Talk:Space group

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"By way of example, the space group of quartz is P3₁21,..."

"In the international short symbol the first symbol (3₁ in this example) denotes the symmetry along the major axis "

This makes no sense at all. Surely "P" is the first symbol. If P isn't being counted as a symbol, then what exactly do we mean by a symbol? I'm trying to gain some understanding of this topic, and this article (like so many others on the subject) seems incomprehensible, to such as extend that even some of the most basic terms like "symbol" seem to be being used in a unconventional way with no explanation of their meaning in this context. — Preceding unsigned comment added by 95.131.110.106 (talk) 11:09, 18 February 2014 (UTC)[reply]

Try to read Hermann–Mauguin notation Bor75 (talk)

How can I mark this page as 'needs to be clearer for a lay person'? Its good, but some of the english seems to be wrong, and not being an expert I am not able to fix it...

"with a threefold screw axis projecting on one face, and two fold rotation axis another."

Is this simply

"with a threefold screw axis projecting on one face, and two fold rotation axis on another."

?

It isn't clear where this information comes from given the space group.

--Dan|^(talk) 17:15, 25 July 2005 (UTC)[reply]

I fixed this. The 3 and 2 are in the notation, the subscript indicates that it is a screw axis.--Patrick 10:33, 2 January 2006 (UTC)[reply]

I've added the attention tag to this article. I believe the article is factually correct, but it is very unclear to me (and I have some knowledge on the subject). To a lay-person it would be completely unreadable. I might get around to doing something about it myself, but I'm sure there are more qualified people out there. If I do it, I might miss out on some of the mathematics. (actually I'm quite sure I would)O. Prytz 15:49, 26 December 2005 (UTC)[reply]

It is an advanced topic. Can you be more specific about what is unclear?--Patrick 02:35, 2 January 2006 (UTC)[reply]

That's a crap reason. Space groups are used daily by chemists, some of whom struggle with calculus, never mind abstract algebra - so I hardly think it's so advanced it can't be made comprehensible. Specific problems with the article: Undefined jargon: enatiomorphus, Bieberbach theorems (also, why it that plural, when it says a single thing?), chiral, Z, semidirect. I can't work out what the difference is between a 'space group' and a 'space group type'.

A space group is a symmetry group or symmetry group type. To avoid ambiguity the term "space group type" can be used. E.g. a change of angle between translation vectors does not affect the space group type if it does not add or remove any symmetry, but the symmetry group is different.--Patrick 02:40, 3 January 2006 (UTC)[reply]

The phrase 'only part of the point groups' is ambigous due to bad grammer (does it mean a section from an arbitry point group is relevent, or that only some of the point groups are?). Who cares about affine space groups? Who cares about symmorphic space groups? The article splits the space groups down by 1D, 2D 3D in about three different places.

For each subject there is a split. A separate article about the 2D case also exists: wallpaper group.--Patrick 02:45, 3 January 2006 (UTC)[reply]

That's just the points I can note since my original draft. There's still a bunch of outstanding issues from that: Schoenflies notation not covered; talking about screw axis and glide plane (are they specific to 3+D?), rather than a list of all the operations (Porobably best done by a simple list, where each term links to an article?).

Screw axis and glide plane are special in not being point groups.--Patrick 02:49, 3 January 2006 (UTC)[reply]

The crystallographic stuff really aught to be clear that it applies to only 3D space groups.

The term crystallographic is also used in other dimensions, such as 2D.--Patrick 02:55, 3 January 2006 (UTC)[reply]

A mention of the magnetic space groups would be useful. Syntax 00:42, 3 January 2006 (UTC)[reply]

All those 1D, 2D 3D verbosities all over the place are a very bad way of writing a readable article. Give a relevant example, if necessary, rather than making such silly lists. Oleg Alexandrov (talk) 01:16, 3 January 2006 (UTC)[reply]

Your objections against discussing and comparing 1D, 2D, and 3D, are weird.--Patrick 02:12, 3 January 2006 (UTC)[reply]

It might work in some places, I don't argue. But you tend to overlist things, and you do it way too often. That destroys continuity in many places. Your edit style transforms a well-written prose into unreadable lists of technicalities, and I am not the only (or the only two or three) persons to complain about it. Oleg Alexandrov (talk) 05:22, 3 January 2006 (UTC)[reply]

If the article contains more information than you currently need, the section and paragraph structure makes it fairly easy to skip parts when studying the article. More verbose prose is not always better than compact list style.--Patrick 10:43, 3 January 2006 (UTC)[reply]

See this old version of space group for how many headings you have.

I am pleased to hear that you are happy with the new headers. Note that your favorite section (recently added old version) is the longest and may need some subheaders.--Patrick 00:35, 4 January 2006 (UTC)[reply]

At least one piece of advice for the future. Do not replace elementary content with more abstract one. Do not rewrite articles from a higher mathematics/group theory point of view. If needed, add to articles. Using your favorite quote, "do not remove information". You seem to react vehemently each time I delete one of your technicalities from articles. So please don't delete elementary information, that one is so much more valuable. Oleg Alexandrov (talk) 16:12, 3 January 2006 (UTC)[reply]

I have not deleted much content, and it was not very clear and elementary either. I agree that an elementary part is useful.--Patrick 01:01, 4 January 2006 (UTC)[reply]

I want to bring back some of the text from an older version of this article which I belive gives a more understandable explanation of the concept of space groups. Most of the current text could be kept in a section on group theroy. Also, I'm wondering if the sections on screw axes and glide planes should be removed. Glide plane has its own article but screw axis doesn't. If no-one objects the next couple of days I'll make the change. O. Prytz 20:08, 1 January 2006 (UTC)[reply]

Be my guest please. The recent version (which you don't like, and me neither) is written by Patrick, who while technically gifted, has a talent for starting with plain words articles and obfuscating them with technical jargon which few people besides himself understand. If you feel that some pieces of that technicality are not salvageable, just delete them, as you said. They will be in the history if anybody wishes to work on that later. Oleg Alexandrov (talk) 21:23, 1 January 2006 (UTC)[reply]

Glide plane does not have its own article, but you can create one, or move the text to glide reflection. Similarly you could split off the article screw axis.--Patrick 01:34, 2 January 2006 (UTC)[reply]

Ah, you're right, glide plane doesn't exist. Should there be a separate glide plane article in addition to the one on glide reflection? O. Prytz 07:29, 2 January 2006 (UTC)[reply]

I think it can be covered in glide reflection.--Patrick 10:52, 2 January 2006 (UTC)[reply]

Please don't be encouraged by Oleg to carry out unhelpful deletions.--Patrick 01:41, 2 January 2006 (UTC)[reply]

One can think later where to move the content. First, of all, let us worry about transforming the article into something readable. Using an older version as a starting point looks like a good idea to me. Oleg Alexandrov (talk) 01:58, 2 January 2006 (UTC)[reply]

Ok, I can see that we're probably going to have a disagreement on this. But Patrick, how do you feel about retrieveing some of the text from the old aricle and using that as the first portion, and then moving most of the current text to a section on 'group theory'? O. Prytz 07:29, 2 January 2006 (UTC)[reply]

Actually I'm going to disagree with the idea of pulling out an old version, and tweaking it. I think that it would be better to design some basic structure, and then draft up an article to match that structure from the content present. Then drop that draft in over the current article. So, as a starting point, howable sections on: Introduction; Mathematical Origin; 3D space groups (subsections: symetry elements, notations); Types of space groups. That does give a lot of prominance to the 3D groups, but I think that is warrented, given thier use. Syntax 00:42, 3 January 2006 (UTC)[reply]

Too late buddy, now we have the old version on top, plus the new version under. So what is needed is how to integrate things well.

I would oppose any huge rewrites without prior discussion on the talk page if anybody would consider doing so. Let us take things easy and discuss things in advance. Oleg Alexandrov (talk) 01:09, 3 January 2006 (UTC)[reply]

Syntax: I think you're being a little too critical here. Some people do care about affine and symmorphic space groups and I think it's good that these things are covered. Personally I don't care much about this, I'm more concerned with practical use. But it's there, let's keep it. Some work _is_ needed on both parts though, I'll see if I have time sometime later today...O. Prytz 06:41, 3 January 2006 (UTC)[reply]

This article makes no mention of the symmetry operation x y z = -x -y -z, called an "inversion center" or "center of symmetry". While it could be considered a subset of improper rotation (rotoinversion rather than rotoreflection, in this case) it is usually classed by itself. The presence (or absence) of an inversion center is an important way to classify space groups in X-ray crystallography for a couple of reasons. The article would be much improved by at least mentioning this. M.Dickman 09:07, 14 August 2007 (UTC)[reply]

I've brought back some old text and put it at the start of the article under the heading 'Space groups in crystallography'. All the text from the previous version is still there, but I've put it under the heading 'Group theory'. I don't propose keeping things exactly as they are now, but Patrick: what do you think of structuring the article along these lines? O. Prytz 07:48, 2 January 2006 (UTC)[reply]

Ok.--Patrick 10:55, 2 January 2006 (UTC)[reply]

Good! I belive this arrangement will work. I also agree with most of the changes you've made to the beginning of the article, but I'll probably make some small changes eventually. O. Prytz 18:45, 2 January 2006 (UTC)[reply]

I would suggest forking off most of the text in "group theory" as mathematical treatment of space groups or something like that, leaving here a shorter, and more elementary version. I believe this is the only way of keeping all the text Patrick wrote, as it is really big, and it does not read as well as the much smaller first section. Oleg Alexandrov (talk) 16:47, 2 January 2006 (UTC)[reply]

And some text could go to screw axis. I believe we have enough material to write a nice article on that, where we can expand a bit beyond what is now in space group about screw axis. Oleg Alexandrov (talk) 16:50, 2 January 2006 (UTC)[reply]

I agree that the group theory text still 'dominates' the article a bit, but I'm not too worried. However, I agree that we should move the parts about screw axis and glide plane. O. Prytz 18:45, 2 January 2006 (UTC)[reply]

Ok, I've had a look at the first part of the article and want to suggest some changes. I haven't made the changes to the article itself yet, but rather copied the text here and edited that copy. I've added sections and rewritten a little bit. Regarding notation: the old version stated that two types of notation are used: the Paterson notation and Schönflies. I believe it should be Hermann-Mauguin and Schönflies. I haven't found refrences supporting the name "Paterson notation", although I don't have the International Tables in front of me. Other than that, I've only corrected an error in the description of glide planes. What do you think? O. Prytz 20:19, 4 January 2006 (UTC)[reply]

I think it is fine to move it over, then we can more easily check the differences. The example should link to Rhombohedral and we should explain how it fits in in that page. More examples would be better too.--Patrick 00:23, 5 January 2006 (UTC)[reply]

Ok, I've changed the first section to my edited version. I haven't added the things you suggest yet, but I agree that more examples would be good. O. Prytz 11:17, 5 January 2006 (UTC)[reply]

I've created the articles Screw axis and Glide plane using some of the text from this article. These articles are stubs and should be expanded somewhat so please go ahead. I hope they don't need to get too mathematical, we should at least keep an introductory text at the level of the current text.

As a follow up, I've removed the screw axis and glide plane paragraphs in the group theory section of this article. Hope that's ok. Patrick: would it be at all possible to shorten the group theory text any further, and instead refer to another article?O. Prytz 12:08, 8 January 2006 (UTC)[reply]

I do not think that is needed, many artices are longer, and there is not a clear subtopic suitable for splitting off.--Patrick 00:28, 10 January 2006 (UTC)[reply]

There are a number of other articles (and redirects) which to me seem to confuse various topics related to crystal symmetry. Here are some examples:

• Bravais lattice redirects to Crystal system. This is incorrect. A crystal system refers to the lengths and angles between the lattice vectors. There are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic (there is an alternative definition giving hexagonal and rhombohedral instead of hexagonal and trigonal). A Bravais lattice is a combination of one of these crystal systems with a lattice centering (P, I, F etc). There are 14 bravais lattices. The two are clearly related, but should have separate aricles. The current Crystal system article mixes the two together.

• Crystal class redirects to crystal system. This is also incorrect. The term crystal class is related to (eqivalent to?) the term point group, or crystallographic point group if you will.

• Unit cell redirects to Crystal structure, which at best is inaccurate. A redirect to Bravais lattice would be more correct, although there probably should be a separate article.

• The Crystal structure article contains several inaccuracies related to the points mentioned above.

I'll probably start editing several of these articles, so this is a heads up to anyone wanting to follow up and check the changes I make. I'll do my best to check all changes with the Iternational tables of crystallography and other sources, but there's bound to be some inaccuracies in the things I do. O. Prytz 16:37, 8 January 2006 (UTC)[reply]

Some time ago I have combined several pages into Crystal system, because of the strong relationships, and the convenience of having all this related information in one place. A redirect is not a claim that two subjects are the same, but it leads to the article where the subject is covered.--Patrick 02:45, 9 January 2006 (UTC)[reply]

I will argue that one has to be really careful with merging articles. You see, a big article has the disadvantage of being hard to look up information in it. Also, you never know when the next editor comes along, and decides to trim some things. That may be a very reasonable decision, but that next editor may not know that plenty of other articles redirect to this one.

That is to say, if you do want to create one single article to treat a lot of stuff, that's fine with me. However, wiping a bunch of non-trivial, well-written articles and making them into redirects to this one, may be a mistake. It never hurts to have clear (even if small) articles about individual concepts, even if there exists also an article reflecting the big picture and duplicating that information to that extent. Oleg Alexandrov (talk) 03:24, 9 January 2006 (UTC)[reply]

That is fine. Some tables can be put in templates which are called from more than one page, see also Talk:Crystal system.--Patrick 11:30, 9 January 2006 (UTC)[reply]

See my comments on the crystal system talk page... O. Prytz 08:15, 9 January 2006 (UTC)[reply]

I removed the attention tag as I think the article has gotten a lot better. Some work is still needed though. What do you guys think? O. Prytz 07:28, 10 January 2006 (UTC)[reply]

The article refers to Space group notation that is not fully explained, i.e. R and others.--Shakujo 04:31, 29 January 2007 (UTC)[reply]

there are almost 230 unique space groups describing all possible crystal symmetries. but when it comes to protein crystallography why there are only 32 space groups considered??? —Preceding unsigned comment added by Intelligeno (talk • contribs) 08:49, August 30, 2007 (UTC)

اناميسي من ليبيا احب بشلونة واتمنا لسبيستون النجاحوالتوفيق واتمنا الفوز لبرشلونة امام هرنان —Preceding unsigned comment added by 41.252.5.62 (talk) 21:57, 9 July 2008 (UTC)[reply]

The key idea is that (naturally occurring) proteins are chiral. Therefore, a protein crystal cannot have exact mirror symmetry, which eliminates most space groups as possibilities.

Proteins are chiral because they're composed of chiral amino acids; the alpha carbon atom (C^α) is chiral in almost all amino acids. Exceptions include glycine (Gly) and some rare amino acids such as 2-aminoisobutyric acid (Aib). Other carbon atoms in amino acids can be chiral, e.g., C^β in threonine and isoleucine.

A crystal of an achiral polypeptide such as (Aib)₃₀ or (Gly)₃₀ or (Aib-Gly)₃₀ could exhibit a space group with a mirror symmetry. However, such molecules generally wouldn't be counted as proteins, since they seem unlikely to fold in the usual sense. The (Aib-Gly)_n molecule might be an interesting system to study, however, since it shares some similarities with silk and might fold into beta sheets. Proteins (talk) 11:02, 4 August 2009 (UTC)[reply]

"A definitive source regarding 3-dimensional space groups is Hahn (2002)."

Is it appropriate to recommend a textbook in the lead? I haven't seen another article have such a line in its lead; it almost seems like an advertisement. Is there any objection to me removing it? JHobbs103 (talk) 22:03, 29 May 2010 (UTC)[reply]

It is more a handbook than a textbook, really. Honestly, there is not much of a market for two compendia of all the spacegroups and their symmetries and whatnot, so I doubt that there is anything comparable (disclosure: I do not think I added that text, but that is the book I use). Certainly coming from the IUCR gives it a certain gravitas. I changed it to the title of the book, but I am not terribly wedded to keeping the mention if you think it really detracts from the article. - 2/0 (cont.) 00:42, 30 May 2010 (UTC)[reply]

I'm considering adding to that table how many:

• Arithmetic crystal classes
• Crystal systems

I propose this explanation of the difference between arithmetic and geometric crystal classes: A geometric crystal class is for a point group considered in isolation, while an arithmetic crystal class is for a point group with some orientation relative to the lattice. The distinction between arithmetic and geometric crystal classes is easier to visualize in two dimensions, in the wallpaper groups. I'll list them by geometric crystal class (point group), with arithmetic crystal classes in sublists:

• C1: p1
• C2: p2
• D1:
  • Reflection line oriented along a lattice direction: pm, pg
  • Reflection line oriented halfway in between lattice directions: cm
• D2:
  • Reflection lines oriented along lattice directions: pmm, pmg, pgg
  • Reflection lines oriented halfway in between lattice directions: cmm
• C4: p4
• D4: p4m, p4g
• C3: p3
• D3:
  • Reflection lines oriented along lattice directions: p31m
  • Reflection lines oriented halfway in between lattice directions: p3m1
• C6: p6
• D6: p6m

A problem with adding these statistics is the headers -- they would grow even larger than they are now. I am thinking of using abbreviations and moving the full headers to a list below the table. That list may itself be a table, with links to the Online Encyclopedia of Integer Sequences and the like. — Preceding unsigned comment added by Lpetrich (talk • contribs) 21:05, 6 January 2013 (UTC)[reply]

Done both. For the small-dimension table, I turned most of the column headers into abbreviations, because the full text makes them too wide. IMO, the resulting table is much cleaner-looking, even though it has more columns.

I have found a problem, however. Discrepancies between various published and stated values in the numbers, especially for 6 dimensions. Could someone with good journal access please review the literature on the subject of these numbers? Some of the relevant articles are paywalled.

Lpetrich (talk) 15:24, 9 January 2013 (UTC)[reply]

I used International Tables (2006), page 720. I think they contain the most updated information. I changed some numbers for 6-dimensional space: 7103 -> 7104, 85308 -> 85311 (+30), 28927915 -> 28927922 (+7052).

I also changed the Table again - I hope, it is better to remove all explanations from the table to space after the table, because anyway, people mostly would be interested in numbers. — Preceding unsigned comment added by Bor75 (talk • contribs) 01:22, 13 January 2013 (UTC)[reply]

In the interest of clarity, it would be helpful to pick a single transliteration of Фёдоров, at least with respect to the internal consistency of the article. There are several reasons to use Fyodorov instead of Fedorov:

• The Cyrillic letter ë has the phonemic value /jo/ or / ʲo/ not /ɛ/
• The version of the article prior to my first edit contained five instances of Fyodorov and three of Fedorov
• The wiki link for Fedorov is marked-up as [[Fedorov|Yevgraf Fyodorov]]
• The wiki link goes to the page titled "Yevgraf Fyodorov"
• The article for the surname is titled "Fyodorov"
• "Fyodorov" lists the person in question and Yevgraf Fyodorov
• Fedorov redirects to Fyodorov
• A Google search for "Yevgraf Fyodorov" OR "Yevgraf Stepanovich Fyodorov" returns 12000 results
• Google auto-corrects "Yevgraf Fedorov" OR "Yevgraf Stepanovich Fedorov" to "Yevgraf Fedorov" OR "Yevgraf Stepanovich Fyodorov"
• The Google search for "Yevgraf Fedorov" OR "Yevgraf Stepanovich Fedorov" returns 29 results

In summary, Fedorov lacks internal consistency within Wikipedia and is by far the rarer spelling.

99.196.225.79 (talk) 03:39, 1 May 2013 (UTC)[reply]

I know that Cyrillic letter ë has the phonemic value /jo/ or / ʲo/ not /ɛ/ , but when Fedorov's works were published in German and English his name was spelled as Fedorov, because that time romanization of Russian was not phonetic.

• http://www.amazon.com/Symmetry-crystals-ACA-monograph-Fedorov/dp/B0006CKHAU - Symmetry of crystals (ACA monograph): E. S Fedorov
• http://www.minersoc.org/pages/Archive-MM/Volume_18/18-84-99.pdf
• Fedorov, E. S. (1895). Theorie der Kristallstruktur.

His name was always spelled as "Evgraf Stepanovich Fedorov"

If you check references you just corrected, you will see, that in both cases it is also written Fedorov, not Fyodorov.

Yes, according to modern rules Wikipedia:Romanization of Russian his last name should be spelled as Fyodorov (phonetically more correct), however, according to rules for persons Wikipedia:Naming conventions (Russia)#Names of persons "If the person is an author of works published in English, the spelling of the name used in such publications should be used.", If the person is the subject of English-language publications, the spelling predominantly used in such publications should be used. A preference is given to publications in the area in which the person specializes. Fedorov is a crystallographer. If you go to site of International Union of Crystallography http://www.iucr.org/iucr and search there you will not find any Fyodorov, but only Fedorov:

http://www.iucr.org/cgi-bin/newiucrsearch?query=fedorov&submit=search gives 19 results

while

http://www.iucr.org/cgi-bin/newiucrsearch?query=fyodorov&submit=Search gives 0 results

In other crystallography journals and in International Tables for Crystallography he is also spelled as Fedorov, not Fyodorov.

• http://link.springer.com/article/10.1134%2F1.1627430

The article about Fedorov also should be renamed and corrected to be consistent with traditional spelling for this scientist - Evgraf Stepanovich Fedorov. I don't understand why somebody wrote there "Yevgraf Stepanovich Fyodorov, sometimes spelled Evgraf Stepanovich Fedorov". This is just a wrong statement.

Search in Google is not relevant for this case, partially because wrong spelling in Wikipedia dramatically influences the whole internet. Many sites just copy information from Wikipedia.

BTW, search "Evgraf Fedorov" OR "Evgraf Stepanovich Fedorov" gives more results, than "Yevgraf Fedorov" OR "Yevgraf Stepanovich Fedorov"

If you go to Google book search

Fedorov crystallography gives 19,400 results

Fyodorov crystallography gives just 190 results

Even links from Fyodorov wiki page mostly give spelling as "Fedorov"

http://www.iucr.org/__data/assets/pdf_file/0020/749/fedorov.pdf

http://www.minsoc.ru/confs.php?cid=488 - Fedorov Session

In summary, I agree that Fedorov lacks internal consistency within Wikipedia and is by far the rarer spelling, but for this particular case spelling should be Fedorov, because it is conventional for this particular scientist to write his name as Evgraf Stepanovich Fedorov.

Bor75 (talk) 06:37, 1 May 2013 (UTC)[reply]

I changed *all* instances of Fyodorov to Fedorov per the historical, stylistic, and bibliographic guidelines.

99.196.225.79 (talk) 12:09, 1 May 2013 (UTC)[reply]

Thank you! Bor75 (talk) 16:30, 1 May 2013 (UTC)[reply]

I compiled a list of space groups with their full names, with spaces for readability, replacing redirect at list of space groups. Tom Ruen (talk) 23:17, 10 February 2014 (UTC)[reply]

I think the introdution should change to

"

In mathematics and physics, a space group is the symmetry group of a configuration in space, usually in three dimensions. A symmetry group of R³ belongs to a space group iff the subgroup of all translations in that group is generated by 3 linearly independent translations and the symmetry group has only finitely many cosets of the subgroup of all translations in it. Two symmetry groups that belong to a space group belong to the same space group iff there exists a linear transformation R such that the function that assigns to each transformation T in the first symmetry group R^-1TR is a bijection from the first symmetry group to the second symmetry group. Note that R^-1TR doesn't necessarily have to be an isometry for all isometries T; it just has to be an isometry for all symmetry operations T of the first symmetry group. In total there are 219 such symmetry groups. For each space group, either all structures that belong to that space group are chiral or none of them are. Some authors consider chiral copies of a space group to be distinct, that is, there define a space group pretty much the same way except for replacing the criterion "there exists a linear transformation R such that" with "there exists a non-inverting linear transformation R such that". 11 of the members of the first definition of a space group can be split into 2 members of the second definition leaving a total of 230 space groups according to the second definition. Those 11 space groups are called chiral space groups. Although all structures that belong to a chiral space group are chiral structures, not all chiral structures that belong to a space group belong to a chiral space group. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space.

In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography (Hahn (2002) harvtxt error: no target: CITEREFHahn2002 (help))." to better explain what a space group is.

Can anyone find sources for any of the information I added? Blackbombchu (talk) 01:17, 24 November 2014 (UTC)[reply]

Comparing the definition of "group" in the context of this article to the mathematical definition of a group, the requirements of "space groups" are stated as properties of certain "group actions" of mathematical groups. For readers only familiar with the properties of a mathematical group, it would helpful to point out that term "group" in this article may often refer to what mathematicians call a "group action". — Preceding unsigned comment added by Tashiro (talk • contribs) 15:55, 13 February 2015 (UTC)[reply]