Talk:Wallpaper group

message: 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)
thread_title:  Re: Guide to recognising wallpaper groups 

message: A simple, easily remembered, illustration of plane isometries requires no images, merely a geometric sans-serif font.
*[d  ] Identity
*[dp] Rotation
*[dd] Translation
*[db] Reflection
*[dq] Glide reflection
I used this in the  Isometries as reflections section of Euclidean plane isometry. KSmrq 14:00, 8 Jun 2005 (UTC)
thread_title:  Re: Illustrating plane isometries 

message: I have completely redone the wallpaper group page. I borrowed some pics and links but most of the text is new. Also there is LOTS of art.
I expanded the section on euclidean plane isometry-ies and moved it into its own article. It has some material in common with Coordinate rotations and reflections. I believe these articles could be profitably merged.
Things to do include:
* stuff on the "to do" list later on this discussion page
* still need to finish labelling the example patterns of Wallpaper group; I've only done the first few. I will finish this off in the next few days. I don't suppose anyone else will have much luck with it; I happen to have the book "Grammar of Ornament" here with me.
* there should be a discussion of lattices somewhere.
* the "stub" bits need to be filled out, especially relationship between informal and formal approaches. I've noticed the text is not very explicit on that point as it stands.
* needs pretty pictures illustrating the various types of euclidean isometries.
* needs REFERENCES
Dmharvey Talk 17:02, 4 Jun 2005 (UTC)
thread_title:  regarding the recent makeover of this page 

message: 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)
thread_title:  request for photographs of wallpaper groups 

message: Someone added the following: 
: Sometimes two categorizations are meaningful, one based on shapes and one also on colors.
No further explanation is given. I don't understand what is meant. Please expand or I will remove it. Thanks Dmharvey Talk 11:19, 24 July 2005 (UTC)
replies: text: E.g. a red p and a black q can together form a symmetric image, but only if we ignore color.--Patrick 23:42, 24 July 2005 (UTC)
replies: text: Similarly, ignoring colors the image shown is p4, otherwise p2, I think.--Patrick 23:47, 24 July 2005 (UTC)
thread_title:  Shapes and colours? 

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

message: 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)
thread_title:  Enumeration proof stub filled 

message: 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)
thread_title:  Article split up? 

message: 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)
replies: text: 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. --KSmrqT 14:09, 22 January 2006 (UTC)
thread_title:  Roger Penrose tiles? 

message: This is a very nice article and I have been enjoying the cultural examples of wall paper groups. I do have a few comments:
It would be nice to more systematically interleave the orbifold and international crystallographic notations. Also,  the diagrams of the "cell structures" of the various groups seem a little misleading: (a) there is usually no canonical cell structure, but you are typically showing generators for the group, so that's no big deal, but (b) there is no difference between, say the symmetry denoted o (in the orbifold notation) and p1 (in the intl cr. notation) and it seems like they should have the same diagram. Same point for the other 16.
The thread above on the crystallographic notation prompts me to write this screed: 
:this discussion really points to the inadequacy of the crystallographic notation. There is nothing canonical about it. Given a new group, you would not be able to guess accurately the name used for the last 100 years. The orbifold notation really does uniquely define a symmetry, in a way that can never be mistaken and can be calculated; it generalizes to all two-dimensional symmetries, of the plane, sphere, and hyperbolic plane, and the frieze groups.
Those participating in this discussion may be interested to know that Conway will be publishing a beautiful book on the subject early in 2007 (I am a co-author). 
Also, Conway has a nice way to understand all the 3D space groups, which will be discussed really for the first time in that book.  Jan 25 2006.
: Dear madam/sir, I'm glad you like the page. Most of the "cultural examples" you speak of are my fault. (Well I didn't draw them obviously, but I collected and organised them). If you have any ideas for the page, you are of course quite welcome to make changes yourself! That's what the "edit" button is for! Don't worry if you get stuck with the wiki syntax, there are plenty of people hanging around here who will help you. You might consider creating an account, since it improves your anonymity slightly (i.e. we can't guess geographical location via your IP address :-)) And of course I'm looking forward to that book. Dmharvey 02:14, 26 January 2006 (UTC)
thread_title:  miscellaneous 

message: 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)
replies: text: Dear anon, that's an excellent point, there's something missing there. Here is my recollection of the situation (it's been a little while, and I don't have references handy). Your definition of "isomorphic" is the correct one, except that besides just rotations and rescalings, you need to include the whole group say H that is generated by GL_2(\reals) and the translations of \reals^2. So, two wallpaper groups G_1 and G_2 are considered isomorphic if there exists some h \in H such that G_1 = h G_2 h^{-1}, or alternatively, they are the same "up to scaling and arbitrary (invertible) linear transformations of the plane". The reason you need to include arbitrary linear transformations is that otherwise there will be infinitely many different versions of say p1, all with slightly different angles between the two independent translations. For similar reasons your definition needs to include the translations in H. Now, the amazing thing, is that after giving this correct "natural" definition, it turns out that two groups are isomorphic in the above sense if and only if they are isomorphic as abstract groups! Obviously you need to do a little bit of work to prove this. I believe there is a discussion of how this all works in the Grunbaum reference, but I'm not 100% sure. Dmharvey 20:13, 28 January 2006 (UTC)
replies: text: Some of these things are also in Space_group#Group_theory (2D case).--Patrick 00:48, 29 January 2006 (UTC)

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

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

message: 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)
replies: text: I would also like to see such a listing. JackSchmidt (talk) 14:11, 10 July 2008 (UTC)
replies: text: 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 be inappropriate to hide them. Ishboyfay (talk) 20:47, 30 October 2009 (UTC)
replies: text: 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 just mathematicians. --seberle (talk) 22:41, 30 October 2009 (UTC)
thread_title: What actual groups are these?

message: 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)
replies: text: 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! --Salix alba (talk) 23:18, 14 August 2008 (UTC)
thread_title:  p3m1 and p31m 

message: 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)
thread_title:  p4 examples 

message: 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)
replies: text: Some symmetry combinations imply translational symmetry in two dimensions, so there is some overlap. However, treatment of individual wallpaper groups is not repeated.--Patrick (talk) 19:40, 12 February 2010 (UTC)

text: The symmetry combinations article is entirely unsourced and appears to violate WP:NOR.  Any verifiable content in the article should be moved here, and the rest should be deleted. Jim.belk (talk) 16:48, 15 November 2010 (UTC)
replies: text: I've now proposed it for deletion. Not sure that there is anything in it which is not already covered here. --Salix (talk): 18:24, 15 November 2010 (UTC)
thread_title:  Merge with [[Symmetry combinations]] 

message: 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? -LesPaul75talk 17:58, 27 April 2011 (UTC)
replies: text: " °̊ " 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. —Keenan Pepper 16:17, 29 April 2011 (UTC)
thread_title:  Degree symbols? 

message: See Wikipedia:Templates_for_discussion/Log/2011_September_15#Template:Wallpaper_group_list. --99of9 (talk) 05:28, 21 September 2011 (UTC)
thread_title:  A template used on this page has been nominated for deletion 

message: 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)
replies: text: 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 no view on whether, or how, this shoild be explained in the article. Maproom (talk) 08:23, 6 October 2012 (UTC)
replies: text: 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 explained. Daaxix (talk) 03:11, 16 January 2017 (UTC)
thread_title:  What is the "F" shape in the pictures 

message: 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)
replies: text: Look at the closeup images - the Egyptian mats differ in details, that's why each one belongs to a different group.--Krótki (talk) 07:32, 11 March 2014 (UTC)
thread_title:  Herringbone examples. 

message: 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)
thread_title:  cm Unit cell picture 

message: I think there are two small errors or typos in the section mentioned in the header:
# "[...] four letters or digits; more usual is a shortened name like c2mm or pg." --> should read as "[...] four letters or digits; more usual is a shortened name like cmm or pg.", i.e. the short notation drops the '2' of c2mm.
# 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
* p4mg (p4mm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45° and
* c2mm (c2mm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis
should be corrected to:
* p4mg (p4g): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45° and
* c2mm (cmm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis
I would be happy, if somebody could confirm and made changes accordingly.
kind regards
Frank --Kohaerenz (talk) 11:38, 14 November 2015 (UTC)
thread_title:  two small errors in the section "Notations for wallpaper groups" 

message: 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)
replies: text: I agree. That paragraph initially looks promising, but descends into gibberish. Maproom (talk) 22:42, 25 May 2018 (UTC)
thread_title:  Deeply confusing 

message: 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.
: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 group.
:IntroductionConsider the following examples:
:Example A: Cloth, Tahiti
:Example B: Ornamental painting, Nineveh, Assyria
:Example C: Painted porcelain, China
: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 the designs, whereas C has a different set of symmetries despite any superficial similarities.
:A complete list of all seventeen possible wallpaper groups can be found below.
:Symmetries of patterns
: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 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.
: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 radially in a circularly symmetric "bar code" in the centre of mass of each tile.
:The types of transformations that are relevant here are called Euclidean plane isometries. For example:
: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 possible shifts are in diagonal directions.
: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 C.
: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 A.
: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 that the wallpaper group of A and B is different from the wallpaper group of C.
--Nø (talk) 15:37, 25 May 2018 (UTC)
replies: text: 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  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. ★NealMcB★ (talk) 15:56, 3 February 2020 (UTC)
thread_title:  New introduction? 

message: 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)
thread_title:  What patterns 

message: In this edit, the beginning of the lead was changed from
:Here a wallpaper is a drawing that covers a whole Euclidean plane by repeating a motif indefinitely ...
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
:Thus, in this context, a wallpaper is a pattern that covers a whole Euclidean plane by repeating a motif indefinitely.
Thoughts?--Nø (talk) 10:48, 13 May 2022 (UTC)
thread_title:  Here, a wallpaper is...