this: Split into two articles. The main Wallpaper group article covering all the theory. A separate Wallpaper group (picture gallery) or Wallpaper group (examples) or something similar, with most of the example images. The main article will have, for each group, the cell diagram, perhaps one pretty example image (perhaps more if some aspect of the theory is best explained by example), and a link to the "examples page" saying "See more examples of this group...".

This will become even more imperative when I add more of the photos that are sitting on my hard drive crying out for inclusion....

Does anyone think this is a good idea?

Dmharvey Talk 13:13, 29 July 2005 (UTC)

Hah! We don't need you to tell us it's long; every time we edit we now get a warning. :-(

The bad news is, the warning is about long text, nevermind the download time for the pictures. Two images per group — one showing the cell structure, and one as an example — sounds fine, splitting the other images to a separate gallery page for each group. I'll give some thought to what might be done about the text. It could use a little reorganizing anyway. KSmrq 21:09, 2005 July 29 (UTC)

If we have a page for each group we can also have all text about that group there, partly copied, partly moved from the main page.--Patrick 21:28, 29 July 2005 (UTC)

The main page still needs a concise, coherent survey of all the groups. Visiting separate pages for image galleries is one thing; doing so when trying to compare groups or understand all 17 as a whole is not so appealing. Of course, the current long scroll makes that awkward as well, which is one reason I've been trying to design small images (maybe like Kali's icons, maybe not) and think about reorganization. Another thing I'd like is more group-theoretic discussion, such as the subgroup relations. KSmrq 22:24, 2005 July 29 (UTC)

[edit] Illustrations

While trying to sort out crystallographic axes, I noticed that the right-hand diagram for p3m1 should be rotated clockwise 30° so that the long diagonal becomes horizontal. The convention elsewhere is a vertical main axis, with perpendicular (horizontal) mirror. KSmrq 21:17, 2005 August 6 (UTC)

Apart from cm and cmm the convention seems to be that one of the translation vectors is horizontal.--Patrick 00:38, 7 August 2005 (UTC)

The left-hand illustration for p3m1 has a horizontal mirror line, and the discussion of axes says we should have a mirror perpendicular to the main axis. The right-hand illustration has neither a horizontal nor a vertical edge as mirror, though there is an internal vertical mirror. I can't swear to it, but I believe the vertical axis is a convention (outside Wikipedia). That's what Tess uses, though a sample of one is hardly conclusive evidence. Believe me, I understand tinkering with figures can be a pain; so I can sympathize if you don't want to change it. KSmrq 03:14, 2005 August 7 (UTC)

I think it is convenient to have the translation cell drawn the same for p3, p31m, and p3m1, as we have now. Anyway, the person to ask first would be the maker, Martin von Gagern.--Patrick 06:35, 7 August 2005 (UTC)

[edit] Article split up?

This article is getting a bit unwieldy!

I created a new article called List_of_Planar_Symmetry_Groups just to show the 17 groups themselves. (Quick&Dirty, but useful for reference!)

It probably wouldn't be a bad idea to move ALL the example images into 17 separate articles, one for each group. I'd do it someday, but a bit overwhelmed at the moment!

I just added articles for the 11 regular/semiregular tiling and symmetry groups for each. Like: Triangular_tiling

ANOTHER nice SHORT article would be for "Spherical symmetry groups" - page symmetry group is equally overwhelming and not clear at all. I'll do this myself when I collect some pictures of the fundamental domains....

Tom Ruen 10:55, 9 October 2005 (UTC)

[edit] Roger Penrose tiles?

I didn't see any reference to Roger Penrose tiling in this article. Penrose put forth many mathematical examples of filling a 2 dimentional plane in a non-repeating (aperiodic) way. I think at least a mention of his contemporary work would be appropriate here as this was the mathematical evelution (AFAIK). Jeff Carr 10:41, 22 January 2006 (UTC)

We don't cover nonperiodic tiling under wallpaper groups because wallpaper groups only describe periodic tilings. An article on tilings or tesselations could raise the issue, as could an article on projections of higher-dimensional periodicity. In fact, my recollection is that such mention is made. --KSmrq^T 14:09, 22 January 2006 (UTC)

[edit] miscellaneous

This is a very nice article and I have been enjoying the cultural examples of wall paper groups. I do have a few comments:

It would be nice to more systematically interleave the orbifold and international crystallographic notations. Also, the diagrams of the "cell structures" of the various groups seem a little misleading: (a) there is usually no canonical cell structure, but you are typically showing generators for the group, so that's no big deal, but (b) there is no difference between, say the symmetry denoted o (in the orbifold notation) and p1 (in the intl cr. notation) and it seems like they should have the same diagram. Same point for the other 16.

The thread above on the crystallographic notation prompts me to write this screed:

this discussion really points to the inadequacy of the crystallographic notation. There is nothing canonical about it. Given a new group, you would not be able to guess accurately the name used for the last 100 years. The orbifold notation really does uniquely define a symmetry, in a way that can never be mistaken and can be calculated; it generalizes to all two-dimensional symmetries, of the plane, sphere, and hyperbolic plane, and the frieze groups.

Those participating in this discussion may be interested to know that Conway will be publishing a beautiful book on the subject early in 2007 (I am a co-author). Also, Conway has a nice way to understand all the 3D space groups, which will be discussed really for the first time in that book. Jan 25 2006.

Dear madam/sir, I'm glad you like the page. Most of the "cultural examples" you speak of are my fault. (Well I didn't draw them obviously, but I collected and organised them). If you have any ideas for the page, you are of course quite welcome to make changes yourself! That's what the "edit" button is for! Don't worry if you get stuck with the wiki syntax, there are plenty of people hanging around here who will help you. You might consider creating an account, since it improves your anonymity slightly (i.e. we can't guess geographical location via your IP address :-)) And of course I'm looking forward to that book. Dmharvey 02:14, 26 January 2006 (UTC)

[edit] Formal definition

This is a beautiful article. But the formal definition doesn't make sense to me:

"A wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane which contains two linearly independent translations."

Everything is explained, except for the "type of". What does it mean for two groups of isometries of the plane to have the same type? Does it mean they are abstractly isomorphic as groups? Or is there something more subtle going on (like that they become the same group of isometries after rotation and rescaling)?

It would be nice if an expert could fix this up. Thanks. 24.82.85.97 19:43, 28 January 2006 (UTC)

Dear anon, that's an excellent point, there's something missing there. Here is my recollection of the situation (it's been a little while, and I don't have references handy). Your definition of "isomorphic" is the correct one, except that besides just rotations and rescalings, you need to include the whole group say H that is generated by and the translations of . So, two wallpaper groups G₁ and G₂ are considered isomorphic if there exists some such that G₁ = hG₂h ^{− 1}, or alternatively, they are the same "up to scaling and arbitrary (invertible) linear transformations of the plane". The reason you need to include arbitrary linear transformations is that otherwise there will be infinitely many different versions of say p1, all with slightly different angles between the two independent translations. For similar reasons your definition needs to include the translations in H. Now, the amazing thing, is that after giving this correct "natural" definition, it turns out that two groups are isomorphic in the above sense if and only if they are isomorphic as abstract groups! Obviously you need to do a little bit of work to prove this. I believe there is a discussion of how this all works in the Grunbaum reference, but I'm not 100% sure. Dmharvey 20:13, 28 January 2006 (UTC)

Some of these things are also in Space_group#Group_theory (2D case).--Patrick 00:48, 29 January 2006 (UTC)

I added them here.--Patrick 09:19, 29 January 2006 (UTC)

[edit] Images showing a rhombus

(see also the section #Crystallographic notation)

The images for cm and cmm are now rhombuses. With regard to the orientation of the images: it is consistent with the other images in having one translation vector horizontal. However, I wonder if it is not easier to have the diagonals horizontal and vertical (e.g., all example images have such an orientation).--Patrick 14:41, 27 February 2006 (UTC)

[edit] A note on p1

User:KSmrq corrected a note I added on p1; thanks for that. Here's KSmrq's text: The two translations (cell sides) can each have different lengths, and can form any angle. Now, could one add something like the following: Thus, if the contents of the unit cell is asymmetric, it is irrelevant whether the shape of the unit cell is symmetrical. And why would I want this? Well, if this mathematical topic is applied to actual wallpapers (you known, things made of paper, glued on the reverse and all that), I find it a bit odd, and hence noteworthy, that

1. wallpapers with square unit cells, and
2. wallpapers with unit cells that are parallellograms with no symmetry at all

are grouped together.--Niels Ø 10:12, 10 April 2006 (UTC)

None of the other groups goes into such detail. The questions you raise are not special to p1. Please read the extensive discussion that precedes the consideration of individual groups.

However, it would be appropriate to note with each group to what extent its translation vectors are constrained. Specifically:

• p1 has independent lengths, any angle
• p2 has independent lengths, any angle
• pm has independent lengths, fixed angle
• pg has independent lengths, fixed angle
• pmm has independent lengths, fixed angle
• pgm has independent lengths, fixed angle
• pgg has independent lengths, fixed angle
• cm, cmm and all other groups have equal lengths, fixed angle

  for the 3-fold and 6-fold groups, the fixed angle is 120°, else 90°

Different groups can have the same cell shape. For example, p1 and p2 both can be any parallelogram, cmm and p4g are both squares, and p31m and p6 are both 120° rhombuses. This is one reason why the "Guide to recognising wallpaper groups" has no mention of cell shape. The symmetries of the pattern are what's important. --KSmrq^T 14:00, 10 April 2006 (UTC)