Talk:Cayley transform

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Mathematics rating:	B Class	Mid Priority	Field: Algebra
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Some suggestions: proceed from elementary to advanced, develop role in Lie theory, enhance lead. Is there more history that we can dig up? See also the discussion below. Geometry guy 20:32, 9 June 2007 (UTC)

[edit] A-class?

This article doesn't have a lead to speak of, should it really be A class?--Cronholm144 04:51, 22 May 2007 (UTC)

Ah, a least a hint. I've seen numerous B-class ratings of articles with no indication of anything that might be improved, including the one here. Apparently, by default, the highest rating is a B, which named offences can lower.

This article includes a little history, thorough references, lovely illustrations, and distinct sections covering disparate (but analogous) forms of Cayley transform. It is not clear that puffing up a lead would add value. I would be delighted to see evidence to the contrary; but even A-class articles can be improved.

Incidentally, I abhor the current FA/GA process, and do not wish to butcher any article I care about by subjecting it to that. If an A-class or Bplus-class rating increases the likelihood of such an assault, I would choose perpetual Start-class instead! --KSmrq^T 06:24, 22 May 2007 (UTC)

Fair enough, but to receive an A rating the article has to go through an in house evaluation before it should be ranked as such. I don't think that outsiders will get involved. However all articles that I have seen go through the GA and FA review process come out better articles. (Nevermind...)(5-26-07) Finally, the article, as per the FA criteria, is to have a lead which summarizes the topic at hand... or some such. Since an A class is considered an almost FA article I think it should have this as a prerequisite. I tend to obey the traditional rules when rating an article, there are many otherwise Bplus articles that I have given a start class rating because the lack a single reference. That is my two cents anyway.--Cronholm144 06:41, 22 May 2007 (UTC)

P.S. I just saw the history of the article... you might be a touch too close to this article to be objective, maybe we should get a third person evaluation? I.E. Oleg, Jitse, Lambiam, Salix, G-guy, etc...--Cronholm144 06:43, 22 May 2007 (UTC)

My opinion derives in part from knowing how much time I spent sprucing up this article (which was tangential to something else I was working on). That included tracking down references (especially on-line ones), making pictures, sorting out the different definitions, and so on. In other words, I tried to include the things I want in an article. I believe the result speaks for itself; compare it to the typical B-class article. That said, it might be a good idea if every rating were certified by three respected editors.

I'm puzzled by your assertion that an A rating requires a formal process; what I read in our description said nothing to suggest this, but only mentioned strengths and weakness typical of each rating. In my view, this article does not have the weaknesses listed for lower ratings. However, I have generally stayed far away from "official" ratings.

I'm open to suggestions, but the intro to this article tackles an awkward challenge: disambiguation. I began editing the article because I needed Cayley's original definition, which concerns matrices; the article had only the complex mapping. Perhaps there is something helpful to say that applies to all the definitions, but I thought it best to merely outline the variations, then dive into each. Ignoring "official" dictates (which are hardly carved in stone), if you feel that important material is missing from the intro, please feel free to add it, or to make a note and downgrade the rating.

Incidentally, my personal guidelines remind me to include intuition (and motivation), examples, counterexamples, and connections. They also recommend pictures and humor (a challenge in an encyclopedia). And despite the wish to be complete, I'm convinced people learn best if they are enticed to work out some details, to think actively about the subject.

I'm tempted to say (much) more about GA/FA and references, but that would lead us away from the article — which is what this talk page is for. :-) --KSmrq^T 09:12, 22 May 2007 (UTC)

This might help, Wikipedia:WikiProject_Mathematics/A-class_rating --Cronholm144 09:30, 22 May 2007 (UTC)

Yes, it does help; thanks. I have downgraded to B+, and will experiment with a peer review for a different article first. --KSmrq^T 11:21, 22 May 2007 (UTC)

Actually, there are a few standard things about Cayley transform that are not even mentioned (such as the Cayley transform in structure theory of semisimple Lie algebras), so by the completeness standard I would argue for B-class, no offense meant. Arcfrk 08:13, 23 May 2007 (UTC)

No offense taken. As I say on my user page, "Well-founded praise and (gentle) criticism of my efforts are welcome."

Some omissions are from inadequate expertise, and some from editorial discretion. There is more I could say about uses of certain forms of the transform, but I chose instead to keep the article more tightly focused. One thing that became clear as I browsed through diverse literature is that each specialization treats its own version as "the" Cayley transform, without so much as a nod to other definitions. If omitting a thorough exploration of implications and uses in the different realms means the article is doomed to a B+ class, I can live with that choice. Imagine if an article on matrices had to discuss all their different interpretations and uses. And this topic is worse, because the different uses seem not to illuminate each other. Perhaps a modest extension here would be appropriate, but it seems unlikely I will do all the writing.

To assist other editors, perhaps you could list any important properties and uses you think deserve mention. (It can be hard to see gaps in my own writing; I commonly set papers aside and come back to them to try to help me see them with fresh eyes.) --KSmrq^T 09:44, 23 May 2007 (UTC)

[edit] Rotation of the Riemann sphere

An article doesn't have to be self contained. Doesn't Riemann sphere provide adequate context? It explains both stereographic projection and the relation to Mobius transformations. So why not add an additional nice tidbit here that helps to visualize this particular mapping? nadav (talk) 09:54, 25 May 2007 (UTC)

You misunderstand my complaint. If we try to understand this as a rotation of the complex plane, as in Euclidean geometry, we have the wrong idea and only confuse ourselves. Adding a point at infinity doesn't help. I know that's not what you meant, but the problem then is what does this mean? Are we meant to model the Riemann sphere as a unit sphere stereographically projected onto the xy plane? (And does it matter if we project from top or bottom?) How is a reader to know that? If they can get that far, they may eventually manage to figure out what a rotation must be that maps, say, (0,0,1) to (1,0,0), (0,−1,0) to (0,0,1), and another (independent) pair.

For our third pair, nothing suitable is mentioned in the article, so we're left to find something on our own. We can reason, for example, that a rotation of the sphere must leave two points fixed, and solve w = ^w−i⁄_w+i to obtain ¹⁄₂(−1+√3)(−1+i) as one of the fixed points. Then using the map from plane to sphere found at stereographic projection, we can convert this to a sphere point, (−r,r,−r)/(2+r), with r = 1+√3, and use that and itself as our third pair. Then with these three pairs we can (if we know how) obtain the matrix

Of course, this also tells us that we could have used (−1,0,0) mapping to (0,1,0), namely −1 mapping to i, as our third pair (a fact I have now added to the article).

For what looks like a one sentence throw-away remark, this rotation idea demands a great deal from a reader. Frankly, I doubt many will find it all that helpful or enlightening, in the form presented. And it would be a structural disaster to try to explain enough at the point where the remark was introduced. Sometimes it is difficult to see things as a reader will; we assume that since we know what we mean, so will they. If only! --KSmrq^T 13:56, 25 May 2007 (UTC)

Your explanation is quite thorough. Point taken and noted. nadav (talk) 14:25, 25 May 2007 (UTC)