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February 2010 (from Likebox talk page)

Thank you for your contributions to Misplaced Pages. When you make a change to an article, please provide an edit summary for your edits. Doing so helps everyone to understand the intention of your edit. It is also helpful to users reading the edit history of the page. Thank you. JohnBlackburne_deeds 19:36, 19 February 2010 (UTC)

I use edit summaries almost always, except under exceptional circumstances or when occasionally I forget. Thanks for the info.Likebox (talk) 19:39, 19 February 2010 (UTC)

See my lengthier follow-up as there's a bit more to it. It's not a big deal but something that will make your edit summaries clearer and probably be easier for you.--JohnBlackburne_deeds 19:47, 19 February 2010 (UTC)

I generally use edit summaries only for main page edits but not talk page edits as is long standing custom here. You can see that all my main page edits have proper edit summaries, while talk page edits usually don't need a summary, since the comment is the summary. I will start to use talk-page summaries from now on.Likebox (talk) 19:49, 19 February 2010 (UTC)

I'll use new section: blah blah blah as edit summary for talk pages, since I don't know what "click +" means. Perhaps you mean new section?Likebox (talk)

By that I mean that one of the tabs at the top of the page is simply marked "+". I have at the top of this page the following tabs

 User  Talk                              Read  Edit  +  History  ☆  ▾

The star adds or removes a page from the watchlist. The triangle is a drop down menu with various tools like 'Move'. The only other symbol is a "+". Click on it to create a new section. It should be there in the default settings, or if you have beta enabled (as I have).--JohnBlackburne_deeds 20:22, 19 February 2010 (UTC)

Rotation_matrix

JohnBlackburne, I recently edited the rotation matrices because I believe they are incorrect. I believe you reversed my change. The - sign should be on the other sine function. This is what I was taught in class, and also what Wolfram Alpha agrees with. Could you explain why you changed it back, or explain what I am misreading? Thanks. Hackcamp (talk) 02:27, 24 February 2010 (UTC)

The reason is by the mathematics on the rest of the page your changes were wrong: more particularly they disagreed with the paragraph immediately below which describes the sense of the rotations precisely. For other rotations it might be different. There was a discussion at Talk:Rotation matrix#Sign error? where I tried to clarify the mathematics behind it. See especially the end of the discussion.--JohnBlackburne_deeds 09:05, 24 February 2010 (UTC)

I am sorry, but I still do not see what you mean. I have been taught this way, everyone I know has been taught this way, Wolfram describes it this way, all my books describe it this way, and http://en.wikipedia.org/Rotation_representation describes it in this way. I am obviously missing some sort of convention you use that differs from the rest of this, and the paragraph under the matrices isn't to helpful. Could you please explain further. Hackcamp (talk) 17:03, 24 February 2010 (UTC)

I mean particularly the reference to the 'Ambiguities' section which touches on the two ways of doing it but doesn't really explain. I would note that Wolfram is a source of varying quality, so I tend not to rely on it for anything. It too gives both ways to rotate though, in 2D. Then decides to focus on the other way in 3D. Rotation Matrix instead describes object rotation: it explicitly says

• "These matrices represent counterclockwise rotations of an object relative to fixed coordinate axes, by an angle of θ",

and then gives the direction. Mathworld is describing the rotation of the axes. But I suggest taking the discussion to the article's talk page, where others are likely to see it and perhaps fix it by e.g. rewriting the ambiguities section so it's clearer what it means.--JohnBlackburne_deeds 17:13, 24 February 2010 (UTC)

Congrats

I just saw that The Wolves in the Walls had been featured on DYK. As a member of the Children's Literature Project I always love to see deserving books get DYK recognition. Congrats on the work you did to get the article there. Barkeep49 (talk) 19:20, 24 February 2010 (UTC)

DYK for St Mary and St Cuthbert, Chester-le-Street

On February 25, 2010, Did you know? was updated with a fact from the article St Mary and St Cuthbert, Chester-le-Street, which you created or substantially expanded. You are welcome to check how many hits your article got while on the front page (here's how, quick check ) and add it to DYKSTATS if it got over 5,000. If you know of another interesting fact from a recently created article, then please suggest it on the Did you know? talk page.

Materialscientist (talk) 12:11, 25 February 2010 (UTC)

DYK nomination of Erlebnispark Tripsdrill

Hi John, thank you for your comments on my DYK nom. I've added Alt 2 to take out the word "daredevil" and the picture is back to the nicer one of the roller coaster. Would you be able to have a look and comment? Thanks! Maedin\ 21:55, 28 February 2010 (UTC)

Thank you, :) Maedin\ 22:54, 28 February 2010 (UTC)

Evil?

In your comment on the currently pending case, you say I called people evil. Why?Likebox (talk) 04:30, 5 March 2010 (UTC)

After scratching my head, I finally found the only place the word "evil" occurs. In case you have a hard time with English idioms--- the "root of the evil" or the "root evil" is the thing that makes other things go bad. It does not imply that any person is evil.Likebox (talk) 04:40, 5 March 2010 (UTC)

It was not a word I made up, and I added a diff later to make that clear. You wrote "I think this is the root of all the evil.", implying some part of this is "the evil" (the phrase, which I'm well aware of, is xx" is the root of all evil", quite different).

More generally processes like this rest on evidence. At the top of the page it says

3. State your request in 500 words or fewer, citing supporting diffs where necessary.

So e.g. if you think an editor has done something you object to say what and post a diff, or multiple diffs as appropriate. Accusations that something is evil or corrupt are very strong, so need strong evidence to back them up. Diffs don't count towards your word count, use as many as you like. Just links to pages are not enough: diffs are needed to show individual editors actions. Look at Sandstein's statement for a good example: you may not agree with his request but he presents his arguments well with diffs.

Long posts without diffs contribute little. Arbitrators do not have time to go to pages, look through long histories and work out who wrote what. Long posts with accusations with no diffs are even worse. They are not evidence, except of the contributions of the poster. i.e your statement with few diffs but with lots of accusations make other editors pay little attention to your posts. Unless there is something in them which they can use as diffs to make a point, as I did.--JohnBlackburne_deeds 10:53, 5 March 2010 (UTC)

"The root of all the evil" does not call any person evil--- you don't need evil people to have evil outcomes.Likebox (talk) 14:58, 5 March 2010 (UTC)

Actually you do, as evil is a matter of intent. But we're getting into what single words mean, which is never good. My more general point still stands:

If you (or any editor) wants to make accusations of evil, or corruption, or stupidity (repeated implications others are simply too ignorant of an old French novel to participate), you should name editors, either explicitly or by simply giving diffs which are easily understood. Without diffs your accusations carry little weight as they are just that - accusations. With diffs other editors can quickly check if what you are accusing editors of is true. You might need to modify your accusations so they're supported by the diffs, but then at least you can be sure what you're saying is well supported. Otherwise your statements are very weak and can be called into question.--JohnBlackburne_deeds 15:19, 5 March 2010 (UTC)

What are you talking about? "The root of all evil" is an expression saying "this is the root of all the bad stuff". There is no implication of bad intent, the "evil" is in the system, not any individual. There are obviously no evil people involved here. As far as Tombe's hyperbole, he may go on too long, but his points are valid.Likebox (talk) 16:19, 5 March 2010 (UTC)

You didn't write "The root of all evil". And on the definition of "evil" see above.--JohnBlackburne_deeds 16:27, 5 March 2010 (UTC)

I wrote "this is the root of all the evil", and the same comments apply. The evil is only getting worse, since ArbCom is intent on enforcing its rules. This is the end of science on Misplaced Pages, I just wish I were wiser and could have seen it coming years ago.Likebox (talk) 16:29, 5 March 2010 (UTC)

Evil to most people means a malicious intent to do a person or creature harm. If you just mean WP will be harmed (or damaged, or disrupted) then say so. Evil means something quite different, and provokes reactions like this.--JohnBlackburne_deeds 17:26, 5 March 2010 (UTC)

It is not my responsibility to imagine every possible incorrect reading of text that I write. Plus, I like the reactions you have. It's case in point.Likebox (talk)

RE: Thanks !!

Not a problem. Kind regards, Spitfire 21:34, 18 March 2010 (UTC)

Talkback

Hell In A Bucket (talk) 15:30, 19 March 2010 (UTC)

Not meaningless

I don't think this is a meaningless statement; it's just not so clearly written. If you take the number of primes less than n, divided by n, you get the ratio of primes to natural numbers, all subject to the constraint of being less than n. The take the limit of that is n grows, and you get 0. I.e. you can make it as close to 0 as you want by making n big enough. That takes some work to prove, but it's fairly elementary and widely known. It is often stated by saying that the density of primes is 0. Michael Hardy (talk) 21:51, 20 March 2010 (UTC)

We took it to the talk page a month ago, and a subsequent edit added it back in better phrased. You should take a look at the article now, as it's changed a few times since, to see if it makes sense now or if it can be improved.--JohnBlackburne_deeds 22:01, 20 March 2010 (UTC)

bavaria

Conversation moved to Talk:Bavaria#images_2

--JohnBlackburne_deeds 00:36, 23 March 2010 (UTC)

Yaw & Pitch axis (music)

• "rv: redlink, and not clear how someone typing "Yaw..." will be looking for the musical term"

On Misplaced Pages there are things called redirects, pages which send readers to articles from an alternative title. Thus if one types "Yaw, pitch, and roll" one is taken to Yaw, pitch, and roll, if one types "Yaw..." one is taken to a search (or blank) page, while if one types "Pitch axis" one is taken to Yaw, pitch, and roll since Pitch axis redirects there. See: Help:What links here.

Thus while it seems unlikely that someone typing "Yaw..." would be looking for the musical term or for "Yaw, pitch, and roll", someone typing "Pitch axis" may actually be looking for Pitch axis (music) but they will be taken to Yaw, pitch, and roll. Thus the hatnote. See: Misplaced Pages:Disambiguation. Hyacinth (talk) 18:53, 6 April 2010 (UTC)

I was going to suggest a disambiguation page is a better solution, and you've made one which I've expanded slightly. Better to stop readers before they get to a page full of maths if all they're looking for is a page on music theory.--JohnBlackburne_deeds 19:11, 6 April 2010 (UTC)

Macau introduction

Could the introduction to the Macau article only have the traditional Chinese and maybe Cantonese Jyutping? Many articles on ROC-related topics begin with only traditional if its form is different from simplified, and the Hong Kong (香港 is same in both forms) article itself does not have any transliterations to begin with. Traditional Chinese and Cantonese are the Chinese norm in Macau and Hong Kong. 华钢琴49 (TALK) 22:40, 12 April 2010 (UTC)

Their inclusion is according to MOS:CHINESE#Simplified and Traditional which suggests that both should be used where different, and 門 and 门 are different. I think that's a bit too broad personally, as I can imagine situations where one or the other would be used almost all the time. But for Macau, both uses are common: Traditional in Macau and HK and Simplified over the border, e.g. on maps like this. So I think both are needed.

I did not add the Simplified Characters; I added the flag to put the traditional characters first. This is according to the same section of the manual of style which says traditional should go first in "contexts involving territories where traditional characters are used", which is definitely the case here.--JohnBlackburne_deeds 23:02, 12 April 2010 (UTC)

um... sorry for my poor wording. I meant that you added them to the introduction... after my reorganisations they were originally within the transliteration box as well. but the full official Chinese name (中華人民共和國澳門特別行政區) should definitely not be included in the introduction, nor should the translation for 'Macau SAR' be there. I see your point with regard to Zhuhai and Shenzhen. I did not imply that for Macau, only the traditional variant is common; obviously the number of people using Simplified globally is many times larger than with Traditional. I guess this is not too much of a deal; I'm just wondering why the change. Just do not be excessive with the fuller names. Thanks, 华钢琴49 (TALK) 23:54, 12 April 2010 (UTC)

You're going to have to provide diffs as I'm not sure what you mean. What I did to the introductory text is this, where I added a flag to the template to put Traditional Chinese first (it defaults to Simplified First). This edit by an anon IP added the Simplified Chinese. But whoever made them both changes are in line with the guidance in the manual of style. My other recent changes were changing the hat notes to use templates, which did not effect how it looked, and undoing vandalism and other inappropriate changes.--JohnBlackburne_deeds 10:32, 13 April 2010 (UTC)

Lagrange's identity and Pythagorean identity (or theorem) in 7D

John: Perhaps you could elaborate on your view at Lagrange's identity as quoted below:

“Yes, the Pythagorean identity holds in 3D and 7D. Or more precisely it is a condition of the cross product in 7D. But that is a different article: that and Lagrange's identity are not the same thing.”

Of course, they are not the same thing, but it does appear to me that in 3D and in 7D the Lagrange identity reduces to the Pythogorean identity. That is Lagrange's identity:

${\displaystyle {\biggl (}\sum _{k=1}^{n}a_{k}^{2}{\biggr )}{\biggl (}\sum _{k=1}^{n}b_{k}^{2}{\biggr )}-{\biggl (}\sum _{k=1}^{n}a_{k}b_{k}{\biggr )}^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2},}$

takes on the particular form which is being referred to as the Pythagorean identity (or theorem):

${\displaystyle |\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}}$

Inasmuch as the article on Lagrange's identity has a section devoted to ℝ where this cross-product formulation is presented, it seems appropriate to either add to this section the comments by David Tombe that the same can be done in ℝ, or to create a new subsection.

However, you reverted David's remarks as "incorrect". Maybe you weren't aware that he was contributing to a subsection on ℝ, and mistook his remarks as seemingly referring to Lagrange's identity in general?

Should not his remarks be restored, or can you provide a more detailed objection why not? Brews ohare (talk) 04:04, 22 April 2010 (UTC)

I've replied there. --JohnBlackburne_deeds 07:49, 22 April 2010 (UTC)

John: I requested further response because I don't find your response quoted above answers my questions. Can't you help? Brews ohare (talk) 14:14, 22 April 2010 (UTC)

Please reply at Talk: Lagrange's identity, where I've tried to explain how David is wrong. It's frustrating to all if discussions take place in two or three places at once (David posted at Talk:Seven-dimensional cross product also)--JohnBlackburne_deeds 14:21, 22 April 2010 (UTC)

I have done that just now. Brews ohare (talk) 15:11, 22 April 2010 (UTC)

Churchill College Buildings

In reference to: (Undid revision 355129314 by 78.151.90.232 (talk)rv; not a RS and not properly referenced.)

Why is the college's own website not a reliable source? I've given the URL if you wanted it to be referenced in a different style surely you could have changed that? 92.26.19.182 (talk) 21:01, 22 April 2010 (UTC)

The alternative prospectus is a light hearted, semi-serious publication largely based on student opinions, not facts, and definitely not independent of the college. At WP:RS it says "Articles should be based on reliable, third-party, published sources with a reputation for fact-checking and accuracy" so it fails on two counts. The quotes seem not to be factual anyway, just amusing opinions. If the criticism was properly sourced and clearly attributed it might be appropriate, but as it was added it was very out of place.

You're right, I could have fixed it to be a proper reference and usually do if I spot an inline link which should be a reference. So it might have been clearer in the edit summary if I only mentioned the reliability issues which were my reasons for removing it.--JohnBlackburne_deeds 21:26, 22 April 2010 (UTC)

Review Big data CFD

You voted on the delete debate for the original entry Big data that was rm by admins. Please reenter the debate? jk (talk) 04:55, 23 April 2010 (UTC)

Speedy deletion nomination of JohnBlackburne/thinBlk

Thank you for experimenting with Misplaced Pages. Your test worked, and the page that you created has been or soon will be deleted. Please use the sandbox for any other tests you want to do. Take a look at the welcome page if you would like to learn more about contributing to our encyclopedia. You may also wish to consider using a Wizard to help you create articles - see the Article Wizard.

If you think that this notice was placed here in error, you may contest the deletion by adding {{hangon}} to the top of the page that has been nominated for deletion (just below the existing speedy deletion or "db" tag), coupled with adding a note on the talk page explaining your position, but be aware that once tagged for speedy deletion, if the page meets the criterion, it may be deleted without delay. Please do not remove the speedy deletion tag yourself, but don't hesitate to add information to the page that would render it more in conformance with Misplaced Pages's policies and guidelines. Lastly, please note that if the page does get deleted, you can contact one of these admins to request that they userfy the page or have a copy emailed to you. Xtzou (Talk) 23:55, 23 April 2010 (UTC)

thanks. It was an accident: should have been in userspace. I noticed quickly and went to CSD it but you got there before me. Please delete it.--JohnBlackburne_deeds 23:58, 23 April 2010 (UTC)

Thanks for your help

Hi John: Thanks for helping me understand some of the ins and outs of WP display and browsers. I'm sure there is much more to this, but I am not yet an aficionado. I understand that it takes patience to deal with the uninitiated, and appreciate your help. Brews ohare (talk) 19:59, 26 April 2010 (UTC)

Discussion: Merging the articles for "Hyperplane" and "Flat"

• Hyperplane
• Flat (geometry)

I'd like to discuss the possibility of merging these two articles. Your opinion on this matter is welcomed: Talk:Hyperplane#Merge to Flat (geometry) Justin W Smith /_stalk 20:39, 5 May 2010 (UTC)

The Seven Dimensional Cross Product

John, Do you agree with the equation,

z₁ + z₂ + z₃ + z₄ + z₅ + z₆ + z₇ = (x₂y₄-x₄y₂) + (x₃y₇-x₇y₃) + (x₆y₅-x₅y₆) + (x₁y₄-x₄y₁) + (x₃y₅-x₅y₃) + (x₆y₇-x₇y₆) + (x₁y₇-x₇y₁) + (x₂y₅-x₅y₂) + (x₄y₆-x₆y₄) + (x₁y₂-x₂y₁) + (x₃y₆-x₆y₃) + (x₅y₇-x₇y₅) + (x₁y₆-x₆y₁) + (x₂y₃-x₃y₂) + (x₄y₇-x₇y₄) + (x₁y₅-x₅y₁) + (x₃y₄-x₄y₃) + (x₂y₇-x₇y₂) + (x₁y₃-x₃y₁) + (x₂y₆-x₆y₂) + (x₄y₅-x₅y₄)

David Tombe (talk) 14:47, 14 May 2010 (UTC)

I've no idea why you mean by "do you agree with...". If it's about an article you would be better off raising it on the article's talk page.--JohnBlackburne_deeds 15:23, 14 May 2010 (UTC)

John, Ultimately it's about an article. But I specifically wanted to raise this issue with yourself. We are now both agreed that the Lagrange identity/Pythagorean identity holds in both 3 and 7 dimensions. I just want to know now whether or not you agree with the above reasoning which led me to agreeing that it holds in 7 dimensions. The 21 terms on the right hand side expand into 84 terms. These 84 terms are in turn what is left of 252 terms. That 252 was made up of three groups of 84 terms. Two of those groups of 84 terms mutually cancelled. Such an amazing cancellation seems to only happen for 7D. It would not happen for 5D or any other D.

You have already satisfied yourself that the Lagrange identity holds in 7D because of pure maths type proofs, such as that of Silagadze which are referenced in the article. You also used the numerical method to prove it. I did not use either of those methods. I was trying to work it out analytically, and that is what I came up with above. As you recall, I once made the error of taking one of those z terms and specifically equating it to three of the xy terms. You pointed out that error. What I now want to know is whether or not you agree that the equation as a whole is valid.

Ultimately, I'm driving at trying to get the article easier to follow for those readers who are not pure maths experts. I'm trying to figure out a way of demonstrating more visibly how the Lagrange identity would hold in 7D. Most sources only show the 3D proof, and from the 3D proof it is far from immediately obvious that it would also hold in 7D. David Tombe (talk) 13:28, 15 May 2010 (UTC)

Please, take it to the talk page of whichever article you think needs improving, and make it clear what changes to the article you want. Then I and other editors can better help with improving the article.--JohnBlackburne_deeds 13:41, 15 May 2010 (UTC)

No John, I won't do that. It was a simple 'yes' or 'no' question which you are obviously not prepared to answer. David Tombe (talk) 15:47, 15 May 2010 (UTC)

Pseudovector

I have nothing against your revert there. It turns out that character takes three bytes in UTF-8, which I was not aware of. So when I tried to look up what character it was, and only looked at the first two bytes, I was confused. When I replaced it with the character I thought it was supposed to be, it didn't change, which confirmed which character it was. — Carl (CBM · talk) 23:15, 21 May 2010 (UTC)

Yes, it's the same char (I checked it myself). kwami replaced an IPA with the proper math symbol, the same as the HTML entity you changed it to. But such entities are best avoided, especially numeric ones, as they are very unfriendly to editors: I recognise few of them by name, and none of them by number. The only one I think's good to use is the non-breaking space, as it's impossible to tell apart from a regular space by inspection. I'm not even sure how I'd look at the raw utf-8: whenever I view or edit a page, or look at the source, I see the unicode. I'm only aware of the extra bytes as it shows up in the page size, when e.g. hyphens are changed to minuses or n-dashes.--JohnBlackburne_deeds 23:39, 21 May 2010 (UTC)

Antiparallel

This is not nearly as interesting as the discussion you're having with Brews, but the 4th definition of antiparallel is "Two vectors that are parallel but have opposite directions." In my experience, the dot product of parallel vectors is +1. MarcusMaximus (talk) 22:27, 23 May 2010 (UTC)

I didn't even notice that. It seems very odd in that article: ungrammatical, unsourced, and inserted by an anon IP who never came back so we can't ask where it's from. The Mathworld article doesn't have that definition, and I've not come across it myself. I've just tagged it to see if anyone can do anything with it. As for parallel to me for both vectors and lines it means the same thing: that if drawn they would never meet, and the formulation that the cross product of parallel vectors is zero is the one I'm familiar with. E.g. it comes up in geometric algebra a lot, where the geometric product of parallel vectors is a scalar, and more generally the product can be used to test for parallelism by e.g. checking for this.--JohnBlackburne_deeds 23:10, 23 May 2010 (UTC)

I don't have an authoritative source on this, but one consequence of your broader definition of parallel would be that parallel vectors have a dot product of +1 or –1. I don't know if that is objectionable or not. MarcusMaximus (talk) 01:41, 24 May 2010 (UTC)

Google books provides plenty sources. I picked one. Cheers - DVdm (talk) 09:36, 24 May 2010 (UTC)

Reconsideration of 7-D cross product?

Hi John:

In the article Cross product is the section Cross product#Alternative formulation based upon the paper by WS Massey. This is the approach he uses for the 7-D cross product.

In the cross-product article also is the subsection Cross product#Lagrange's identity, which follows (for the 3D cross product) exactly the paradigm I have suggested for the 7-D cross product. In particular, it combines the documented relations:

${\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2},}$

and

${\displaystyle \sum _{1\leq i<j\leq n}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}=\|\mathbf {a} \|^{2}\ \|\mathbf {b} \|^{2}-(\mathbf {a\cdot b} )^{2}\ ,}$

to obtain the cross product in terms of components of a and b (changed to 7D below):

${\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}=\sum _{1\leq i<j\leq 7}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}\ .}$

I wonder if this 3-D example might have persuaded you to support including this result in the article Seven-dimensional cross product? Brews ohare (talk) 16:03, 3 June 2010 (UTC)

Nothing's changed: we need a source for this, i.e. one on the 7D cross product. If you can point to a source that "this result" is from it will be clear how it relates to what's there already. Otherwise it's original research, so should not be included.--JohnBlackburne_deeds 16:09, 3 June 2010 (UTC)

(edit conflict) and in future please take time to review your comments before posting, not after. It is most annoying that I have to edit my reply twice because you've edited yours in the interim.--JohnBlackburne_deeds 16:09, 3 June 2010 (UTC)

OK, John. It is still my opinion that no source is necessary. Both the equations:

${\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2},}$

which I might shorthand as:

${\displaystyle x(a,b)=y(a,b)\ \mathrm {for\ every\ choice\ of} \ a,b,}$

and

${\displaystyle \sum _{1\leq i<j\leq 7}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}=\|\mathbf {a} \|^{2}\ \|\mathbf {b} \|^{2}-(\mathbf {a\cdot b} )^{2}\ ,}$

which I might shorthand as:

${\displaystyle z(a,b)=y(a,b)\ \mathrm {for\ every\ choice\ of} \ a,b,}$

are documented in the cross product article and in previous discussion between us. The result I'd like to see in the 7-D cross product article is then:

${\displaystyle x(a,b)=z(a,b)\ \mathrm {for\ every\ choice\ of} \ a,b\ ,}$

which I cannot understand as a "new" result unsupported by sources, because a simple mathematical substitution of equivalent results is simply Routine calculation, and not at all Synthesis to advance a position: it doesn't involve my judgment on an issue, but merely replaces one expression of a mathematical quantity with another one in different form, sourced as being identical. Brews ohare (talk) 19:33, 3 June 2010 (UTC)

John, I am going to copy this discussion to 7D cross product, where it is more germane. Brews ohare (talk) 19:36, 3 June 2010 (UTC)

Role of Lagrange's identity

Hi John:

I summarized my take on Lagrange's identity here. I believe we can agree on the following:

• ${\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}+(\mathbf {a} \cdot \mathbf {b} )^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2},}$

may be viewed as part of the definition of the cross product.

• Lagrange's identity in n-dimensions says:

${\displaystyle \sum _{1\leq i<j\leq n}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}=\|\mathbf {a} \|^{2}\ \|\mathbf {b} \|^{2}-(\mathbf {a\cdot b} )^{2}\ ,}$

and this relation must be satisfied by the components of any pair of vectors in whatever dimension n.

In 3-D these two relations, the definition and Lagrange's identity can be combined to provide:

• ${\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}=\sum _{1\leq i<j\leq 3}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}\ .}$

This relation is satisfied provided the cross product of the unit vectors satisfy:

${\displaystyle \mathbf {\hat {e}} _{i}\mathbf {\times {\hat {e}}} _{j}=\mathbf {\hat {e}} _{k}\ ,}$

with {i, j, k} a cyclic permutation of {1, 2, 3}.

At this point we part company, I guess. One might ask what the implication of Lagrange's identity is for 7-D. Clearly it must be satisfied. What is your view of its impact?

We disagree here, but my take is that it sets the requirement upon the cross-product that:

${\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}=\sum _{1\leq i<j\leq 7}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2},}$

and it is this condition (plus the orthogonality properties) that results in turn in the requirement:

${\displaystyle \mathbf {\hat {e}} _{i}\mathbf {\times {\hat {e}}} _{i+1}=\mathbf {\hat {e}} _{i+3}\ ,}$

with {1, 2, 3, 4, 5, 6, 7} permuted cyclically and translated modulo 7.

Perhaps we differ upon the origin of the restriction to this table? What do you think is the role of the Lagrange identity? My guess is that one could track down the restriction to n=3 and n=7 to precisely the impossibility of satisfying this Lagrange's identity requirement for other n. In other words, that would be another approach to Hurwitz' theorem.

What do you think? Brews ohare (talk) 20:33, 4 June 2010 (UTC)

BTW, it appears to me that this source has used as definition of the cross product the orthogonality requirement and the requirement:

${\displaystyle \|\mathbf {a_{1}\times a_{2}\times \dots a_{r}} \|^{2}=det(<\mathbf {a_{i},a_{j}} >)}$

which may correspond to using the Lagrange restriction:

${\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}=\sum _{1\leq i<j\leq 7}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2},}$

as definition of the cross product instead of the "Pythagorean theorem":

${\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}+(\mathbf {a} \cdot \mathbf {b} )^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2}.}$

I hope I have understood their notation. Brews ohare (talk) 21:20, 4 June 2010 (UTC)

That's the same as the definition of the more general (i.e. not just 2 vectors) given by Lounesto on page 98, except his is a lot clearer.--JohnBlackburne_deeds 21:39, 4 June 2010 (UTC)

Hi John: I believe you are right. What's more, I believe it is trivial to show the Gram determinant form of cross product is exactly the same as the Lagrangian form:

${\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}=\sum _{1\leq i<j\leq 7}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2},}$

${\displaystyle ={\frac {1}{2}}\sum _{i,j=1}^{n}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2},}$

${\displaystyle =\sum _{i}a_{i}^{2}\sum _{j}b_{j}^{2}-\sum _{i,j}a_{i}a_{j}b_{i}b_{j}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2}-(\mathbf {a\cdot b} )^{2}\ }$

In other words, contrary to my earlier thoughts, the formulation found using the Lagrange identity contributes nothing new to the situation, and contrary to your thoughts, it is trivially identical with the "Pythagorean theorem" or "Gram determinant" formulation. Consequently, either can be used at will. It is the combination of this result with orthogonality that leads to the restrictions upon n . Whattaya say? Brews ohare (talk) 00:29, 5 June 2010 (UTC)

Query on inner product spaces

Hi John: I wonder if you would take the time to comment upon norms in an inner product space. The various articles indicate that the "natural" norm of a normed inner product space is the one based upon the inner product itself, namely:

${\displaystyle \|x\|^{2}=\langle x,\ x\rangle \ .}$

• My understanding is that the natural norm is equivalently called the Euclidean norm, and such a space a Euclidean space. Is that correct terminology?

From my reading I have found statements that any norm that is to be expressed as an inner product must satisfy the parallelogram equality, so a p-norm with p≠ 2 won't be expressible like this.

• Does use of a space with an inner product with, say, the p-norm instead of the natural norm, lead to conflict with the natural norm? For example, does orthogonality based upon the inner product conflict with the p-norm?

• Is there an example where that is done?

Thanks for your insight. Brews ohare (talk) 23:00, 20 June 2010 (UTC)

not something I know much about I'm afraid: I know a lot about the inner product on e.g. real spaces but not the abstractions of it. You could ask on the Maths project page.--JohnBlackburne_deeds 23:05, 20 June 2010 (UTC)

Talkback: Nils von Barth: Misplaced main and see-also links

Appeal filed

John: I have filed an appeal of Sandstein's ban following up on your enforcement request. Brews ohare (talk) 16:48, 4 August 2010 (UTC)