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[Original question at User talk:Lbeben --Bob K31416 (talk) 05:47, 10 January 2010 (UTC)].
Thank you sir for your question, I am pleased to oblige on two primary sources.
5/4/94 for the Smithsonian Specifically referring to Dr. Peskins' lengthy answer regarding "Mathematical Collagen Fibers"
Peskin quotes a Dr. Carolyn Thomas who apparently was an anatomist in New England in the 1950s. Her early drawings of the porcine heart have led to the focusing of several Kray mainframes on the work of the myocardium.
Buckberg extensively quotes the late Dr. Francisco Torrent-Guasp regarding myocardial band theory.
In composing these two paragraphs, I hope to better illuminate these concepts to non-medical readers of a web-based encyclopedia.
Study of physiologic Compliance suggests mathematical proof of what is not Afterload. ["A new noninvasive method for the estimation of peak dP/dt" Circulation 1993]--lbeben 03:37, 5 January 2010 (UTC)
Sir I sincerely appreciate your counsel, discussion and edits regarding afterload, I remain hopeful we may continue to discuss other areas relevant to heart disease in the future. The ECG, atrial fibrillation, heart failure and Chagas Disease all remain topics of great interest to me. Best wishes —Preceding unsigned comment added by Lbeben (talk • contribs) 02:31, 14 January 2010 (UTC)
Noticed that you reverted them, what are your feelings about it? AniRaptor2001 (talk) 05:44, 21 January 2010 (UTC)
Hey, Bob. I have used this source for the Titanic (1997 film) article. Streetdirectory.com is a reliable source, and the place the source is from seems reliable, as well as the author of the piece, but I am worried about it being an anecdote. Some of this guy's retailing of events can be backed up by more reliable sources, and is in a few parts in the article, but would you say that this source is appropriate to use?
On a side note, Titanic is playing on TNT right now where I am; it will be over soon, though. I feel that they are mainly playing it right now, because Avatar is about to beat it. LOL. With Avatar set to become the highest-grossing film of all time, it makes sense that they would show the previous highest-grossing film also by Cameron. Flyer22 (talk) 00:15, 24 January 2010 (UTC)
Please see Talk:Avatar#Literary antecedents (Themes and inspirations) -- Jheald (talk) 21:15, 28 January 2010 (UTC)
Just a quick note Bob. I know we've crossed swords a few times on the Avatar talk page but it was improper of me to accuse you of a nationalist bias, but you know, it was late and I was tired and I was a bit tetchy. You haven't given me any reason to doubt you haven't got the best interests of the article at heart. The main thing for the article is that it remains stable and any disputes stay on the discussion page which is a principle we both seem to respect. Betty Logan (talk) 23:56, 2 February 2010 (UTC)
Please see WP:NPA, WP:CIVIL, and WP:DEADHORSE. —Preceding unsigned comment added by 203.122.242.126 (talk) 05:36, 9 February 2010 (UTC)
Hello, Bob K31416. If you are interested, there is a request for amendement regarding this matter. I was told that you were interested.Likebox (talk) 04:06, 11 February 2010 (UTC)
Hi Bob:
Hi Bob: Yep, I shouldn't take it personally, and I do not. I take it as evidence that the appeal process has not worked: suggestions and evidence have been ignored completely, without excuse. That is sad for WP, as discussion on Talk pages is headed toward bus-stop conversation:
Unfortunately, I think Wikid and North8000 may not have as good of a grasp on what needs to be done to correct this wiki policy problem as you do. I think you and I will probably have to lead the way. Scott P. (talk) 04:03, 19 April 2010 (UTC)
Bob: I really don't know what the >> symbols are "for." Haven't looked too closely. As always, am interested in content much more than formatting. If you'd like to see them removed, then go to town !!!! Calamitybrook (talk) 18:38, 19 April 2010 (UTC)
I have requested for Avatar (2009 film) to be peer reviewed. Since I saw you were one of it's top contributors, I thought I should let you know. Feel free to to fix any objections on the peer review page. Thanks.Guy546(Talk) 22:53, 16 May 2010 (UTC)
Bob, I'll answer your question about the triangle in more detail here, since you seem to be genuinely interested in getting to the bottom of the issue. I am fully aware of the fact that a triangle sits in a two dimensional plane. But you are overlooking a number of factors. Trivially, there is the fact that every two dimensional plane has an associated perpendicular. Two dimensions don't ever exist in the absence of the third dimension.
Secondly, you correctly pointed out that the Pythagoras theorem is a special case of the cosine rule. The cosine rule is about angles, and so therefore is Pythagoras's theorem about angles. The triangle is all about three angles. Those angles all require the third dimension in order to have any meaning. We cannot have a rotation about a point, as someone has suggested. We need a perpendicular direction.
But the full argument can all be very neatly summed up in the three dimensional version of the Lagrange identity which is effectively Pythagoras's theorem in the form,
This equation clearly contains both an inner product and an outer product, yet the section in the main article on Pythagoras's theorem called 'inner product spaces' is trying to treat Pythagoras's theorem in the absence of any mention of the outer product.
Anyway, I thought that you were also driving at the fact that the sources that deal with the 'n' dimensional Pythagoras theorem only talked about it in terms of being a definition of distance. You were hinting earlier that you didn't think that the 'n' dimensional cosine rule should be in the article. What actually is your own viewpoint on the matter? David Tombe (talk) 23:17, 18 May 2010 (UTC)
Bob, On your point number (1), I simply can't imagine a rotation without a rotation axis. I can't really say anything more on that matter. It's just a question of belief. On your point number (2), the section 'inner product spaces' is actually written up very well indeed, and it is very clear. It certainly ties in with the sources. But that is not the point. Somebody wanted to highlight the fact that the actual interpretation of Pythagoras's theorem actually changes when we generalize it to inner product spaces. That emphasis has now been removed from the article by virtue of the title change and the fact that that section was moved away to a different location. On your point number (3), the bottom line is that cross product (outer product) is intricately linked up with Pythagoras's theorem, whereas there seems to be a focus in the article on the inner product and a tendency to brush the outer product aside. I've replied to Carl again on the Pythagoras's theorem talk page. Perhaps you should take a look at that reply. David Tombe (talk) 09:27, 19 May 2010 (UTC)
Bob, I know all about the above manipulations. But are you aware of the fact that the cross product only exists in 0, 1, 3 and 7 dimensions? That is certainly a well sourced fact. David Tombe (talk) 12:55, 19 May 2010 (UTC)
Bob, There is a wikipedia article entitled seven dimensional cross product and it contains alot of sources. Nobody was ever disputing the fact that cross product only holds non-trivially in 3 and 7 dimensions. The prolonged debate on the talk page at that article was about whether or not the equation,
is valid in seven dimensions. Initially, I wrongly believed that it wasn't valid in 7 dimensions, but after trying out numbers, I finally had to concede that it is indeed valid in seven dimensions. However, nobody at that page was ever arguing that its validity extended outside 3 or 7 dimensions. David Tombe (talk) 15:45, 19 May 2010 (UTC)
Bob, Yes indeed, it is alot of reading, and in retrospect I can see now that there is alot of chaff and unneccessary wrangling over terminologies. I'll now give you a brief summary.
(1) Everybody agreed from the beginning that the cross product only holds in 0,1, 3, and 7 dimensions. This has been known since the 19th century, but a formal proof has only existed since the 1960's. I discovered about the 7D cross product only many years after leaving university, and that was while browsing through a 'history of maths' article in an Encyclopaedia Britannica. It told me that no formal proof yet existed for the fact that cross product only exists in 0,1,3, and 7 dimensions, but that one was nearing completion, and that it was very complex.
So the first argument on the talk page was over the fact that the reason given in the article for the proof of the fact that cross product only exists in 0,1,3,and 7 dimensions, was not adequate. Since then, sources have been provided which give the modern proofs, but these proofs themselves still don't appear in the article as such, not that that necessarily matters as such.
(2) Then came the issue of the equation,
I had sources, including two wikipedia articles, which showed that this equation was the special 3D case of the Lagrange identity. And so it is. But John Blackburne claimed that it held in 7D as well. I disagreed initially. But finally John Blackburne told me to use numbers to test it out. I did so and was forced to concede that John Blackburne was correct. I then re-examined the analysis and figured that in 7D, the right hand side of the equation contains 7 terms which expand into 252 terms. That is 3 groups of 84. Two of these groups mutually cancel, and we are left with 84 terms that reduce to 21 terms in brackets anti-distributively. The 21 terms are the terms needed to make the equation valid with the magnitude of the 7D cross product. This only works in 3D and 7D. The 3D case is easy because the right hand side contains only three terms and the cross product relationship is self evident.
Anyway, the point is that cross product only holds non-trivially in 3 and 7 dimensions, and everybody is agreed about that. Hence Pythagoras's theorem can be shown to be a special case of the Lagrange identity, but only in 3D, with 7D being a special half-way house case. David Tombe (talk) 19:54, 19 May 2010 (UTC)
Bob. Good question. The first questions that naturally arise when one first contemplates the concept of a vector cross product in dimensions higher than 3D is 'What does it look like? How do we do it?'. We are all very familiar with everything to do with the three dimensional cross product and all the inter-relationships that you have already manipulated above. But then comes the question of what a seven dimensional cross product would look like. Well as you know, the embryo of the 3D cross product began with an inspiration by Sir William Rowan Hamilton in 1843 as he walked along the tow path of the Royal Canal in Dublin. He was so excited about it that he inscribed the result on the wall at Brougham Bridge. This result is the effective basis of the later result of Gibbs that,
z = x × y | x × y |
---|---|
i | j×k |
j | k×i |
k | i×j |
The sine relationship then follows on.
As regards the seven dimensional cross product, the situation is more complicated because each unit vector can be the product of three distinct pairs from amongst the other 6. Here is one example of how it might look.
z = x × y | x × y |
---|---|
i | j×l, k×o, and n×m |
j | i×l, k×m, and n×o |
k | - i×o, j×m, and l×n |
l | i×j, k×n, and m×o |
m | i×n, j×k, and l×o |
n | -i×m, k×l, and j×o |
o | i×k, j×n, and l×m |
However, the seven dimensional cross product does not obey either the vector triple product relationship or the Jacobi identity. But both the 3D and the 7D cross products allow the 'n'D Lagrange identity to take on the form,
The proof of this in 3D is quite straightforward and most sources are misleading in that they would tend to give the impression that this equation is uniquely the 3D version of the Lagrange identity. And with the sine relationship added, this equation then of course becomes Pythagoras's theorem.
The argument on the talk page at seven dimensional cross product was because initially I couldn't see how this equation could possibly apply in 7D. But John Blackburne finally forced me to look closer by pointing out that substitution of numbers will adequately confirm the fact. If you look at the talk page at Lagrange identity you will see how I eventually came to accept it. Like I said yesterday, in the 7D case, the right hand side is seven terms that expand into 252 terms. These 252 terms form three groups of 84, two of which are mutually cancelling. That leaves 84. The 84 contract down to 21 terms in brackets and these 21 terms make the equation work.
However, the 7D cross product cannot be related to 'sine', because it doesn't fit with the Jacobi identity. The conclusion is that Pythagoras's theorem is the 3D version of the more general 'n'D Lagrange identity.
The answer to your specific question above is that the 3D cross product, whether written in 'sine' form or not, only holds in 3D. David Tombe (talk) 09:57, 20 May 2010 (UTC)
Brews, there is a bit of irony here. If you look at the edit history of seven dimensional cross product, you will see that I actually tried to remove the name 'Pythagorean identity' from that equation. But now I have changed my mind. It's a tricky issue. The equation is accurately called the Lagrange identity for the special cases of 3 and 7 dimensions. But in my opinion it is also exactly Pythagoras's theorem in the 3D case. In the context of the article, it is first presented as an equation which needs to hold for the purposes of defining the cross product. As such, in the context, we can't call it the Lagrange identity initially because it doesn't reveal itself as being the Lagrange identity in 3 or 7 dimensions until after it has been shown that the equation only works in 3 or 7 dimensions. However, using the name 'Pythagorean identity', as Lounesto does, immediately incorporates the spirit of why that equation is desirable in the first place as a starting point. Nevertheless, I don't think that 'Pythagorean identity' is necessarily a good name for the 7D case.
The 3D case is however unambiguous. Pythagoras's theorem is clearly the special 3D form of the Lagrange identity. The cross product is merely a transitionary mathematical tool which is used in demonstrating that linkage. Clearly Pythagoras's theorem is a 3D theorem. It is a theorem about a 2D triangle in a 3D space. Lagrange's identity tells us unequivocally that Pythagoras's theorem is not a theorem in a 2D space. David Tombe (talk) 17:00, 20 May 2010 (UTC)
Brews, The bottom line is that Pythagoras's theorem is the Lagrange identity in three dimensions. Cross product is merely a tool which enables this fact to be exposed. See the reply which I am about to give to Carl on his talk page.
Meanwhile, the relevance of the Jacobi identity in all of this is to rule out the same argument for seven dimensions, because the sine relationship is dependent on the Jacobi identity which does not hold in 7D. David Tombe (talk) 09:20, 21 May 2010 (UTC)
Hi David, I looked at the article Seven-dimensional cross product and here's an excerpt for the case of n-dim,
So according to the Wikipedia article, this relation that held in 3-dim also holds in n-dim. Am I understanding this correctly? Regards, --Bob K31416 (talk) 15:50, 21 May 2010 (UTC)
Regarding Pythagoras' theorem, I think we have the often seen occurrence of a semantic difficulty. My present take is that Pythagoras' theorem means square of magnitude is sum of squares of orthogonal components, and as such is divorced entirely from cross-product. It is therefore readily applied to arbitrary dimension n. It is the cross-product that provides the dimensional requirements, and can be used only in 3D and 7D, where it just so happens it can be used as an alternative expression of Pythagoras' theorem. To combine this point with your request for a connection to rotation, because the existence of cross product also means an axis of rotation can be found, I'd hazard that if we require angle to be connected to rotation, then you are perfectly right that Pythagoras applies only in 3D and 7D. But if we allow angle to be a meaningless expression that says only the dot product has a maximum value of ||a|| ||b||, then Pythagoras' theorem can be used anywhere. Brews ohare (talk) 20:48, 21 May 2010 (UTC)
Hi David, I'll put aside any followup I have regarding the cross product and sine for now, since it appears that we need to discuss the more basic concept of angle first. Regarding your remark, "I would argue that angle is a 2D concept that only has meaning providing that there exists a singular third dimension" and a previous remark of yours "I simply can't imagine a rotation without a rotation axis."
What are your thoughts regarding the second and third sentences of the lead of the Wikipedia article Rotation?
Regards, --Bob K31416 (talk) 19:21, 21 May 2010 (UTC)
David, I'll put aside any followup regarding angle for now, since it appears that we need to discuss the more basic concept of a 2-dimensional space first. From your comments, it seems that you believe that a 2-dimensional space cannot exist mathematically without a 3-dimensional space that it is part of. Am I understanding you correctly? --Bob K31416 (talk) 20:07, 21 May 2010 (UTC)
Bob, That's basically it. Anything that we assume about a 2D space is based on our observations of 2D geometry in a 3D space. It's impossible to know anything at all about the realities of a purely 2D space, because the idea is purely imaginary. David Tombe (talk) 00:07, 23 May 2010 (UTC)
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Bob, That's been exactly my point all along. There is no correspondence to physical space in any of the definitions relating to higher dimensions. It's all pure mathematics. But even the pure maths restricts Pythagoras's theorem to 3 and 7 dimensions. This fact is made quite clear through the Lagrange identity. There only remains the issue of eliminating the 7D case through the Jacobi identity. See the comment that I am about to make on the talk page at Pythagoras's theorem. David Tombe (talk) 14:04, 23 May 2010 (UTC)
Bob, We can certainly discuss it. But it will all be pure speculation that will no doubt be heavily prejudiced by our knowledge of a 2D plane in a 3D space. There is nothing stopping us from defining a 2D space or a 15D space. There is nothing stopping us from defining a theorem in the likeness of Pythagoras's theorem to apply in these imaginary mathematical constructs. But we should not assume that these defined 'n' dimensional Pythagoras's theorems, which are purely mathematical constructs, should be equated with the very real Pythagoras's theorem, which is actually a proveable theorem in 3D space. David Tombe (talk) 16:10, 24 May 2010 (UTC)
Bob, You are equating this mathematical 2D space with a 2D plane in a 3D space. A purely mathematical 2D space has go no connection whatsoever with areas or geometry. The Lagrange identity in 2D gives us no linkage whatsoever with 3D Euclidean space. We cannot assume that a purely mathematical 2D space has got any connection with a 2D plane in a 3D space. Only 3D space can be linked to Euclidean geometry, and it's the 3D Lagrange identity which leads us to Pythagoras's theorem.
The answer to your question above is 'yes', but only if we are dealing with a 2D plane in a 3D space. David Tombe (talk) 19:17, 25 May 2010 (UTC)
Bob, You gave a plane geometrical interpretation to the mathematical concept of a 2D space. I can't imagine such an interpretation. I can only imagine a 2D plane in a 3D space. So you are asking me a question about a scenario which I can't imagine unless I assume it to equate to a 2D plane in a 3D space. What you must remember here is that I am banned from discussing physics on wikipedia. And If I were to fully give you justice on your question, I would have to branch into physics. I do have a better answer for you, but I am disqualified from stating it. I have tried my best to answer you within the confines of mathematics/geometry. David Tombe (talk) 11:07, 26 May 2010 (UTC)
Bob, The problem is because I can't imagine any such concept as a plane geometrical 2D plane in the absence of a third dimension. If we want to simply assume that such a 2D plane can exist, and then import all the rules and visualizations from a 2D plane in a 3D space, then of course I would have to concede that we can have angle. But we will run into trouble when we discover that we can't use the cross product to describe rotational phenomena.
Just as an aside, have you ever thought about 7D curl? David Tombe (talk) 11:48, 26 May 2010 (UTC)
Bob, There's a set of relationships that are all fully compatible in 3D. These are Pythagoras's theorem, cross product, dot product, the Lagrange identity, and the Jacobi identity. Try moving outside of 3D and you get a glitch with the Jacobi identity for all other dimensions, and you also get a glitch with the Lagrange identity for all other dimensions apart from 7. Pythagoras's theorem emerges from the 3D Lagrange identity. With the exception of the controversial case of 7D, it certainly doesn't emerge from the Lagrange identity in any other dimensions. You seem to be assuming that a mathematical 2D space can be represented by a 2D plane as we understand such in a 3D space. Are you confident that you can make that assumption? David Tombe (talk) 16:14, 26 May 2010 (UTC)
Bob, You are still making the assumption that a purely 2D space can be represented by plane geometry, whereas in fact it is merely an algebraic contruct. You have already told me that you can't imagine a 2D plane where no 3rd dimension exists? So how do you know that the algebra of a 2D space would relate to such a concept if you can't even imagine it? In 3D, we can clearly see that the algebra of a 3D space relates to 3D geometry, and 3D geometry involves 2D planes. David Tombe (talk) 20:07, 26 May 2010 (UTC)
Hi Bob: It looks like the editing of this page has begun to attract the WP crazies. It's time to leave. In a month or so it'll quiet down again, and if we are still motivated, we can clean up the wreckage they have left behind, like a bunch of janitors after the party has ended. Brews ohare (talk) 14:40, 20 May 2010 (UTC)
I thought you meant to give it to alt text, since there is no alt text there. My bad. Guy546(Talk) 21:03, 23 May 2010 (UTC)
Hey, Bob. Your thoughs on the Critical reception section length for the Avatar article and why it is designed the way it is may be helpful in the peeer review. Flyer22 (talk) 17:10, 26 May 2010 (UTC)
I wasn't sure if you still wanted an answer, even though you removed this question from my talk page:
Hi Bob:
I have never been involved in a Featured Article procedure, and thought you might have some words on the subject. The Pythagorean theorem article looks pretty good to me at this point. Obviously I have made many revisions of this article, so I am rather too close to it at this point to have an objective view. Would you be willing to participate? Brews ohare (talk) 15:32, 7 June 2010 (UTC)
Hi Bob: Did that. Thanks for the suggestion. Brews ohare (talk) 21:53, 7 June 2010 (UTC)
I have added the "reviewers" property to your user account. This property is related to the Pending changes system that is currently being tried. This system loosens page protection by allowing anonymous users to make "pending" changes which don't become "live" until they're "reviewed". However, logged-in users always see the very latest version of each page with no delay. A good explanation of the system is given in this image. The system is only being used for pages that would otherwise be protected from editing.
If there are "pending" (unreviewed) edits for a page, they will be apparent in a page's history screen; you do not have to go looking for them. There is, however, a list of all articles with changes awaiting review at Special:OldReviewedPages. Because there are so few pages in the trial so far, the latter list is almost always empty. The list of all pages in the pending review system is at Special:StablePages.
To use the system, you can simply edit the page as you normally would, but you should also mark the latest revision as "reviewed" if you have looked at it to ensure it isn't problematic. Edits should generally be accepted if you wouldn't undo them in normal editing: they don't have obvious vandalism, personal attacks, etc. If an edit is problematic, you can fix it by editing or undoing it, just like normal. You are permitted to mark your own changes as reviewed.
The "reviewers" property does not obligate you to do any additional work, and if you like you can simply ignore it. The expectation is that many users will have this property, so that they can review pending revisions in the course of normal editing. However, if you explicitly want to decline the "reviewer" property, you may ask any administrator to remove it for you at any time. — Carl (CBM · talk) 12:33, 18 June 2010 (UTC) — Carl (CBM · talk) 13:31, 18 June 2010 (UTC)
Hi Bob: You may recall the suggestion you made that is summarized here. A nagging question in my mind is whether the proof that "the shortest distance between two points is a straight line" doesn't require Pythagoras' theorem. In other words, perhaps a different justification than that provided is necessary to avoid circular reasoning? What is your view? Brews ohare (talk) 16:50, 19 June 2010 (UTC)
For example, here is a source. Brews ohare (talk) 16:57, 19 June 2010 (UTC)
Sure thing. This observation is a crucial point in establishing the inequalities used in Staring's proof. Although it is intuitively obvious that the hypotenuse is longer than either of the adjacent sides of a right triangle, to put this inequality on a firm axiomatic footing one needs something akin to the Cauchy-Schwarz inequality. That is, the intuitive obviousness of this point stems from our everyday experience with normal 3-D space, and actually cannot be established without Pythagoras' theorem. Without a way to establish this fact, Staring's proof begs the question. Brews ohare (talk) 17:26, 19 June 2010 (UTC)
Differently put, Pythagoras' theorem is a consequence of the Euclidean axioms, and it is stated somewhere in the article that it is tantamount to the non-intersection of parallel lines. What we need to make Staring's proof non-circular is to identify the axiom that leads to the required inequality, so that it is plain that we are not simply using Pythagoras' theorem itself to establish the theorem. Brews ohare (talk) 17:31, 19 June 2010 (UTC)
I need some help here because my grounding in the axioms is not deep enough. For example, the assumption that the shortest distance between two points is a straight line is a consequence of Pythagoras' theorem. Cauchy-Schwarz inequality establishes this point for function spaces, and it appears that that is sufficient to establish the function space as Euclidean. So it appears that Staring's proof could be taken as establishing Pythagoras by assuming Cauchy-Schwarz inequality. But that doesn't seem to be a big accomplishment if it already is known by logical deduction. Brews ohare (talk) 17:41, 19 June 2010 (UTC)
In short, all that Staring has proved may be that if Pythagoras' theorem holds, then a da + b db = c dc. That seems to be a result more readily established by direct application of differentiation. Brews ohare (talk) 17:48, 19 June 2010 (UTC)
Hi Bob: Question: There is a question of logical sequence here. If a theorem is to be used to establish Pythagoras, then it mus be logically prior to Pythagoras. By that I mean that the logical chain leading to the theorem must not include Pythagoras in the chain of reasoning supporting its proof. So the question is just where does this Theorem fit in? It is dependent upon a definition of angle, which might (I don't know) involve Pythagoras' theorem somehow. For example, angle is often defined using the Cauchy-Schwarz inequality. I can read this reference over, but maybe you know the answer? Brews ohare (talk) 18:46, 19 June 2010 (UTC)
Here's my attempt to answer this question. A viable axiomatic basis for a geometry: postulate the properties of points and lines, and define a distance function d = PQ for the distance between points P and Q:
Then one can define a variety of distance functions that satisfy these postulates. Some such distance functions also satisfy the triangle inequality, which becomes an additional postulate of a geometry:
This inequality appears to me to say "a straight line segment is the shortest line segment that can be drawn between two points". Do you agree? However, although necessary, the triangle inequality appears insufficient to establish Pythagoras' theorem: Example: Euclidean distance function
Example: Taxicab geometry:
Both satisfy the triangle inequality.
Thus, Pythagoras' formula appears not to follow from these axioms, but to require more, because taxicab distance also satisfies the requirements.
How does Staring's proof force us into a Euclidean distance function instead of a taxicab distance (say)? Brews ohare (talk) 19:59, 19 June 2010 (UTC)
Bob: Here's an interesting discussion. Brews ohare (talk) 20:23, 19 June 2010 (UTC) And here's another one. Brews ohare (talk) 20:26, 19 June 2010 (UTC)
The triangle inequality also applies in spherical geometry where the distance between any two points P and Q is the angle they subtend at the center of the sphere, and shortest distances are great circles. As I don't have access to your source, I don't know what it is about their proof that would make it inapplicable to spherical geometry where Pythagoras' theorem doesn't work. If it does apply in spherical geometry, what is it about Staring's proof that doesn't work? Brews ohare (talk) 21:15, 19 June 2010 (UTC)
Hi Bob: I believe we agree on many points above. I'd try to summarize my uncertainties as follows. It is agreed by all sources that Pythagoras' theorem is equivalent to the parallel postulate, which takes on many equivalent forms. Staring's proof must therefore do the same as all other proofs of Pythagoras' theorem: start with some assumption equivalent to Pythagoras' theorem and thereby derive the equivalence of that assumption to Pythagoras' theorem. My guess is that it is the assumption expressed in the footnote that is equivalent to Pythagoras' theorem in one of its many guises. I'd like to pin that down to one of the established postulates that is already known to be the same as Pythagoras' theorem. Would you agree that this is culprit, or does it show up elsewhere in Staring's proof? Brews ohare (talk) 15:57, 20 June 2010 (UTC)
Hi Bob: I've changed my mind on the origin of the assumption leading to Pythagoras' theorem in Staring's proof. I now understand it to stem from the use of similar triangles in order to establish cosθ and cosφ. Establishing similarity of triangles involves showing that all the angles are the same, and that involves using the Triangle postulate, known to be equivalent to the parallel postulate and hence equivalent to Pythagoras' theorem. Hence, Staring's proof is of the same ilk as the other proofs using similarity, in particular this one and this one. Comments? Brews ohare (talk) 22:20, 20 June 2010 (UTC)
Hi Bob: I think we are on different wavelengths here. Here's my perspective, which I think differs form where you are coming from. The geometry is an axiomatic development à la Hilbert's axioms or Euclid's. A proof of Pythagoras' theorem amounts to showing how the theorem stems from the axioms. If one wishes, one can turn any such proof around, remove an axiom (the parallel postulate, say), make Pythagoras' theorem an axiom and deduce the removed axiom (the parallel postulate).
My initial worry was that Staring's proof was trivial, making an assumption of Pythagoras to deduce Pythagoras. However, I grew away from that idea to the question of just what axiom equivalent to the parallel postulate was used in Staring's proof to obtain Pythagoras. There is a list of propositions equivalent to the parallel postulate, and Pythagoras is one of them. The proofs using similar triangles use the Triangle postulate, which is a recognized equivalent to the parallel postulate. Staring uses similar triangles in order to establish expressions for cosθ and cosφ, so I conclude that this is the bridge he is building: similar triangles → Pythagoras. Since parallel postulate → Triangle postulate → similar triangles, I see why Staring's proof works, and am reassured that it is not trivially circular. Brews ohare (talk) 00:39, 21 June 2010 (UTC)
Bob: Sure. There are two issues involved in this particular case. (i) Is Staring's proof reliable? I guess you could say that its publication is circumstantial evidence that it is reliable, and WP would go along with that idea unless there was a contradictory publication. In this case, Cut the knot Proof #40 makes two erroneous claims that appear to contradict Staring's publication - first they say his proof is the same as Hardy's, which isn't so. Second, they piss on Hardy's proof as needing a "grain of salt", and by implication also upon Staring's proof. I see no basis for saying Staring's proof requires a grain of salt, but this negative view led me to worry a bit. Having sifted through this proof several times, I am of the opinion that no grain of salt is necessary. (ii) How does Staring's proof depend upon the parallel postulate? This question is not simply a question of whether his proof is correct, but is also a question that can be asked of any proof of Pythagoras' theorem. It is a matter of curiosity because we know that all proofs of Pythagoras must involve the parallel postulate somehow, and it is of interest to know just where that postulate is smuggled into the proof. If the proof does not involve the parallel postulate in some guise, it is either trivial (circular) or erroneous.
I don't know where OR enters this discussion. Maybe the identification of the use of similar triangles as the point where the parallel postulate enters the proof? I haven't proposed introducing this point into the article, and until that happens the matter is just a discussion, and OR is an inapplicable criticism. Brews ohare (talk) 21:34, 21 June 2010 (UTC)
BTW, Hardy's proof as presented by Cut-the-knot Proof #40 is not erroneous, it simply left out some steps (as 90% of proofs do because they don't want to go all the way back to Euclid's axioms). I provided some extra steps here, as you may recall. I believe those steps remove the Cut-the-knot objections, but you objected that it was OR, which is what led to introduction of Staring's proof here. Brews ohare (talk) 21:40, 21 June 2010 (UTC)
Hi Bob:
I gather you're not altogether happy with some of our exchanges. However, for my part, I've enjoyed working with you over the Staring proof and some other issues on Pythagorean theorem. Development of an article is not a seamless process, and sometimes things don't develop easily. A number of sections didn't go quite as I wanted originally. However, I think the end result was fine.
Where do we stand? Brews ohare (talk) 03:34, 29 June 2010 (UTC)
(I copied the following message of mine from User talk:DCGeist.)
Hi. Thanks for your edits. What I was trying to get at, was that putting quotes around words is like using weasel words. That is, the quotation marks themselves have the same effect as weasel words by giving the impression that someone said what was in the quotes, just like weasel words give the impression that the text associated with them was said by someone. --Bob K31416 (talk) 02:51, 9 July 2010 (UTC)
Hi, am not sure if you meant to delete your recent talk page contribution: [2]. Perhaps you are still thinking about it? I like the fix you suggested. Articles should be based on reliable, third-party (independent), published sources with a reputation for fact-checking and accuracy. This prevents unverifiable claims from being added to articles, and citing those sources helps prevent plagiarism and copyright violations. (In the first sentence I would say "prevents" rather than "helps prevent.")
Regarding "with a reputation for fact-checking and accuracy", note proposal 5 higher up on that talk page. This phrase may -- arguably, I would be grateful for feedback -- become redundant if proposal 5 is implemented. --JN466 23:50, 6 October 2010 (UTC)
Bob, I had an edit conflict with you at WP:V talk and somehow -- I don't understand how -- appear to have accidentally reverted an edit you had made to your post. I've undone it. [3][4]. Very sorry. --JN466 09:29, 28 October 2010 (UTC)
Hi, I stumbled upon your village pump 2008 proposal and was just wondering if you found a way to accomplish this? -PrBeacon (talk) 03:01, 9 November 2010 (UTC)
... the Ringo Starr quote on your user page :-) - Cheers - DVdm (talk) 22:28, 9 November 2010 (UTC)
We don't list "mentions". We only list in-depth references mentioned in a third-party source to establish notability. See Wikipedia:"In popular culture" content. Yworo (talk) 15:33, 11 November 2010 (UTC)
I have nominated speed of light for FAC. As a major contributor, please leave your 2cents on the review page.TimothyRias (talk) 16:07, 6 December 2010 (UTC)