Talk:Wallpaper group

3167:. As the article says, "The group p1 contains only translations; there are no rotations, reflections, or glide reflections." If you choose some feature of the wallpaper in the picture, say one of the largest flowers, and look for a rotated version of it (the same flower a different way up), you don't find one; so there are no rotations. And if you look for a mirror-image of it, you also don't find one; so there are no reflections (either straightforward or glide). p1 is the only wallpaper group with no rotations and no reflections. 3557: 91: 81: 53: 2202:

article. Removing a level of structure around the discussion of notation is also a step in the wrong direction. Then we come to the wording. I have high standards of writing, and most Knowledge contributions fall short, including some parts of this article. However, we need to respect mathematics as well as rhetoric, and I was uncomfortable with what I saw happening. Lacking the time and lacking the inclination to nitpick every sentence change, I simply reverted all.

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little disk around it. It should be clear that all that's possible is to rotate the point, or reflect it, or have a whole bunch of reflections making a kaleidoscope there. For the rest of it, remember that we're recording the topology of the orbifold surface, and surfaces can be sorted out pretty straightforwardly by topology, into their constituent parts: cross-caps (x), handles (o) and boundaries (*). That's precisely what the notation is recording.

3490:

vertical stripes horizontally by one stripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns that repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored.

3681: 320: 312: 3673: 3136: 869: 2688:(a) If we stick with just rotations: these have cost 1/2, 2/3, 3/4, 4/5, 5/6, 6/7 etc. (i) Can't have 6/7 or higher: Clearly can't have two such, since at least 12/7 and smallest cost available is 1/2. Can't have one such either, since remaining cost is greater than 1 but no more than 8/7; 1/2+1/2 is too small and 1/2+2/3 is too big. Similarly can't have 4/5. So we're left with 1/2, 2/3, 3/4 and 5/6. 1387:, 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. 22: 157: 67: 1229:

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.

1677:

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.

2279:

for the number, as I think many readers will, and I tried to condense the essence of the simplest argument I could find. I'm especially surprised that you propose merging this with a discussion of notation, as both the topics and audiences are distinct. If we can make things clearer for lay readers, that would be great.

1495:, 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. 3489:

A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (shifted) some finite distance and appear unchanged. Think of shifting a set of

3479:

Examples A and B have the same wallpaper group; it is called p4m in the IUC notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4g or 4*2 . The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of

2523:

Everything remaining must either start with a "*", or have just digits and no "*". Let's assume it starts "*6...". Then there can't be another "6" because that only leaves 1/6, which is less than 1/4 and nothing can be added to reach 2. Similarly there can't be a "4". But there can be a "3", in which

1712:

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

1671:

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

1658:

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

1395:

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

1320:

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

1292:

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

828:

I agree that the article is too long, and some of the images are poor. This is most true of the "computer-generated" images, which are sometimes hard to make out - I feel they should all be removed. Some of the other images are in my view too small-scale, with too many repeats - these could easily be

418:

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

347:

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

3702:

However, I note that each pair of red and blue tiles can be considered a single tile, in which case there is no rotation or reflection at all, which would make this group p1 (o) using an oblique cell structure. In fact, that is how I created the image, first by creating a pair or red and blue tiles,

2201:

I appreciate that you would like to improve the article; thanks for that. Unfortunately, you made a number of edits recently with mixed results. Use of a gallery in the intro was a good idea; I have retained that. Removing images is not such a good idea; the assortment is one of the strengths of the

1420:

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

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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

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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.

982:

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

3500:

If we shift example B one unit to the right, so that each square covers the square that was originally adjacent to it, then the resulting pattern is exactly the same as the pattern we started with. This type of symmetry is called a translation. Examples A and C are similar, except that the smallest

3143:

I must say I found this article deeply confusing and unhelpful. That may be because I am interested in wallpaper and not in mathmatics. None of the huge number of illustrations are much use to describe the actual issues in large pattern wallpaper design at all. What group is the picture illustrated

3097:

The "F" shape on a tile is there to destroy any internal symmetry of that tile, and to show how the symmetries of the group relate that tile to the other tiles. So the orientation of the first "F" in a diagram is arbitrary, what matters is how it relates to the orientation of the other "F"s. I have

2849:

OK, but let's not assert that the only mathematical content here involves the abstract groups. The subject of wallpaper groups is a mathematical subject, namely: realizing abstract groups as certain groups of motions in the plane. The images are essential to understanding this subject, and it would

2557:

That exhausts the groups with a "*" and all the rest have only digits. If there is a "6", there can't be another "6" because that only leaves 1/3, which is too big for any feature as long as there's no "*" in sight. Similarly there can't be a "4". However, there can be a "6" and a "3" in which case

1676:

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

1639:

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.

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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

640:

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

2278:

When we do, I would love to hear your views on the notation subsections, and on the "why 17" explanation. Those are appealing for me to discuss because it's mostly my writing. "Obfuscated" surprises me; my sources were scarce and rather more difficult. I had always wanted an accessible explanation

2253:

On the subject of the structure of the notation, I strongly feel for technical reasons that the discussion is incorrectly structured and overstructured. My thinking on this point is not fully fleshed out, because I did not yet revise the obfuscated section on "why there are 17 wallpaper groups".

1286:

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

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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

769:

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

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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

520:

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

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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 Knowledge syntax than I would in XHTML/CSS2. Admittedly,

3512:

However, example C is different. It only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a diagonal line, we do not get the same pattern back; what we do get is the original pattern shifted across by a certain distance. This is part of the reason

2864:

I agree with Ishboyfay. It would be very good to add the isomorphic groups, but please do not hide the figures. The real world connection for this topic is important and having several examples is very helpful for visualizing the transformations. Remember too that your audience includes more than

2234:

I understand the general point that it's nice to have a lot of illustrations, moreover that this article in particular merits a catalogue of examples. In my opinion, the article goes overboard. You would surely agree that there is such a thing as too many examples; the question then is where to

2019:

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

1666:

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

1513:

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

1011:

Something that isn't clear to me is why are 2,3,4,6,*,x,o the only "features" allowed? I can buy that once you restrict yourself to these, there are only 17 combinations that add up to 2 with that formula... but I'm left scratching my head as to whether we've missed possible wallpapers because of

599:

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.

3752:

Convention would be, for such a diagram, to consider it p1 - but it is one of the things often left unsaid when discussing symmetry groups that makes the subject confusing (for me, at least - and I may have gotten it wrong!) Another confusing thig is that it is p1 whether the unit cell is square

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Sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry. In black and white there are also 17 wallpaper groups; e.g., a colored tiling is equivalent with one in black and white with the colors coded

2681:

1) Why are the only features o,x,*, numbers. (Note, don't rule out 5, etc, a priori. This comes for free a little later) There are two kinds of feature, those that fix a point and those that don't. If a point is fixed, we can understand the feature that fixes it by looking at what happens to a

1302:

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

3112:

I think it should be better explained. I have a maths background (but not in group theory) and it has taken me awhile to understand the diagrams. They also say "on the right" for the diagrams when I think they mean "on the left." Overall the intro to the group diagrams is confusing and not well

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which lists all the possible (non-hyperbolic) 2 dimensional orbifolds. You can enumerate all the possible orbifolds, and calculate their Euler characteristics, most will turn out to be hyperbolic, some are spherical and some, the one we are interested in, are parabolic. For example those with a

1228:

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

713:

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

1698:

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

3081:

Can someone please give an explanation of the "F" shape in the tiles. It is used in a lot of illustrations, it tantalizingly seems to contain some information, but it is never explained anywhere what that information is. It is especially frustrating because it isn't the sort of thing one can

2239:

Then we come to the wording. I have high standards of writing, and most Knowledge contributions fall short, including some parts of this article. However, we need to respect mathematics as well as rhetoric, and I was uncomfortable with what I saw happening. Lacking the time and lacking the

1464:

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

2811:

I think this mathematical information would be more useful and relevant than the multiple images to illustrate every wallpaper group. Ok, the images are very pretty, and it would be nice to keep them; maybe they could live inside "hide" boxes, so that readers who are more interested in the

1943:

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

955:

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

623:

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

2187:

This is a nice article, but in my view it belabors its subject. I think that it would teach people more if we simplified the wording, and removed remarks which are redundant or nearly trivial. I spent some time this morning simplifying the article, but I think that more could be done.

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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

654:

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.

397:

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

3504:

If we turn example B clockwise by 90°, around the centre of one of the squares, again we obtain exactly the same pattern. This is called a rotation. Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for

2473:

Or there could be an "x" and multiple features that sum to 1. We can't have both "x" and "6" because that's already 1 + 5/6, and there's no feature whose value is as small as 1/6. Similarly, we can't have both "x" and "4" or "x" and "3", so the only group we can get this way is

475:

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.

3899:

For any given wallpaper image, you can select an infinite number of differently-shaped "unit cells", with a various amounts of symmetry in their shape, and all of them will share the property of being able to re-build the original pattern by repeating the cell. Image 2 in the

793:

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. --

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in that subsection! In a general encyclopedia, you really do need to show a little more working and not assume quite so much mathematical knowledge. Somewhere there needs to be something that shows explicitly where the 17 comes from, not leaving it to readers to infer.

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Here, on the other hand, I think that you are being unfair. I am a professional mathematician and I certainly do have complete respect for the mathematics as well as the rhetoric. I made changes that I thought improved the explanations simultaneously at the lay level

351:

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.)

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at the technical level. I don't think that it's reasonable for you to revert these changes just because you lack the time and inclination to nitpick, because that is exactly what I did find time for. To be fair, you may not have realized how I thought about these

3285:

The section on crystallographic notation describes how the letter c denotes a face-centred cell. However the cm cell picture shows only the primitive cell. Would it be useful to have a picture that showed clearly the face-centred cell from which the name is taken?

1275:

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

3699:, this tiling has 180° rotations in half the tiles, and no reflections, but it does have a glide, with each row of the same color displaced a little bit to the left of the one below it. Because it has no glide reflection, the table says it's in group p2 (2222). 3455:

so that the article in effect had a new long introduction followed by the old short introduction, including repeated material. I have reverted to the old short introduction, but it may be a good idea to integrate parts of the following text in a proper fashion.

3873:

section is supposed to help you with this assignment. This image either has no rotational symmetry (if you take colours into account) or has 180° rotational symmetry (if you would remove them), and that property alone determines whether it may belong to p1 or

1628:

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).

3838:

Nø's explanation basically said the same thing I did in my initial comment, that depending on how you look at it, the group is p1 or p2. My question remains unanswered. Suppose this image was to be included in the article. Where would it fit? In p1 or p2?

2655:

Based on that definition, a notation of "x*" is not legitimate, and "326" is essentially the same as "632". This seems to answer the question asked. I'm not an expert here, but if others agree with my definition, I think it should be added somewhere.

1252:

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!

401:

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

2574:

We could try putting in both a "4" and a "3" but that just doesn't work out (try it). Similarly we can't have a "4" and a "2" without there being a second "4". So the rest of the groups don't have "4"s. If there's a "3" then the only possibility is

843:

I went ahead and replaced the old "cell structure" images with the newer SVG versions. I also think you're right about the "computer-generated" images. They don't add much to the article beyond what the "cell structure" images already illustrate.

2605:

This still leaves a lot of questions for the layperson (such as myself). Why isn't, for example "x*" an 18th wallpaper group? (you only have "*x" listed above)? Also, this pretty much just transfer the question to, why are there only 7 features?

1428:

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

2716:

b) Now here's a great trick! Kaleidoscopic numbers, red in the book, those after *, cost half as much as rotation ones. Since * costs $ 1, and since half of $ 2 is $ 1, we can take any of the above symbols and get one with all numbers after *:

3532:

Thank you for preserving this alternate text. I've just rearranged and reworked the lede to clarify it for a lay audience, and tried to incorporate some of these themes. I referred to p1 in it, and think that adding the more familiar p1 image

628:. 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. 3400:, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal 2216:

and I were collaborating to design for the article. Unfortunately, real life intruded (he's a doctoral student in mathematics at Harvard) before we settled on a final version. In other words, I have a long-term involvement in improving the

1659:

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.

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uncomfortable that I (reluctantly) decided it was better to revert. I really would like to follow the one step back with two steps forward. In the real world, I've got some distractions just now, and in Knowledge I'm trying to finish up a

3904:

section gives an example of two different "unit cells" with an area "a" on a Pythagorean tiling, one with symmetry, the other without. The shape of a "unit cell" has absolutely no bearing on what group the particular pattern belongs to.

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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

3590:

In short, I propose to describe at the beginning of article what is a repetitive pattern for the article, and insert the present image. To have a minimal area, a parallelogram‑shaped pattern shall be contructed from two translations

3241:

My understanding is that a given wallpaper cannot be classified in two different ways. That this classification scheme creates a partition of all wallpapers, is this correct. If so, then there is an issue with the images on the page.

3026:. I'll copy/paste it here: °̊ (wiki seems to accept it...) Is this some sort of mathematical "double degree" notation that I don't recognize, or just an editing error, or is Firefox doing something strange with the generated HTML? - 1405:

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

3508:

We can also flip example B across a horizontal axis that runs across the middle of the image. This is called a reflection. Example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for

191:

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).

337:. 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. 1559:

is a convention (outside Knowledge). 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.

1128:. Another is that the notation discussion considerably precedes the diagrams. Actually , 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. 1303:

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 CaCO

1038:

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?

810:

It looks like this discussion ended somewhere in early 2006, but the article has been left in a state where every group has two different "cell diagram" images. Using one or the other would clean up and shorten the article.

3460:

Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same

2532:

It's impossible to have "*6..." and then no "3", because there's no way to make that add to 2, so let's now assume it starts "*4...". There is 5/8 left over, and the only way to make that is another "4" and a "2", yielding

427:

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.

3568:, two occurrences before its first images. And never we do know really which repetitive patterns they are. To begin it should be written that any repetitive pattern of the article will have a minimum area, and that 2924:

One of the examples needs to be fixed. Someone has added an "Errata". First of all, it should be "Erratum". Secondly, there shouldn't be any errata at all. If the example is misclassified, please move it to the correct

641:

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

359:

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.

3631:

to the same without the "Here". But obviously, a wallpaper is a finte roll of paper with a printed pattern to be glued to a wall. I don't find the "Here" a good solution, but I don't think we can simply do without.

2490:

Now let's do the ones that have one "*". "6" can't be to the left of "*" because that's already 1 + 5/6, but "4" can be to the left of "*". In that case the remaining feature must be a "2" on the right, so we have

2254:

That section, if revised, could be merged with the section on orbifold notation. But it will be difficult for me to edit this article if people undo changes just because they don't have time to think about them.

936:

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.

3753:(i.e., highly symmetric in shape) or a parallellogram (minimal symmetry) - unless the content of the unit cell is symmetrical, the symmetry of the cell's shape is considered irrelevant for this classification. 2053:

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

340:

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.

1219:

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?--

1968:

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.

2286:

and I were working on are long overdue, and this would be a stimulus for me to finish them off and put them in place. (The best ones do not appear on my talk page.) So, meet you back here on February 25?

2904:

Yes I think your right. The key difference is that one has a three fold rotation without a reflection through the point. I've now delinked the images. Well done for spotting it, only 4 year in error! --

1131:

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

1059:

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

391:

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.

2747:

My favorite symmetry. Again reversible, so again, that's it. Since * and x are too expensive to have two of with numbers in the same symbol, and o can't be combined at all, that's it for numbers.

273:; 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. 3043:" °̊ " is two unicode characters: a degree symbol, and a "combining ring above" (\u030a). No idea how it got that way, but I think I just fixed it by replacing them with simple degree symbols. — 3318:

if I understand it correctly, the bold and blue (with links) symbols are the short and the bold and black symbols in brackets are the full/long symbols? If that is true, then the two entries

147: 2306:

The "Why there are exactly 17 groups" subsection is not great for the layperson, as nowhere in its explanation does it actually say "There are 17 because". Indeed, the number 17 isn't even

1173:

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".

3726:

As for including it as an example, I think a photo of the actual flagstone tiling would be more fittng for an encyclopedia. We already have two artificially-generated exampes under p1. --

2235:

draw the line. I recommend removing some of the uglier or more indistinct examples. But I concede room for reasonable disagreement on this point and I do not mind this reversion so much.

1344:

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.

3059: 1695:"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." 3536:

would be good, but I'm not quite sure how to do that. I think the lede could still use more work, e.g. to clarify what it means to be talking about groups here, not just symmetries.

2263:

Thanks for your thoughts. Perhaps we're not too far apart on what we're looking for. I did see some things I liked in your rewordings, so I don't mean to suggest it was all negative.

3185:, which a reader could use to identify the wallpaper group of any particular wallpaper. It would probably start with the question "does the wallpaper have any rotational symmetry?" 1772: 1566:

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.--

1937: 1801: 3786:

Texas tile is a better approximation of the state's shape than that Texas tile), but I have no idea of Reddit's licensing terms and it's unlikely we could use that photograph.

1446:

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.

1183:

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.--

2951:

seems to be a neologism coined by the article's creator and the resulting article is a content fork of this article, though written at a slightly more elementary level.--

1396:

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.

2966:

Some symmetry combinations imply translational symmetry in two dimensions, so there is some overlap. However, treatment of individual wallpaper groups is not repeated.--

2205:

Please don't take the reversion as a slight; under other circumstances I would try to work with you paragraph by paragraph. Maybe we can come back to this in the future.

1595:

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!

1537:

should be rotated clockwise 30° so that the long diagonal becomes horizontal. The convention elsewhere is a vertical main axis, with perpendicular (horizontal) mirror.

3949: 2507:

We can have a "2" to the left of the "*", and in this case there is more than one possibility. There is 1/4 left over, so there can be either another "2" to the left:

1881: 1855: 1828: 1351:

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

2640:

That isn't very helpful, actually. It doesn't specify what is allowed for orbifold notation. From what I can tell, a proper orbifold notation is made of 4 parts:

2191:

Likewise I think that four photographic examples is plenty for each of the symmetry groups. The page is too "tall", partly because it has so many illustrations.

1962: 1731: 3869:

This has nothing to do with conventions. The assignment to a wallpaper group is directly and only based on the image's rotational and mirror symmetries, and the

2591:

This exhausts the possible Conway orbifolds of genus 2 and therefore the planar symmetry groups. These seventeen are all there are and all there shall ever be. —

2771:

Something I would like to see here is a listing of what groups these actually are, abstractly. What groups they are isomorphic to, if you like. For example, I

2341:

Um, I can show where the number 17 comes from, but I don't think it will help much, because that's not the difficult part of the derivation. Anyway, here goes:

1337:. 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. 1124:

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

1667:

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.

2728:

c) An even cooler trick: Can swap any pair of identical numbers after the * for a single one in front of the star, costing twice as much. Thus *3 33 yields

2482:

That's all the groups that contain either "o" or "x", so all the others must have only digits and "*". If there are two "*"s, then that's it for the group:

785:

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

714:

instead of a fundamental region, but that's a graphics challenge. Meanwhile, using different colors for non-isomorphic features (centers, axes) may suffice.

3639:

saying that a wallpaper group is the symmetry group of a wallpaper, but rather that it is the symmetry group af a "repetitive plane pattern", or the like.

2808:, etc. (Don't trust me, I may have got this wrong; but these are examples of the kind of information I would expect to see, for all 17 wallpaper groups.) 165: 2323:

I agree. It's been two years. Can somebody editing this page please put it in plain English, or if it just isn't that easy, change the section heading?

956:

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.

3944: 137: 747:

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.--

1145: 419:

disagree with the wisdom of this, I have left the previous list version in the source, commented out. Admittedly, just shuffling the deck chairs. :-)

959:

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

3414:(Tree? Where from? Three type of Triangles? As I read above, we talk about only two types of triangles... Groups? Why three, not four, five, six?) 667:

I think it is sometimes helpful to have both versions, showing sometimes a different orientation, a different choice of fundamental domain, etc.--

3743:

the colours, focusing on the outline of the Texases only, the figure has an 180° symmetry that takes a red Texas into a blue Texas, and it is p2.

1964:. Now, the amazing thing, is that after giving this correct "natural" definition, it turns out that two groups are isomorphic in the above sense 3779: 330:

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

113: 3939: 2607: 1518:

icons, maybe not) and think about reorganization. Another thing I'd like is more group-theoretic discussion, such as the subgroup relations.

1019: 256: 3855:

Clearly, as conventions are, it would be p1 only. It is, I believe, not a figure that would add anything valuable to the article, though.

363:

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.

248:

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.

3575:

On the present image, various patterns are parallelograms constructed each from two translations under which the wallpaper is invariant.

3083: 3412:

are equal we have p6, if they are each other's mirror image we have p31m, if they are both symmetric we have p3m1; if two of the three

1148:; that is the smallest possible cell that is repeated by translation, while, as usual, having the translation vectors as its sides.-- 344:

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

3692:, I encountered a small plaza paved with Texas-shaped flagstones. I just spent some time figuring this out, and created this image. 2889:

The "computer generated" examples for p31m and p3m1 seem to be switched, according to the definitions. Can someone verify this? --

1504:

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.--

104: 58: 1468:

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

719:

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

3452: 348:

office ceiling"). We might as well keep reasonably high-res versions available (the examples I have given are approx 1MB jpegs).

994:

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.

926:

It is a little confusing that the orientation of the diagram is sometimes different from that of the computer-generated image.--

3778:

the same picture. Especially if the colors are removed. I did this from memory, it was many years ago. Looking around, I found

2082:

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

1608:

is equally overwhelming and not clear at all. I'll do this myself when I collect some pictures of the fundamental domains....

1208:

text, whether caption or otherwise, is to refer to the rhombus, it should be visible. I believe a diamond shape will work for

781:

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

1589: 1312:

Now the challenge is, how to say just enough to explain wallpaper group notation without dragging in all of crystallography!

3824:

has basically said the same thing, and I guess his explanation seems sufficient. Is there anything left to explain here? --

1655:

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

287:

relationship between informal and formal approaches. I've noticed the text is not very explicit on that point as it stands.

2324: 571: 33: 2678:

Having had the pleasure of being a coauthor of the "Symmetries of Things" can address a couple of issues raised here.

1457: 3648:

Thus, in this context, a wallpaper is a pattern that covers a whole Euclidean plane by repeating a motif indefinitely.

3265:

Look at the closeup images - the Egyptian mats differ in details, that's why each one belongs to a different group.--

986:

One benefit is that anyone can enumerate the groups, and can do a sanity check on orbifold notation. For example, is

497:

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

486:

Also could you please fix up the text to clarify exactly how the terms dihedral and cyclic are used in this context.

327:

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

612: 3749:

the colours, it only has a parallellogram (oblique) unit cell (possibly square), and no further symmetry; thus p1.

3635:

Perhaps we should rewrite? Maybe we should start by saying what a "Wallpaper group" is, not what a wallpaper is -

394:

would be nice eventually to have a (possibly informal) discussion on how to prove that there are exactly 17 groups

3592: 458:

Grid lines show up for me. Could be your browser or the new WikiMedia software. The lines do assist readability.

2703:

If we now restrict ourselves to 1/3 and 1/2, clearly can't mix them, and we get 1/2+1/2+1/2+1/2 and 1/3+1/3+1/3

1461: 625: 521:

the "proof of 17" stub with this idea, but am a little worried about the background required. The rule is that "

2611: 1023: 704: 252: 233: 229: 3172:"Wallpaper group" is what mathematicians call these things. And non-mathematicians don't normally talk about " 3000:

I've now proposed it for deletion. Not sure that there is anything in it which is not already covered here. --

3087: 2499:

We can have a "3" to the left of the "*", in which case the remaining feature must be a "3" on the right, so

885:

No further explanation is given. I don't understand what is meant. Please expand or I will remove it. Thanks

3603: 3581: 2626:

feature 5,7,8,... or *5,*7,*8,... can't be parabolic as a triangle with these angles can't tile the plane.--

2948: 2941: 3556: 1001:

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

795: 691:

is a wonderful tool for making symmetry examples, and wanted to make a new set of small images, based on "

646: 448:

Hey what happened to the table? It used to have grid lines, where did they go? Looked much better before.

3379: 1736: 642: 578:, you could include them here as well. The diagrams are somewhat different than the ones currently used. 3845: 3802: 3712: 2840: 2312: 2275:

rewrite of one particular article. My head is full. Could we possibly come back to this in, say, a week?

1940: 39: 3626:

Here a wallpaper is a drawing that covers a whole Euclidean plane by repeating a motif indefinitely ...

3497:

The types of transformations that are relevant here are called Euclidean plane isometries. For example:

3355:): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45° and 3330:): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45° and 1886: 459: 90: 2661: 2566:

It's impossible to have a "6" and no "3". If there is a "4", then there can be another "4" and a "2":

2344:

Let's start from the features with the highest value and work our way down. The possible features are:

1703: 3145: 3005: 2909: 2631: 1777: 1545:

Apart from cm and cmm the convention seems to be that one of the translation vectors is horizontal.--

1287:

the article where we have not yet (if ever) introduced the background needed for a proper discussion.

1015: 728:

quality. Also, it doesn't quite visually match the images. It may be a temporary server problem, but

3723:

This tiling does not have 180° symmetry. If you rotate it by 180°, it is no longer the same picture.

2449:

If the "no symmetry" feature "o" is there, that's 2 by itself, so no other features can be present.

1176:

I've introduced an extended description of the cell distinction, split out as a separate paragraph.

366:

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

21: 3910: 3829: 3731: 3270: 3173: 3032: 2894: 2855: 2725:

Since the trick is reversible, these are the only symbols with a single * followed by all numbers.

2255: 2192: 1240:

Second, the discussion of notation no longer refers to a rhombus, for reasons I gave above. If the

850: 817: 756:

Re missing images: I fixed that, it was due to the use of group names with an extra m at the end.--

2541:

If it starts "*3...", then there is 2/3 left over and the only way to make that is two more "3"s:

112:

on Knowledge. If you would like to participate, please visit the project page, where you can join

3599: 3577: 3287: 3182: 3114: 3044: 2991: 2592: 1477: 941: 889: 659: 632: 490: 480: 452: 432: 300: 169: 96: 1598:

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

80: 52: 2986:. Any verifiable content in the article should be moved here, and the rest should be deleted. 2643:

1. An unordered group of integers (corresponding to gyration points that do not lie on mirrors)

2230:

Removing images is not such a good idea; the assortment is one of the strengths of the article.

562: 3541: 3480:

the designs, whereas C has a different set of symmetries despite any superficial similarities.

3436: 3222: 3204: 3190: 3153: 3103: 2971: 2930: 2870: 2825: 2817: 1599: 1355:"if there is a mirror perpendicular to a translation axis we choose that axis as the main one" 1012:

left out a "feature", called 'f', for example, so that 2f2 could be another wallpaper group.

900:

E.g. a red p and a black q can together form a symmetric image, but only if we ignore color.--

834: 615:, so feel free to ask. But beware, It's a crude job in some places, so don't expect too much. 575: 2649:

3. Another unordered group of integers (corresponding to gyration points that lie on mirrors)

2220:

I've enjoyed some of my past joint efforts on this, and hope to have more as time permits. --

3840: 3797: 3707: 3421: 3291: 3118: 2956: 2836: 2079:. Please read the extensive discussion that precedes the consideration of individual groups. 2075:

None of the other groups goes into such detail. The questions you raise are not special to

1860: 204:

A simple, easily remembered, illustration of plane isometries requires no images, merely a

3901: 3870: 3796:

Yes, you basically stated the source of my confusion, and why I started this discussion. ~

3696: 3365:): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis 3340:): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis 3311:

should read as " four letters or digits; more usual is a shortened name like cmm or pg.",

3255: 3247: 3243: 3214: 3067: 3023: 3001: 2905: 2627: 2209: 2169: 2039:

The two translations (cell sides) can each have different lengths, and can form any angle.

1833: 1806: 1533:

While trying to sort out crystallographic axes, I noticed that the right-hand diagram for

1453: 724:

The examples themselves look clean to me, but the (old?) key looks like a crude scan, not

270: 193: 3860: 3758: 3658: 3522: 2743:

d) Finally, any * not followed by numbers can be converted to a x without trouble, hence

2740:. Since the trick is reversible, these are the only symbols of the form numbers*numbers. 1486:

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

880:

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

585:

Different equivalence classes of symmetry elements are colored (and rotated) differently

3906: 3825: 3769: 3727: 3680: 3375: 3266: 3027: 3022:

In several places, the ° symbol is displayed oddly in my browser (FF4). An example is

2890: 2851: 1630: 1625: 1605: 1515: 845: 812: 3060:

Knowledge:Templates_for_discussion/Log/2011_September_15#Template:Wallpaper_group_list

2457:

Next let's try "x". That's 1, so there's 1 left over. If there's another "x", we have

2172:" has no mention of cell shape. The symmetries of the pattern are what's important. -- 1947: 1716: 970:

I promise, to atone for this silly fidgeting I will fill in another section stub. :-)

319: 311: 3933: 3572:. So a shape of pattern of minimal area will be choosen for each type of wallpaper. 3082:

successfully google. It would probably be a good idea to explain it in the article.

2987: 2983: 2283: 2213: 1970: 1678: 1474: 938: 886: 656: 629: 487: 477: 449: 429: 370: 297: 3135: 1662:

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

1592:

just to show the 17 groups themselves. (Quick&Dirty, but useful for reference!)

868: 708:

nice to accompany the classification table, so we can see all the patterns together.

533:; after, half that. Both "*" and "x" themselves count as 1, and "o" counts as 2. So 3537: 3464: 3432: 3305:

I think there are two small errors or typos in the section mentioned in the header:

3218: 3200: 3186: 3149: 3099: 2967: 2926: 2866: 2821: 2813: 2021: 1992: 1983: 1611: 1567: 1546: 1505: 1430: 1407: 1277: 1220: 1184: 1149: 1100: 1051: 927: 910: 901: 830: 757: 748: 668: 574:

of this article. For this, I created a new set of diagrams. As those images are on

3672: 3310:" four letters or digits; more usual is a shortened name like c2mm or pg." --: --> 1692:

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

3494:

radially in a circularly symmetric "bar code" in the centre of mass of each tile.

3408:

are not equal, not each other's mirror image, and not both symmetric (if the two

3431:

I agree. That paragraph initially looks promising, but descends into gibberish.

3417: 2952: 2657: 1979: 1939:, or alternatively, they are the same "up to scaling and arbitrary (invertible) 385: 109: 3513:

that the wallpaper group of A and B is different from the wallpaper group of C.

2982:

The symmetry combinations article is entirely unsourced and appears to violate

3856: 3821: 3791: 3754: 3695:

I am having trouble determining what group it's in. According to the table in

3654: 3518: 3483:

A complete list of all seventeen possible wallpaper groups can be found below.

3251: 3063: 2515:

Or there can be digits on the right, and it turns out the only possibility is

2288: 2221: 2173: 2034: 1641: 1560: 1538: 1519: 1496: 1422: 1388: 1345: 1338: 1313: 1254: 1230: 1177: 1138: 1093: 1043: 1005: 971: 775: 741: 550: 442: 420: 405:

would be nice if this article was more in sync with the frieze groups article.

237: 86: 66: 3396:

Imagine a tessellation of the plane with equilateral triangles of equal size

3213:

I realise now that the article already provides a key. It forms the section

290:

needs pretty pictures illustrating the various types of euclidean isometries.

2652:

4. An unordered group of x's and o's (x = sliding mirror, o = no symmetries)

255:-ies and moved it into its own article. It has some material in common with 2061:

wallpapers with unit cells that are parallellograms with no symmetry at all

1604:

ANOTHER nice SHORT article would be for "Spherical symmetry groups" - page

1144:

I think the figures of cm and cmm should show a rhombus, as I indicated on

1329:

and mean that the mirror is perpendicular to the first cell axis, just as

909:

Similarly, ignoring colors the image shown is p4, otherwise p2, I think.--

3369:

I would be happy, if somebody could confirm and made changes accordingly.

2622: 2066: 3914: 3864: 3850: 3833: 3807: 3762: 3735: 3717: 3706:

Which would it be? I'd like to include this as an example if possible. ~

3662: 3607: 3585: 3545: 3526: 3440: 3425: 3404:, while both types have rotational symmetry of order three, but the two 3383: 3295: 3274: 3259: 3226: 3208: 3194: 3157: 3122: 3107: 3091: 3071: 3047: 3037: 3009: 2995: 2975: 2960: 2934: 2913: 2898: 2874: 2859: 2844: 2829: 2665: 2635: 2615: 2595: 2336: 2315: 2291: 2258: 2224: 2195: 2176: 2069: 2024: 1995: 1986: 1973: 1706: 1681: 1644: 1633: 1614: 1570: 1549: 1508: 1480: 1433: 1410: 1280: 1223: 1187: 1152: 1120:

Ideally, any explanation should make sense in 3D as well, since this is

1103: 1054: 1042:

Anyway, the section is finally filled. (That's two in two days!) Enjoy.

1027: 944: 930: 913: 904: 892: 855: 838: 822: 798: 760: 751: 671: 662: 649: 635: 373: 205: 2037:

corrected a note I added on p1; thanks for that. Here's KSmrq's text:

3098:

no view on whether, or how, this shoild be explained in the article.

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And finally if it starts "*2..." then there must be four "2"s total.

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inclination to nitpick every sentence change, I simply reverted all.

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a wallpaper group? No, the sum is 1/2+1+1/4, which is too small. Is

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

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It might be useful if the article contained what botanists call a

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It says "Here are all the names" and then 11 groups are listed.--

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I used to create those images, I'm willing to release them under

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two small errors in the section "Notations for wallpaper groups"

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mathematics only get to see one image for each wallpaper group.

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2) Why 17? The proof above is pretty long. Here's a faster way:

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I don't understand your comment. If I rotate it by 180°, I get

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I see you found something to say about the distinction between

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My proposal to remedy this: Split into two articles. The main

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and if there are only "2"s, then there must be four of them:

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of the unit cell is asymmetric, it is irrelevant whether the

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The following material was added before the introduction by

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A template used on this page has been nominated for deletion

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Different groups can have the same cell shape. For example,

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It would be nice if an expert could fix this up. Thanks.

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Orientation of diagram vs. that of computer-generated images

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reason I've been trying to design small images (maybe like

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figure has a rhombus that is visible, though subtle. The

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improved, by constructing "zoomed-in" versions of them.

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refer to a rhombus, which is essentially invisible for

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still need to finish labelling the example patterns of

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stuff on the "to do" list later on this discussion page

259:. I believe these articles could be profitably merged. 1137:

error in lengths and angles); less so the symmetries.

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Wallpaper group#Guide to recognizing wallpaper groups

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are both 120° rhombuses. This is one reason why the "

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and all other groups have equal lengths, fixed angle

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

108:, a collaborative effort to improve the coverage of 2646:

2. A number of asterisks (corresponding to mirrors)

774:should always come last, and should be lower case. 3703:and then using that pair as a single larger tile. 3570:patterns of minimal area can have different shapes 1956: 1931: 1875: 1849: 1822: 1795: 1766: 1725: 1449:My two cents is: The article is now way too long. 1146:commons:Image talk:Wallpaper group diagram cmm.png 2697:If we use 3/4, then we can only use 3/4+3/4+1/2: 2041:Now, could one add something like the following: 3560:Various repetitive patterns of a same wallpaper. 3473:Example B: Ornamental painting, Nineveh, Assyria 725: 3250:have the same image used as examples for both. 2713:Those are the only symbols with all rotations. 1857:are considered isomorphic if there exists some 355:Note that it is virtually impossible to get an 3416:apply then the third also, and we have p6m). 2691:If we use 5/6, then we must have 1/2 and 2/3: 2011: 441:part of that was climbing the learning curve. 3621:, the beginning of the lead was changed from 3199:Ok, that's some help. A key would be useful. 386:http://www.mi.sanu.ac.yu/vismath/ana/ana6.htm 8: 2787:, and has presentation <x,y|xyx⁻¹y⁻¹: --> 1456:article covering all the theory. A separate 3501:possible shifts are in diagonal directions. 3453:user:2402:3a80:6ab:2bee:8cf5:45d1:c8f3:1122 1491:The bad news is, the warning is about long 307:request for photographs of wallpaper groups 3564:A lot of occurrences of “patterns” in the 2208:If you visit my talk page, you will see a 244:regarding the recent makeover of this page 47: 3688:Many years ago, while in a large city in 3315:the short notation drops the '2' of c2mm. 2835:I would also like to see such a listing. 2552:13. *2222 (1 + 1/4 + 1/4 + 1/4 + 1/4 = 2) 2524:case there must also be a "2", so we get 1949: 1920: 1910: 1894: 1888: 1862: 1841: 1835: 1814: 1808: 1787: 1783: 1782: 1779: 1757: 1756: 1747: 1738: 1718: 1359:This gives us no guidance for the groups 687:Coincidentally, I've recently discovered 187:Re: Guide to recognising wallpaper groups 3163:This picture shows wallpaper with group 2357: 1585:This article is getting a bit unwieldy! 3950:Featured articles on Mathematics Portal 3597:under which the wallpaper is invariant. 2266:As I said, I saw enough places where I 1471:Does anyone think this is a good idea? 1293:is an example of a calcite description: 541:) adds up as 3/4 + 1 + 1/4. Similarly, 388:, would be nice to include some of this 283:the "stub" bits need to be filled out, 49: 19: 3684:Actual shape of Texas, for comparison. 3144:here? Is "wallpaper group" really the 2058:wallpapers with square unit cells, and 1307:, with more copies of the oxygen atom. 3871:Guide to recognizing wallpaper groups 3812:No. The tiling has 180° symmetry not 3676:Tessellation with Texas-shaped tiles. 3402:(Oops! Triangles are not equal size?) 3393:Read section p3 (my comment in bold) 3215:Guide to recognizing wallpaper groups 3077:What is the "F" shape in the pictures 2170:Guide to recognising wallpaper groups 1244:figure is modified, we then have the 414:Although the text said the following 7: 2586:17. 2222 (1/2 + 1/2 + 1/2 + 1/2 = 2) 2124:has independent lengths, fixed angle 2118:has independent lengths, fixed angle 2112:has independent lengths, fixed angle 2106:has independent lengths, fixed angle 2100:has independent lengths, fixed angle 1767:{\displaystyle GL_{2}(\mathbb {R} )} 1624:I didn't see any reference to Roger 1004:But I do like filling the stub. :-) 257:Coordinate rotations and reflections 102:This article is within the scope of 3476:Example C: Painted porcelain, China 2527:10. *632 (1 + 5/12 + 1/3 + 1/4 = 2) 1169:rhombus. On close inspection I can 581:Main Advantages in my opinion are: 510:, look at the fundamental region. ( 38:It is of interest to the following 3782:on Reddit (and I must say I think 3643:that, we could say something like 2544:12. *333 (1 + 1/3 + 1/3 + 1/3 = 2) 2536:11. *442 (1 + 3/8 + 3/8 + 1/4 = 2) 2094:has independent lengths, any angle 2088:has independent lengths, any angle 1932:{\displaystyle G_{1}=hG_{2}h^{-1}} 14: 3945:Mid-priority mathematics articles 2518:9. 2*22 (1/2 + 1 + 1/4 + 1/4 = 2) 1978:Some of these things are also in 1458:Wallpaper group (picture gallery) 765:The files need to be uploaded to 525:" before a "*" or "x" counts as ( 384:historical links to follow up at 200:Re: Illustrating plane isometries 122:Knowledge:WikiProject Mathematics 3816:if the colours are removed, but 3467:Consider the following examples: 2623:Orbifold#2-dimensional orbifolds 2212:of technical illustrations that 2051:of the unit cell is symmetrical. 1796:{\displaystyle \mathbb {R} ^{2}} 603:If someone is interested in the 588:Higher resolution, available as 276:there should be a discussion of 125:Template:WikiProject Mathematics 89: 79: 65: 51: 20: 2850:be inappropriate to hide them. 2152:both can be any parallelogram, 1588:I created a new article called 1554:The left-hand illustration for 1401:Problem with italics in headers 142:This article has been rated as 3072:05:28, 21 September 2011 (UTC) 2935:01:21, 26 September 2009 (UTC) 2465:and if there's a "*", we have 1761: 1753: 1590:List_of_Planar_Symmetry_Groups 875:Someone added the following: 333:I am looking specifically for 1: 3384:11:38, 14 November 2015 (UTC) 3209:16:20, 22 December 2012 (UTC) 3195:10:47, 22 December 2012 (UTC) 3158:02:24, 22 December 2012 (UTC) 3010:18:24, 15 November 2010 (UTC) 2996:16:48, 15 November 2010 (UTC) 2976:19:40, 12 February 2010 (UTC) 2961:12:15, 12 February 2010 (UTC) 2767:What actual groups are these? 2578:16. 333 (2/3 + 2/3 + 2/3 = 2) 2569:15. 442 (3/4 + 3/4 + 1/2 = 2) 2561:14. 632 (5/6 + 2/3 + 1/2 = 2) 2292:09:37, 18 February 2007 (UTC) 2282:Also, those revised graphics 2259:19:00, 17 February 2007 (UTC) 2225:07:56, 17 February 2007 (UTC) 2196:17:29, 13 February 2007 (UTC) 2025:14:41, 27 February 2006 (UTC) 978:Enumeration proof stub filled 799:01:40, 27 February 2006 (UTC) 116:and see a list of open tasks. 3940:C-Class mathematics articles 3902:What this page calls pattern 3820:if the colours are removed. 3608:16:30, 11 January 2022 (UTC) 3546:15:56, 3 February 2020 (UTC) 3296:20:43, 20 October 2014 (UTC) 3123:03:11, 16 January 2017 (UTC) 2875:22:41, 30 October 2009 (UTC) 2860:20:47, 30 October 2009 (UTC) 2820:) 11:40, 10 July 2008 (UTC) 2596:08:03, 22 January 2010 (UTC) 2337:20:52, 21 January 2010 (UTC) 2316:00:25, 3 December 2007 (UTC) 1996:09:19, 29 January 2006 (UTC) 1987:00:48, 29 January 2006 (UTC) 1974:20:13, 28 January 2006 (UTC) 1707:19:43, 28 January 2006 (UTC) 1682:02:14, 26 January 2006 (UTC) 1645:14:09, 22 January 2006 (UTC) 1634:10:41, 22 January 2006 (UTC) 1192:Ah. First, the captions for 770:issues: As I understand it, 3586:11:34, 9 January 2022 (UTC) 3227:20:45, 2 January 2013 (UTC) 3108:08:23, 6 October 2012 (UTC) 3092:05:22, 6 October 2012 (UTC) 2914:23:18, 14 August 2008 (UTC) 2899:01:07, 14 August 2008 (UTC) 1803:. So, two wallpaper groups 1699:rotation and rescaling)? 1615:10:55, 9 October 2005 (UTC) 1563:03:14, 2005 August 7 (UTC) 1541:21:17, 2005 August 6 (UTC) 1421:be bothered with doing it? 1391:19:47, 2005 August 6 (UTC) 1348:08:36, 2005 August 6 (UTC) 1341:07:00, 2005 August 6 (UTC) 1180:04:16, 2005 August 3 (UTC) 1141:13:39, 2005 August 2 (UTC) 570:I'm currently working on a 3966: 3915:10:16, 15 March 2024 (UTC) 3865:17:58, 14 March 2024 (UTC) 3851:17:34, 14 March 2024 (UTC) 3834:17:13, 14 March 2024 (UTC) 3808:14:40, 14 March 2024 (UTC) 3763:09:32, 14 March 2024 (UTC) 3736:05:57, 14 March 2024 (UTC) 3718:19:33, 13 March 2024 (UTC) 3275:07:32, 11 March 2014 (UTC) 3260:19:39, 10 March 2014 (UTC) 3048:16:17, 29 April 2011 (UTC) 3038:17:58, 27 April 2011 (UTC) 2666:19:42, 14 March 2018 (UTC) 2558:there must also be a "2": 2510:8. 22* (1/2 + 1/2 + 1 = 2) 2502:7. 3*3 (2/3 + 1 + 1/3 = 2) 2494:6. 4*2 (3/4 + 1 + 1/4 = 2) 2477:4. 22x (1/2 + 1/2 + 1 = 2) 2177:14:00, 10 April 2006 (UTC) 2070:10:12, 10 April 2006 (UTC) 2012:#Crystallographic notation 1571:06:35, 7 August 2005 (UTC) 1550:00:38, 7 August 2005 (UTC) 1462:Wallpaper group (examples) 1425:12:01, 2005 July 29 (UTC) 1316:17:38, 2005 August 4 (UTC) 1281:06:13, 4 August 2005 (UTC) 1257:17:58, 2005 August 3 (UTC) 1248:to discuss that aspect of 1233:23:07, 2005 August 3 (UTC) 1224:20:32, 3 August 2005 (UTC) 1188:07:14, 3 August 2005 (UTC) 1153:22:33, 2 August 2005 (UTC) 1096:11:39, 2005 July 29 (UTC) 1046:20:32, 2005 July 27 (UTC) 1008:10:22, 2005 July 27 (UTC) 974:11:37, 2005 July 26 (UTC) 856:17:58, 21 March 2011 (UTC) 839:10:04, 16 March 2011 (UTC) 823:22:18, 15 March 2011 (UTC) 778:06:10, 2005 July 26 (UTC) 744:11:50, 2005 July 25 (UTC) 626:Wallpaper group (examples) 471:orbifold notation examples 251:I expanded the section on 2845:14:11, 10 July 2008 (UTC) 2830:11:40, 10 July 2008 (UTC) 2636:17:29, 15 July 2010 (UTC) 2616:15:59, 15 July 2010 (UTC) 1522:22:24, 2005 July 29 (UTC) 1509:21:28, 29 July 2005 (UTC) 1499:21:09, 2005 July 29 (UTC) 1481:13:13, 29 July 2005 (UTC) 1434:19:50, 29 July 2005 (UTC) 1411:21:56, 28 July 2005 (UTC) 1104:19:54, 29 July 2005 (UTC) 1055:00:01, 29 July 2005 (UTC) 1034:Crystallographic notation 1028:15:53, 15 July 2010 (UTC) 983:than nothing? I hope so. 945:18:11, 25 July 2005 (UTC) 931:10:49, 25 July 2005 (UTC) 914:23:47, 24 July 2005 (UTC) 905:23:42, 24 July 2005 (UTC) 893:11:19, 24 July 2005 (UTC) 761:22:37, 25 July 2005 (UTC) 752:22:37, 25 July 2005 (UTC) 672:22:46, 25 July 2005 (UTC) 663:13:30, 23 July 2005 (UTC) 650:22:58, 22 July 2005 (UTC) 636:19:59, 22 July 2005 (UTC) 553:10:51, 2005 July 25 (UTC) 493:28 June 2005 18:07 (UTC) 483:28 June 2005 18:07 (UTC) 455:28 June 2005 16:15 (UTC) 445:05:43, 26 Jun 2005 (UTC) 423:02:17, 11 Jun 2005 (UTC) 230:Isometries as reflections 206:geometric sans-serif font 163: 141: 74: 46: 3663:10:48, 13 May 2022 (UTC) 3527:15:37, 25 May 2018 (UTC) 3470:Example A: Cloth, Tahiti 3441:22:42, 25 May 2018 (UTC) 3426:16:55, 23 May 2016 (UTC) 3344:should be corrected to: 2732:and *22 22 yields first 2004:Images showing a rhombus 1980:Space_group#Group_theory 1774:and the translations of 1333:does, and similarly for 967:, depending on context. 705:Euclidean plane isometry 549:) yields 2/3 + 1 + 1/3. 462:28 June 2005 17:23 (UTC) 435:11:22, 23 Jun 2005 (UTC) 374:21:32, 31 May 2005 (UTC) 303:17:02, 4 Jun 2005 (UTC) 253:euclidean plane isometry 240:14:00, 8 Jun 2005 (UTC) 234:Euclidean plane isometry 196:04:03, 6 Jun 2005 (UTC) 148:project's priority scale 3613:Here, a wallpaper is... 3594:that generate the group 2865:just mathematicians. -- 2721:*632, *442, *333, *2222 2065:are grouped together.-- 1216:, which might suffice. 1099:Thanks. I don't know.-- 1064:is listed as short for 105:WikiProject Mathematics 3685: 3677: 3561: 3486:Symmetries of patterns 3140: 2160:are both squares, and 2010:(see also the section 1958: 1941:linear transformations 1933: 1877: 1876:{\displaystyle h\in H} 1851: 1824: 1797: 1768: 1727: 1357: 1084:are different. Why is 872: 567: 324: 316: 262:Things to do include: 160: 28:This article is rated 3683: 3675: 3559: 3237:Herringbone examples. 3138: 2949:Symmetry combinations 2942:Symmetry combinations 1959: 1934: 1878: 1852: 1850:{\displaystyle G_{2}} 1825: 1823:{\displaystyle G_{1}} 1798: 1769: 1733:that is generated by 1728: 1353: 871: 565: 322: 314: 159: 3406:(Triangles? Groups?) 3281:cm Unit cell picture 2763:round out the list. 1991:I added them here.-- 1948: 1887: 1861: 1834: 1807: 1778: 1737: 1717: 1620:Roger Penrose tiles? 740:are missing for me. 619:Your diagrams are a 410:Discrimination table 128:mathematics articles 3668:What group is this? 1092:; or should it be? 864:Shapes and colours? 643:File upload service 600:What do you think? 558:Diagrams on Commons 369:Thank you so much. 228:I used this in the 3686: 3678: 3562: 3148:for these things? 3141: 2210:number of examples 1954: 1929: 1873: 1847: 1820: 1793: 1764: 1723: 1442:Splitting the page 1416:Fascinating; that 873: 595:F-shaped tile mark 568: 566:p4mm as an example 325: 317: 170:Mathematics Portal 161: 97:Mathematics portal 34:content assessment 3849: 3806: 3716: 3447:New introduction? 3139:This is wallpaper 2485:5. ** (1 + 1 = 2) 2468:3. *x (1 + 1 = 2) 2460:2. xx (1 + 1 = 2) 2436: 2435: 1957:{\displaystyle H} 1726:{\displaystyle H} 1688:Formal definition 1600:Triangular_tiling 1581:Article split up? 1018:comment added by 796:Martin von Gagern 703:" already on the 647:Martin von Gagern 576:Wikimedia Commons 379:other stuff to do 184: 183: 180: 179: 176: 175: 3957: 3843: 3800: 3795: 3773: 3710: 3398:(Ok, equal size) 3389:Deeply confusing 3176:" in this sense. 3131:Deeply confusing 3030: 2358: 2334: 2333: 2329: 1963: 1961: 1960: 1955: 1944:translations in 1938: 1936: 1935: 1930: 1928: 1927: 1915: 1914: 1899: 1898: 1882: 1880: 1879: 1874: 1856: 1854: 1853: 1848: 1846: 1845: 1829: 1827: 1826: 1821: 1819: 1818: 1802: 1800: 1799: 1794: 1792: 1791: 1786: 1773: 1771: 1770: 1765: 1760: 1752: 1751: 1732: 1730: 1729: 1724: 1713:whole group say 1122:crystallographic 1030: 848: 815: 727: 702: 698: 694: 293:needs REFERENCES 224:Glide reflection 166:selected article 130: 129: 126: 123: 120: 99: 94: 93: 83: 76: 75: 70: 69: 68: 63: 55: 48: 31: 25: 24: 16: 3965: 3964: 3960: 3959: 3958: 3956: 3955: 3954: 3930: 3929: 3789: 3767: 3670: 3615: 3566:current article 3554: 3449: 3391: 3303: 3283: 3239: 3133: 3079: 3056: 3035: 3028: 3020: 3018:Degree symbols? 2945: 2922: 2887: 2806: 2769: 2676: 2608:129.105.122.208 2331: 2327: 2325: 2304: 2185: 2032: 2020:orientation).-- 2006: 1946: 1945: 1916: 1906: 1890: 1885: 1884: 1859: 1858: 1837: 1832: 1831: 1810: 1805: 1804: 1781: 1776: 1775: 1743: 1735: 1734: 1715: 1714: 1690: 1653: 1622: 1583: 1531: 1454:Wallpaper group 1444: 1403: 1306: 1299: 1036: 1020:129.105.122.208 1013: 980: 953: 924: 866: 853: 846: 820: 813: 700: 696: 695:" rather than " 692: 560: 473: 412: 381: 309: 271:Wallpaper group 246: 202: 189: 127: 124: 121: 118: 117: 95: 88: 64: 61: 32:on Knowledge's 29: 12: 11: 5: 3963: 3961: 3953: 3952: 3947: 3942: 3932: 3931: 3928: 3927: 3926: 3925: 3924: 3923: 3922: 3921: 3920: 3919: 3918: 3917: 3886: 3885: 3884: 3883: 3882: 3881: 3880: 3879: 3878: 3877: 3876: 3875: 3787: 3750: 3744: 3738: 3724: 3669: 3666: 3651: 3650: 3629: 3628: 3614: 3611: 3598: 3576: 3553: 3550: 3549: 3548: 3515: 3514: 3510: 3506: 3502: 3498: 3495: 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1273: 1272: 1271: 1270: 1269: 1268: 1267: 1266: 1265: 1264: 1263: 1262: 1261: 1260: 1259: 1258: 1238: 1237: 1236: 1235: 1234: 1212:as it did for 1174: 1165:figure has no 1129: 1118: 1088:not short for 1035: 1032: 979: 976: 952: 949: 948: 947: 923: 920: 919: 918: 917: 916: 907: 883: 882: 865: 862: 861: 860: 859: 858: 851: 818: 808: 807: 806: 805: 804: 803: 802: 801: 754: 721: 720: 716: 715: 710: 709: 684: 683: 679: 678: 677: 676: 675: 674: 665: 597: 596: 593: 586: 572:german version 559: 556: 555: 554: 517: 516: 499: 498: 472: 469: 468: 467: 466: 465: 464: 463: 437: 436: 411: 408: 407: 406: 403: 399: 395: 392: 389: 380: 377: 308: 305: 295: 294: 291: 288: 281: 274: 267: 245: 242: 226: 225: 222: 219: 216: 213: 201: 198: 188: 185: 182: 181: 178: 177: 174: 173: 162: 152: 151: 140: 134: 133: 131: 114:the discussion 101: 100: 84: 72: 71: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 3962: 3951: 3948: 3946: 3943: 3941: 3938: 3937: 3935: 3916: 3912: 3908: 3903: 3898: 3897: 3896: 3895: 3894: 3893: 3892: 3891: 3890: 3889: 3888: 3887: 3872: 3868: 3867: 3866: 3862: 3858: 3854: 3853: 3852: 3847: 3842: 3837: 3836: 3835: 3831: 3827: 3823: 3819: 3815: 3811: 3810: 3809: 3804: 3799: 3793: 3788: 3785: 3781: 3777: 3771: 3766: 3765: 3764: 3760: 3756: 3751: 3748: 3745: 3742: 3739: 3737: 3733: 3729: 3725: 3722: 3721: 3720: 3719: 3714: 3709: 3704: 3700: 3698: 3693: 3691: 3682: 3674: 3667: 3665: 3664: 3660: 3656: 3649: 3646: 3645: 3644: 3642: 3638: 3633: 3627: 3624: 3623: 3622: 3620: 3612: 3610: 3609: 3605: 3601: 3600:Arthur Baelde 3596: 3595: 3588: 3587: 3583: 3579: 3578:Arthur Baelde 3573: 3571: 3567: 3558: 3552:What patterns 3551: 3547: 3543: 3539: 3535: 3531: 3530: 3529: 3528: 3524: 3520: 3511: 3507: 3503: 3499: 3496: 3492: 3488: 3485: 3482: 3478: 3475: 3472: 3469: 3466: 3463: 3459: 3458: 3457: 3454: 3446: 3442: 3438: 3434: 3430: 3429: 3428: 3427: 3423: 3419: 3415: 3411: 3407: 3403: 3399: 3394: 3388: 3386: 3385: 3381: 3377: 3364: 3360: 3357: 3354: 3350: 3347: 3346: 3345: 3339: 3335: 3332: 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1572: 1569: 1565: 1564: 1562: 1557: 1553: 1552: 1551: 1548: 1544: 1543: 1542: 1540: 1536: 1529:Illustrations 1528: 1521: 1517: 1512: 1511: 1510: 1507: 1503: 1502: 1498: 1494: 1490: 1489: 1485: 1484: 1483: 1482: 1479: 1476: 1472: 1469: 1466: 1463: 1459: 1455: 1450: 1447: 1441: 1435: 1432: 1427: 1426: 1424: 1419: 1415: 1414: 1413: 1412: 1409: 1400: 1394: 1393: 1392: 1390: 1386: 1382: 1378: 1374: 1370: 1366: 1362: 1356: 1352: 1349: 1347: 1342: 1340: 1336: 1332: 1328: 1324: 1315: 1311: 1310: 1301: 1300: 1291: 1290: 1285: 1284: 1283: 1282: 1279: 1256: 1251: 1247: 1243: 1239: 1232: 1227: 1226: 1225: 1222: 1218: 1217: 1215: 1211: 1207: 1203: 1199: 1195: 1191: 1190: 1189: 1186: 1182: 1181: 1179: 1175: 1172: 1168: 1164: 1160: 1156: 1155: 1154: 1151: 1147: 1143: 1142: 1140: 1135: 1130: 1127: 1123: 1119: 1115: 1111: 1107: 1106: 1105: 1102: 1098: 1097: 1095: 1091: 1087: 1083: 1079: 1075: 1071: 1067: 1063: 1058: 1057: 1056: 1053: 1049: 1048: 1047: 1045: 1040: 1033: 1031: 1029: 1025: 1021: 1017: 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