Talk:Point group

Rotoreflection?

Is there a difference between Rotoreflection and Improper rotation, in the opening of the article? (And since this is about higher dimensional geometry, shouldn't this be removed?--GuenterRote (talk) 17:05, 27 January 2013 (UTC)

I don't think there is a difference. I simplified the sentence and linked improper rotation, unless someone else can clarify it. From Coxeter's reflection POV, a rotation is the product of two reflections, and an rotoreflection or improper rotation is the product of 3 reflections, so this applies to any dimensional geometry 3 or higher. Tom Ruen (talk) 00:41, 28 January 2013 (UTC)

Override of redirect

Point group had been redirected to crystallographic point group, suggesting either that these were synonyms, or that the crystallographic version was so much more important that the more general concept did not warrant discussion. In fact, that article does not even acknowledge the distinction.

Yet the more general point group is essential for purposes outside crystallography. Chemistry, in particular, describes the symmetries of molecules and bonds, where the crytallographic restriction is an unwanted intrusion. Mathematics also needs the full generality to talk about finite symmetry groups with a fixed point, and it is unacceptable to omit the icosahedral group, say, because of the crystallographic restriction.

Perhaps some of the pages that link to point group ought to be edited to link to the more appropriate page. Of course, crystallographic point group could be made a subset of this article if it has little additional data to offer. But the status quo ante was an inappropriate absorption.

KSmrq 2005 July 4 23:52 (UTC)

Merge?

I've suggested that the article on point groups in three dimensions be merge into this one. See the talk page of that article for discussion. O. Prytz 00:02, 6 June 2006 (UTC)

Sorry; just saw this suggestion. Even though I originally wrote this article, I took it off my watchlist after Patrick persistently mangled its accuracy and chopped it to pieces. You may find that an early version would be more satisfactory. I certainly would. For example, the current article says "The 3D discrete point groups are heavily used in chemistry", introducing a mistake not found in the original, which says "The 3D point groups are heavily used in chemistry". What's the difference? The symmetry of the hydrogen atom, for one! (Or of any linear molecule, such as acetylene.) I tired of arguing with Patrick over numerous pages, and could no longer bear to look at the butchery here. (If you scan through the archives of his talk page, you'll find he's still a persistent source of trouble.) The meek may inherit the Earth, but the bullheaded seem to win on Wikipedia. --KSmrq^T 17:02, 27 June 2006 (UTC)

Ignoring the above (what else can I do?) I would suggest that the whole area of symmetry space group point group crystal system etc. etc. needs re-engineering from the ground up. Its a problem with the Wiki approach that one perspective on a broad topic is not the same as another, but I feel we need one overall perspective into which to fit the individual articles, allowing a convenient fit with the fields of mathematics, chemistry and protein crystallography. In conclusion, don't merge; redesign. Lets start a list of articles that need to at least coordinate their content. Perhaps a hierarchical TOC for the whole 'topic', then we can decide what information goes into what articles... Its a big job. --Dan|^(talk) 07:51, 30 June 2006 (UTC)

Right. Sorry for the rant; I prefer to look for solutions and ways forward, not just vent frustrations.

Reorganization and some rewriting could be helpful. The good thing is, we have a fair amount of content to work with. The bad thing is … the same!

There are interesting challenges in how to factor, because mathematics, physics, chemistry, and other disciplines are entangled. I got sucked in through the article on wallpaper groups, as a little recreational diversion. Then it turned out we needed an article on the crystallographic restriction theorem, so I wrote that. Along the way I found that point group was being redirected to crystallographic point group (Patrick's myopic perspective), so I killed the redirect and created new content. I've had a recreational interest in minerals and crystals (and a friend with a rock shop), so I've got some awareness of that area; I also looked into crystallography as used for biochemistry, physics, and so on; and I'd been reading on the use of group theory and symmetry in physical chemistry. At the heart of it all is the mathematics — for me, anyway. But we have to (and I want to) write for all these interests.

I'll do what I can to support a redesign, but I do have some other Wikipedia things I'm working on, as well as a life. ;-)

If you want to take the lead in organizing a list of articles, great. I think that would be a good thing even if nothing else is done. I wonder if somewhere there is a tool that can reveal the link structure of articles; but we can always crawl through and assemble a map of connections by hand if we must.

Actually, it would be good to check with existing projects for mathematics, physics, chemistry, or whatever and see if something already exists that we don't know about. And if not, figure out where to collect the results. Drop me a note on my talk page when you're ready for help. --KSmrq^T 15:49, 30 June 2006 (UTC)

A reworking of the articles relating to crystallography is sorley needed. I've been tinkering a little here and there, but I feel sorry for anyone new to crystallography reading these articles. A prototype for a hierarchy for the crystallography articles might look something like this:

Crystal structure

Crystal system

Cubic crystal system

Tetragonal crystal system

etc...

Bravais lattice

Crystallographic point group

Space group

If we think along these lines, the crystal structure article should briefly introduce the four sub-topics, and tie all of them together. The articles for each sub-topic should, of course, be more in-depth. The articles we have today aren't completely useless, but they're confusing and inconsistent, and I'd be happy to help in reworking them. My problem is that I have very little interest in the mathematics of e.g. group theory, so any contributions I make would be more on the practical use of these concepts in crystallography. Furthermore, I feel pretty strongly that any mathematics should be kept to a minimum. Others don't agree. The way I see it, many articles might have to consist of two quite separate parts (description of practical use, and the mathematical basis), or perhaps split into two articles. In a way I'm quite content that we have two articles on point groups: crystallographic point group and point group (although the third on point groups in three dimensions is probably redundant). Should we start by working on the hierarchy above and try to agree on a set of articles that we want to work on? O. Prytz 19:59, 3 July 2006 (UTC)

I sympathize with a practical interest, but it is not practical to write for an encyclopedia without addressing a broad range of interests. Even were we to focus on crystallography alone, a great deal of mathematics would be natural, even necessary.

Sometimes the solution is to split articles. However, the style manual for mathematics recommends that a mathematical article should do its best to begin with material suitable for a wider audience, then build in abstraction and generality as it goes. For many articles that may suffice. This demands good writers who are sensitive to novices, practitioners, and higher mathematicians, but it can be done.

Be sure to look at the category, and at the crystallography article itself. Some time back I added a number of external links to the article. This one should be of particular interest to mineralogists, while this one (found through the IUCr link) would be of more interest to biochemists. I did not spend time on the contents of the article, which remains weak. If I might suggest a miniproject, go to work on that. --KSmrq^T 00:52, 4 July 2006 (UTC)

"discrete"

This is mentioned twice near the beginning of the article as a modifier to "point group", but is never defined. I think this should be done so, or removed. Sounds like from the previous discussion it is important; I take it excludes groups like C_∞, yes? Baccyak4H 19:15, 2 November 2006 (UTC)

duoprism groups

Tom, I don't get it: the table seems to say that changing the colors of the dots changes the group order. —Tamfang (talk) 05:10, 16 July 2012 (UTC)

Sorry for the slow reply(!) There's color and node indices. So if two nodes have the same index (or color), that means there's a symmetry operation that maps one onto the other, and doubles the symmetry (or factorial order increase if multiples). So like in Coxeter notation o o o o --> [2,2,2] has order 16, and [4,3,3] has order 384, 24 times higher. But with 4 same-"colored/indexed? nodes, this is represented as [3,3[2,2,2]]=[4,3,3], where the order [3,3] symmetry family exists in all permutations of 4 nodes. And the simpler case of doubling, like [ [3,3] ]=[1[3,3]]=[4,3], with [1] as a bilateral symmetry within [3,3] that raises it to [4,3]. Tom Ruen (talk) 20:45, 10 January 2013 (UTC)

E6×2

The E6 Coxeter diagram has mirror symmetry, so there exists an E6×2 group , which is not included in the page.

The 1_22 polytope has this symmetry, for example. --Goomba1729 (talk) 21:57, 18 March 2021 (UTC)

"Conway" notation?

What's "Conway" notation, which has its own table column name in Point group#Three_dimensions? It links to Conway the guy, not the notation. It doesn't appear related to Conway notation (knot theory). Orbifold notation already has its own column. This entire table was added here in 2011 by 75.146.178.58. Please add a reference or link for this notation if it is current; otherwise I'd like to remove the column. –MadeOfAtoms (talk) 03:45, 2 October 2021 (UTC)

I removed the Conway notation column. –MadeOfAtoms (talk) 06:45, 23 November 2021 (UTC)

one point fixed

I had difficulty working through the first sentence, which at this reading says:

In geometry, a point group is a mathematical group of geometric symmetries (isometries) that keep at least one point fixed.

Symmetries of what? Isometries between what and what? Keep one point of what fixed? Does it have to be the same point for all symmetries in the group, or can different symmetries qualify because they fix different points? Does the criterion of keeping the point fixed define the group, or can I leave out some symmetries that fix the point?

After looking at enough examples, it sort of filtered through to me that these are all self-isometries of Rⁿ for some fixed n, and the "keep one point fixed" clause is supposed to mean one particular point, which is the same for all elements of the group, and can be taken to be the origin. Then we're looking at subgroups of the group of all isometries that fix the point; we don't have to take all such symmetries in a particular group.

But I don't think that's obvious at all. By omitting mention of what the symmetries are on, it invites readers to spin various possibilities. For example, suppose I look at self-isometries of the torus T². Then a 180-degree rotation in the plane of the torus does not fix any point of T², so it can't be in any point group? Apparently that's not what is intended, but it's not so clear from the text.

I'm not sure how best to reword. It seems to me that the use model being aimed at is, you have say some molecule or crystal represented as a subset S of R³, and you want the group of self-isometries of R³ that fix S setwise. It's really a setwise stabilizer, whereas the current language suggests a pointwise stabilizer. Maybe we could clarify by focusing on that? --Trovatore (talk) 00:07, 19 March 2022 (UTC)

Please see if this rewording helps, and repair it or complain as appropriate.

As usual for the lead, it's a balancing act between being precise, being complete, and giving the general idea to readers unfamiliar with the subject. For perspective, several related articles have similar issues and ideas for how to explain things: see Space group, Symmetry group, Isometry group, Point groups in 2D, in 3D, in 4D, and Molecular symmetry.

I'm not sure if "self-isometry" would be understood as an isometry to the same metric space, so I hope that the context now makes it clear enough.

Point groups in spaces other than Euclidean space, such as in C^d, must be of interest too but I don't have knowledge or references on those, so need to leave it to others.

A subset of R^d is a common use model of an object, and I agree that a point group is a setwise stabilizer of such a set, when the set has that symmetry. The same is true of more general space groups. And it is also common to use more general object models, such as an arbitrary function of R^d, which allows things like vector fields and any properties at each point to be specified. Discussion of the relationship between subsets of R^d and the point groups that describe them is interesting (and not so simple: any subset having C_∞ symmetry also has C_∞v) but I'm unsure where it belongs. –MadeOfAtoms (talk) 08:35, 3 April 2022 (UTC)

Thanks MadeOfAtoms; I think it's a lot clearer now. I'm still not clear on the reason to emphasize a common fixed point for the isometries; it'll be true for setwise stabilizers of compact sets in Rⁿ but it doesn't seem like a defining characteristic. But if this is the way the subject is typically introduced then I suppose that's fine.

On the general issue of lead sections, I would say that precision can be sacrificed but accuracy cannot; lies to children have no place in Wikipedia. But I don't see any such issue here (and indeed I didn't even before; my concern was not that the lead was over-simplified but that it was hard to understand what it was getting at at all). --Trovatore (talk) 18:16, 3 April 2022 (UTC)