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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)
A simple, easily remembered, illustration of plane isometries requires no images, merely a geometric sans-serif font.
I used this in the Isometries as reflections section of Euclidean plane isometry. KSmrq 14:00, 8 Jun 2005 (UTC)
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:
Dmharvey Talk 17:02, 4 Jun 2005 (UTC)
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)
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)
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)
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:
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.
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. - LesPaul75 talk 22:18, 15 March 2011 (UTC)
Someone added the following:
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)
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)
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)
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)
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 left out a "feature", called 'f', for example, so that 2f2 could be another wallpaper group. —Preceding unsigned comment added by 129.105.122.208 ( talk) 15:53, 15 July 2010 (UTC)
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)
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)
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
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)
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)
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)
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)
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)
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)
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:
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.
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)
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)
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
are grouped together.-- Niels Ø 10:12, 10 April 2006 (UTC)
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.
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. Greg Kuperberg 17:29, 13 February 2007 (UTC)
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 mentioned 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. 81.153.111.37 00:25, 3 December 2007 (UTC)
Feature: | o | x | * | 6 | 4 | 3 | 2 | (*)6 | (*)4 | (*)3 | (*)2 |
Value: | 2 | 1 | 1 | 5/6 | 3/4 | 2/3 | 1/2 | 5/12 | 3/8 | 1/3 | 1/4 |
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? 129.105.122.208 ( talk) 15:59, 15 July 2010 (UTC)
Having had the pleasure of being a coauthor of the "Symmetries of Things" can address a couple of issues raised here.
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 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.
2) Why 17? The proof above is pretty long. Here's a faster way:
(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.
If we use 5/6, then we must have 1/2 and 2/3: 632
If we use 3/4, then we can only use 3/4+3/4+1/2: 442
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 2222 and 333
Those are the only symbols with all rotations.
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 *:
*632, *442, *333, *2222
Since the trick is reversible, these are the only symbols with a single * followed by all numbers.
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 3*3 and *22 22 yields first 2*22 and then 22*. Since the trick is reversible, these are the only symbols of the form numbers*numbers.
d) Finally, any * not followed by numbers can be converted to a x without trouble, hence 22x 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. xx, **, x* and o round out the list.
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 think that p1 is Z×Z, and has presentation <x,y|xyx⁻¹y⁻¹>; p2 is (Z×Z)⋊C2, and has presentation <x,y,t|xyx⁻¹y⁻¹,xtxt,ytyt>, 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.)
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 mathematics only get to see one image for each wallpaper group. Maproom ( talk) 11:40, 10 July 2008 (UTC) Maproom ( talk) 11:40, 10 July 2008 (UTC)
The "computer generated" examples for p31m and p3m1 seem to be switched, according to the definitions. Can someone verify this? -- Lasunncty ( talk) 01:07, 14 August 2008 (UTC)
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 category.-- seberle ( talk) 01:21, 26 September 2009 (UTC)
The term Symmetry combinations 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.-- RDBury ( talk) 12:15, 12 February 2010 (UTC)
In several places, the ° symbol is displayed oddly in my browser (FF4). An example is this table. 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? - LesPaul75 talk 17:58, 27 April 2011 (UTC)
See Wikipedia:Templates_for_discussion/Log/2011_September_15#Template:Wallpaper_group_list. -- 99of9 ( talk) 05:28, 21 September 2011 (UTC)
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 successfully google. It would probably be a good idea to explain it in the article. 76.175.173.23 ( talk) 05:22, 6 October 2012 (UTC)
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 here? Is "wallpaper group" really the WP:COMMONNAME for these things? Johnbod ( talk) 02:24, 22 December 2012 (UTC)
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. Group p2 and Group pg have the same image used as examples for both. Cliff ( talk) 19:39, 10 March 2014 (UTC)
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? Ben476 ( talk) 20:43, 20 October 2014 (UTC)
I think there are two small errors or typos in the section mentioned in the header:
should be corrected to:
I would be happy, if somebody could confirm and made changes accordingly.
kind regards
Frank --
Kohaerenz (
talk)
11:38, 14 November 2015 (UTC)
Read section p3 (my comment in bold)
Imagine a tessellation of the plane with equilateral triangles of equal size (Ok, equal size), 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 (Oops! Triangles are not equal size?), while both types have rotational symmetry of order three, but the two (Triangles? Groups?) are not equal, not each other's mirror image, and not both symmetric (if the two (Triangles?) 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 (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?) apply then the third also, and we have p6m). Jumpow ( talk) 16:55, 23 May 2016 (UTC)
The following material was added before the introduction by user:2402:3a80:6ab:2bee:8cf5:45d1:c8f3:1122 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.
-- Nø ( talk) 15:37, 25 May 2018 (UTC)
A lot of occurrences of “patterns” in the current article, 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 patterns of minimal area can have different shapes. So a shape of pattern of minimal area will be choosen for each type of wallpaper.
On the present image, various patterns are parallelograms constructed each from two translations under which the wallpaper is invariant.
Arthur Baelde (
talk)
11:34, 9 January 2022 (UTC)
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
that generate the group under which the wallpaper is invariant.
Arthur Baelde (
talk)
16:30, 11 January 2022 (UTC)
In this edit, the beginning of the lead was changed from
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.
Perhaps we should rewrite? Maybe we should start by saying what a "Wallpaper group" is, not what a wallpaper is - not 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. Following that, we could say something like
Thoughts?-- Nø ( talk) 10:48, 13 May 2022 (UTC)
Many years ago, while in a large city in Texas, I encountered a small plaza paved with Texas-shaped flagstones. I just spent some time figuring this out, and created this image.
I am having trouble determining what group it's in. According to the table in Wallpaper group#Guide to recognizing wallpaper groups, 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).
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, and then using that pair as a single larger tile.
Which would it be? I'd like to include this as an example if possible. ~ Anachronist ( talk) 19:33, 13 March 2024 (UTC)
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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)
A simple, easily remembered, illustration of plane isometries requires no images, merely a geometric sans-serif font.
I used this in the Isometries as reflections section of Euclidean plane isometry. KSmrq 14:00, 8 Jun 2005 (UTC)
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:
Dmharvey Talk 17:02, 4 Jun 2005 (UTC)
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)
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)
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)
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:
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.
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. - LesPaul75 talk 22:18, 15 March 2011 (UTC)
Someone added the following:
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)
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)
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)
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)
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 left out a "feature", called 'f', for example, so that 2f2 could be another wallpaper group. —Preceding unsigned comment added by 129.105.122.208 ( talk) 15:53, 15 July 2010 (UTC)
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)
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)
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
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)
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)
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)
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)
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)
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)
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:
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.
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)
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)
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
are grouped together.-- Niels Ø 10:12, 10 April 2006 (UTC)
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.
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. Greg Kuperberg 17:29, 13 February 2007 (UTC)
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 mentioned 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. 81.153.111.37 00:25, 3 December 2007 (UTC)
Feature: | o | x | * | 6 | 4 | 3 | 2 | (*)6 | (*)4 | (*)3 | (*)2 |
Value: | 2 | 1 | 1 | 5/6 | 3/4 | 2/3 | 1/2 | 5/12 | 3/8 | 1/3 | 1/4 |
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? 129.105.122.208 ( talk) 15:59, 15 July 2010 (UTC)
Having had the pleasure of being a coauthor of the "Symmetries of Things" can address a couple of issues raised here.
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 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.
2) Why 17? The proof above is pretty long. Here's a faster way:
(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.
If we use 5/6, then we must have 1/2 and 2/3: 632
If we use 3/4, then we can only use 3/4+3/4+1/2: 442
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 2222 and 333
Those are the only symbols with all rotations.
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 *:
*632, *442, *333, *2222
Since the trick is reversible, these are the only symbols with a single * followed by all numbers.
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 3*3 and *22 22 yields first 2*22 and then 22*. Since the trick is reversible, these are the only symbols of the form numbers*numbers.
d) Finally, any * not followed by numbers can be converted to a x without trouble, hence 22x 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. xx, **, x* and o round out the list.
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 think that p1 is Z×Z, and has presentation <x,y|xyx⁻¹y⁻¹>; p2 is (Z×Z)⋊C2, and has presentation <x,y,t|xyx⁻¹y⁻¹,xtxt,ytyt>, 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.)
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 mathematics only get to see one image for each wallpaper group. Maproom ( talk) 11:40, 10 July 2008 (UTC) Maproom ( talk) 11:40, 10 July 2008 (UTC)
The "computer generated" examples for p31m and p3m1 seem to be switched, according to the definitions. Can someone verify this? -- Lasunncty ( talk) 01:07, 14 August 2008 (UTC)
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 category.-- seberle ( talk) 01:21, 26 September 2009 (UTC)
The term Symmetry combinations 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.-- RDBury ( talk) 12:15, 12 February 2010 (UTC)
In several places, the ° symbol is displayed oddly in my browser (FF4). An example is this table. 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? - LesPaul75 talk 17:58, 27 April 2011 (UTC)
See Wikipedia:Templates_for_discussion/Log/2011_September_15#Template:Wallpaper_group_list. -- 99of9 ( talk) 05:28, 21 September 2011 (UTC)
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 successfully google. It would probably be a good idea to explain it in the article. 76.175.173.23 ( talk) 05:22, 6 October 2012 (UTC)
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 here? Is "wallpaper group" really the WP:COMMONNAME for these things? Johnbod ( talk) 02:24, 22 December 2012 (UTC)
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. Group p2 and Group pg have the same image used as examples for both. Cliff ( talk) 19:39, 10 March 2014 (UTC)
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? Ben476 ( talk) 20:43, 20 October 2014 (UTC)
I think there are two small errors or typos in the section mentioned in the header:
should be corrected to:
I would be happy, if somebody could confirm and made changes accordingly.
kind regards
Frank --
Kohaerenz (
talk)
11:38, 14 November 2015 (UTC)
Read section p3 (my comment in bold)
Imagine a tessellation of the plane with equilateral triangles of equal size (Ok, equal size), 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 (Oops! Triangles are not equal size?), while both types have rotational symmetry of order three, but the two (Triangles? Groups?) are not equal, not each other's mirror image, and not both symmetric (if the two (Triangles?) 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 (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?) apply then the third also, and we have p6m). Jumpow ( talk) 16:55, 23 May 2016 (UTC)
The following material was added before the introduction by user:2402:3a80:6ab:2bee:8cf5:45d1:c8f3:1122 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.
-- Nø ( talk) 15:37, 25 May 2018 (UTC)
A lot of occurrences of “patterns” in the current article, 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 patterns of minimal area can have different shapes. So a shape of pattern of minimal area will be choosen for each type of wallpaper.
On the present image, various patterns are parallelograms constructed each from two translations under which the wallpaper is invariant.
Arthur Baelde (
talk)
11:34, 9 January 2022 (UTC)
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
that generate the group under which the wallpaper is invariant.
Arthur Baelde (
talk)
16:30, 11 January 2022 (UTC)
In this edit, the beginning of the lead was changed from
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.
Perhaps we should rewrite? Maybe we should start by saying what a "Wallpaper group" is, not what a wallpaper is - not 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. Following that, we could say something like
Thoughts?-- Nø ( talk) 10:48, 13 May 2022 (UTC)
Many years ago, while in a large city in Texas, I encountered a small plaza paved with Texas-shaped flagstones. I just spent some time figuring this out, and created this image.
I am having trouble determining what group it's in. According to the table in Wallpaper group#Guide to recognizing wallpaper groups, 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).
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, and then using that pair as a single larger tile.
Which would it be? I'd like to include this as an example if possible. ~ Anachronist ( talk) 19:33, 13 March 2024 (UTC)