Talk:Pythagorean theorem/Archive 3

A metric interpretation
I looked at the sources used in the section A metric interpretation and I didn't see any mention of the Pythagorean Theorem in the relevant parts.

Also, please note this excerpt from WP:NOR,
 * To demonstrate that you are not adding original research, you must be able to cite reliable published sources that are directly related to the topic of the article, and that directly support the material as presented.

It doesn't appear that the material in the relevant parts of the sources used in the section, are about the Pythagorean Theorem since there is no mention of it. I put up an OR template. --Bob K31416 (talk) 13:34, 17 May 2010 (UTC)


 * Bob: I don't think you looked carefully at the sources. For example, see Berberian, Lax, Eq 1, Saville and Wood (see Figure), Folland. There isn't any OR in this section, and IMO it is adequately sourced. Brews ohare (talk) 16:16, 17 May 2010 (UTC)


 * Hi Brews, Let's take it one step at a time. Regarding the Berberian p. 31 reference for the statement before the first equation of the section, p.31 doesn't appear to say that the equation is a form of the  Pythagorean Theorem. --Bob K31416 (talk) 17:21, 17 May 2010 (UTC)

Bob, I think that we really need to try and get a full understanding of this issue first, and then try and reconcile it with the references afterwards. This is clearly one of these tricky topics where the waters have been muddied by inconsistent usage of terminologies throughout the literature. I think we need to look past terminologies for the time being.

Lounesto rightly or wrongly refers to this equation,


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

as the Pythagorean identity. He is not technically correct in doing so, and he appears to be alone in the literature in doing so. But nevertheless, there is merit to Lounesto's usage of the term which needs to be fully explored. The equation in question is actually a particular form of the Lagrange identity which only applies in 3 and 7 dimensions due to the fact that the vector cross product only exists in 3 and 7 dimensions. The equation is not Pythagoras's theorem as such, but it is uncannily similar in substance to Pythagoras's theorem. It is indeed so similar to Pythagoras's theorem that I am not prepared to criticise Lounesto for using the name.

We only have to introduce the sine relationship in the vector cross product, and Pythagoras's theorem will fall out before our eyes. And what does this tell us? It should tell us that Pythagoras's theorem only holds in 3 and 7 dimensions, providing that the vector cross product obeys the sine relationship.

Then we come to the Jacobi identity. The sine relationship in the vector cross product is dependent on the Jacobi identity. But the Jacobi identity only holds in 3D. The conclusion is that Pythagoras's theorem only holds in 3D.

But we shouldn't have needed to introduce all these arguments. It is obvious from reading the article that Pythagoras's theorem is only proved from 3D geometry.

If you are trying to say that Pythagoras's theorem doesn't hold in 'n'D in general, then I am inclined to agree with you. David Tombe (talk) 19:41, 17 May 2010 (UTC)
 * Re "I think that we really need to try and get a full understanding of this issue first, and then try and reconcile it with the references afterwards." Nope, I don't think so. The issue is simple. A statement was made that doesn't appear to be backed up by the source. Here's my point again.
 * Regarding the Berberian p. 31 reference for the statement before the first equation of the section, p.31 doesn't appear to say that the equation is a form of the  Pythagorean Theorem.
 * Regards, --Bob K31416 (talk) 01:58, 18 May 2010 (UTC)

Bob, it seems then that we are both objecting to the same material, but for different reasons. You are objecting to that material on the grounds that it doesn't correspond to the cited references. I didn't even look at the references. I am objecting to it on the grounds that 'n' dimensional definitions in imitation of Pythagoras's theorem are not Pythagoras's theorem at all. Pythagoras's theorem is a theorem in 3D geometry which can be proved. 'n' dimensional facsimiles are not a theorem at all. They are merely a definition which can neither be proved nor disproved.

But I wanted to try and reach a consensus as regards the understanding of the subject. I didn't want to get involved in issues like 'no original research' merely on the grounds of terminologies which can be somewhat ambiguous. It's not a big jump between Pythagoras's theorem and the cosine rule.

While the debate over the validity of the 'n' dimensional case continues, I'll re-word the contentious material to bring it more accurately into line with accepted terminologies. David Tombe (talk) 08:41, 18 May 2010 (UTC)
 * Thanks. I just looked at your rewording and it's an improvement but it still has the problem that the source doesn't refer to it as the Pythagorean Theorem. That may be because it is a complex inner product space instead of a real one, though I can't really say what was in that author's mind. Hmmm. Looking at it some more, as far as I could see, the reference on p.31 didn't even give the expression that it reduces to for the orthogonal case. Seems like whoever put that into the article originally, was trying to include the complex case in the Pythagorean Theorem, which so far I haven't noticed in any of the sources that we have. --Bob K31416 (talk) 13:51, 18 May 2010 (UTC)
 * Have you tried actually searching for "pythagorean theorem hilbert space" on google books? There are lots of references there. For example, the "induction" result in this article is exactly proposition 3.10 of Banach algebra techniques in operator theory by Ronald G. Douglas. &mdash; Carl (CBM · talk) 14:20, 18 May 2010 (UTC)
 * Well, then why don't you get it and put it in and we'll see what it looks like.
 * But I should say that, I think the traditional 3-sided triangle Pythagorean Theorem is the one that is rich with ideas. The others don't seem to contribute much and only seem to use the name Pythagorean Theorem. I think we should mention that these other relations have been called the Pythagorean Theorem, but there doesn't seem to be much else about them that is very informative about the basic idea of the theorem. The ones in more generalized n-dim vector spaces, including ones with functions as vectors, don't seem to add anything to the basic concept of the Pythagorean Theorem (c2=a2+b2). They seem to just be definitions of length squared in the n-dim vector spaces.
 * But again, let's see what you have in mind for the article. And I'd be interested in seeing the Prop 3.10 that you mentioned. Could you give the link to it and the other info that you referred to? Thanks a bunch. --Bob K31416 (talk) 14:30, 18 May 2010 (UTC)
 * (1) The ordinary Pythagorean theorem says that the sum of the squares of the lengths of the two sides of a right triangle is the same as the square of the length of the hypotenuse. Triangles are inherently two-dimensional; it makes no difference what the dimension of the ambient space is. The only time that the dimension of the ambient space might make a difference is for formulas that involve a cross product, which is not defined in spaces of arbitrary dimension.
 * (2) As I pointed out, you can find the reference by Douglas on google books. I was just pointing out that the use of the term "Pythagorean theorem" to refer to things like that is actually easy to find in the functional analysis literature. &mdash; Carl (CBM · talk) 14:39, 18 May 2010 (UTC)
 * Well, I guess I should take back my "Thanks a bunch" since you don't want to contribute what you suggested to the article and you don't want to share the links you found to the particular items you mentioned at Google books. Have a nice day. --Bob K31416 (talk) 14:47, 18 May 2010 (UTC)
 * Go to google books, type in "Banach algebra techniques in operator theory pythagorean theorem" and click the first result. &mdash; Carl (CBM · talk) 14:49, 18 May 2010 (UTC)
 * I just looked for Prop 3.10 in it at Google books and it doesn't appear to be one of the pages they included for viewing online. --Bob K31416 (talk) 15:00, 18 May 2010 (UTC)
 * Maybe this will work - I don't really trust these links either . &mdash; Carl (CBM · talk) 15:05, 18 May 2010 (UTC)
 * That's as far as I got too. Prop 3.10 is apparently somewhere in pages 39-58 which they don't show. --Bob K31416 (talk) 15:08, 18 May 2010 (UTC)
 * It's on page 60.—Emil J. 15:17, 18 May 2010 (UTC)
 * Here's page 60. The url above had a "|" too many in the page parameter. DVdm (talk) 15:29, 18 May 2010 (UTC)
 * Thanks everyone. I made the corresponding changes in the article. --Bob K31416 (talk) 15:55, 18 May 2010 (UTC)
 * Thanks everyone. I made the corresponding changes in the article. --Bob K31416 (talk) 15:55, 18 May 2010 (UTC)

If that's OK now, let's move on to the next line in the section,
 * "Using mathematical induction, the previous result can be extended to any finite number of pairwise orthogonal vectors.[34}
 * 34.

The reference doesn't seem to support the part re "Using mathematical induction". The reference does say "repeated use of the Pythagorean Theorem" but that isn't a proof by mathematical induction. Also, the source is about Euclidean space, which involves real numbers, not complex numbers. And the source doesn't refer to the result as the Pythagorean Theorem, but only says that the Pythagorean Theorem is repeatedly used to get the result. --Bob K31416 (talk) 16:09, 18 May 2010 (UTC)
 * The finite sum formulation is exactly what the Douglas reference states as Prop. 3.10. Any computation like that requires mathematical induction; it shows up in the Douglas book in the second equality, when the rules for inner products are used repeatedly. Indeed, "repeatedly" is just an informal way of saying "by induction". Moreover, the entire Douglas book (like most functional analysis books) is in the context of complex inner product spaces. See page 58 . The same thing is given (with the same proof) on p. 165 of Folland's "real analysis", which also uses complex inner products. &mdash; Carl (CBM · talk) 16:37, 18 May 2010 (UTC)
 * Re "The finite sum formulation is exactly what the Douglas reference states as Prop. 3.10." - I moved the 'Douglas1998' citation to replace the Lax citation. --Bob K31416 (talk) 17:12, 18 May 2010 (UTC)


 * Note everyone, it looks like you can find the remaining material from the section "A metric interpretation" in the section Inner product spaces, which is the new title, etc from CBM. --Bob K31416 (talk) 17:23, 18 May 2010 (UTC)

Proof
"Pythagoras's theorem not holding in 'n'D in general"? See for instance Theorem 1.6. DVdm (talk) 10:27, 18 May 2010 (UTC)


 * They are not talking about the Pythagorean theorem. They are talking about some other identity involving cross products. &mdash; Carl (CBM · talk) 13:10, 18 May 2010 (UTC)
 * Look at part b where it says, "proving the Pythagorean theorem." --Bob K31416 (talk) 13:17, 18 May 2010 (UTC)
 * When I said "they" I mean you and someone else higher on this talk page. The book is talking about the actual Pythagorean theorem. The discussion higher up is about cross products. &mdash; Carl (CBM · talk) 13:26, 18 May 2010 (UTC)
 * Thanks for clarifying, and so far I haven't contributed anything to a discussion of cross products. Anyhow, thanks to you and DVdm for your comments. Regards, --Bob K31416 (talk) 13:36, 18 May 2010 (UTC)

Cross products
Bob, if you look at page 50 of the Sterling Berberian reference you will see a reference to a generalised Pythagorean theorem in 'n' dimensions. References to 'n' dimensional Pythagorean identity do exist. You have said on my talk page that just because sources exist, it doesn't mean that the material has to go into the article. I wasn't the person who put the material in, but nevertheless, I think I fully understand why it was inserted. See my reply to CBM below. David Tombe (talk) 17:02, 18 May 2010 (UTC)

"spaces of n dimensions
What is the point of this section title? The "metric interpretation" is for arbitrary inner product spaces, including infinite-dimensional ones. And "The Pythagorean theorem is a specific statement about triangles in two dimensions." is somewhat odd because all triangles are two-dimensional, regardless of the dimension of the ambient space. I am going to hack at it some. &mdash; Carl (CBM · talk) 16:44, 18 May 2010 (UTC)


 * I moved the stuff on inner products to the end, as the other stuff in that section seems to be more concretely focused. I also removed the stuff about Fourier analysis, which seems a little remote from the topic of the article. &mdash; Carl (CBM · talk) 16:54, 18 May 2010 (UTC)

CBM, I am the one who introduced cross products to the discussion. I did so deliberately because Lounesto refers to this version of the Lagrange identity,


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

as the Pythagorean identity. He may be in a minority in doing so, and the expression may not technically be the Pythagorean identity. But nevertheless, this equation becomes Pythagoras's theorem as soon as we introduce the sine relationship in the cross product. This tends to hint at the fact that Pythagoras's theorem can only hold in 3 or 7 dimensions. And since the sine relationship is closely connected with the Jacobi identity, this tends to further narrow it down to 3D only.

But you are correct that there are plenty of references in the literature to Pythagoras's theorem in 'n' dimensions. In my opinion, these references are talking about a concept that is a mere definition, and something that is a mere definition should not be confused with an actual theorem, such as Pythagoras's theorem, simply because the definition has been created in the likeness of an extrapolation of Pythagoras's theorem to 'n' dimensions.

In 3D, Pythagoras's theorem is a theorem that can be proved. It can be proved using areas in Euclidean geometry. The generalised Pythagorean identity in 'n' dimensions on the other hand is merely a definition which can neither be proved nor disproved.

Therefore the article was correctly written up yesterday, in line with the sources, by having segregated Pythagoras's theorem into an 'areal interpretation' and a 'metric interpretation'. There is an important distinction between the two cases. Unfortunately there was a trivial error whereby the 'cosine rule' was referred to as 'Pythagoras's theorem', but that did not require a 'no original research' banner to be erected. That error has now been corrected. David Tombe (talk) 17:03, 18 May 2010 (UTC)


 * The "unprovability" claim you have made is not true. The following result is generally called "Pythagorean theorem" in functional analysis books and can indeed be proved:
 * Given any finite set $$\{ u_1, \ldots, u_n\}$$ of pairwise orthogonal vectors in an inner product space, $$||\textstyle\sum_i u_i||^2 = \textstyle\sum_i ||u_i||^2\,$$.
 * However, if we stick with just Euclidean spaces ($$\mathbb{R}^n$$), the Pythagorean theorem holds in all of them, in the form
 * (*) The sum of the squares of the lengths of the sides of a right triangle equals the square of the length of the hypotenuse.
 * This is because every triangle sits in a plane, and every plane is isometric to the xy-plane. So there is no triangle in 6D, 15D, or 17677D space that does not satisfy (*). &mdash; Carl (CBM · talk) 17:12, 18 May 2010 (UTC)


 * A while ago there was a similar discussion (well, "discussion") -- with the same people -- on Seven-dimensional cross product where they objected that angles are meaningless in Rn, that is, until they were pointed to a source that explicly stated that two vectors in Rn span a two dimensional subspace in which the angle stares at theit face, so to speak. See refs (3) and (4). It looks like they are trying to import their synthesis in here now. DVdm (talk) 17:18, 18 May 2010 (UTC)

Carl, Yes indeed, a triangle sits in a two dimensional plane. The angle between any two sides of the triangle relates to the degree of rotation about an axis in the third dimension. But if we have a triangle in a four dimensional space, how do we know which axis of rotation to define the concept of angle from?

From what I can recall, all matters relating to Pythagoras's theorem in 'n' dimensions concern the definition of distance. It can't therefore be the same thing as that which bears the same name in 3D. The classical 3D Pythagoras's theorem is a theorem. It is a theorem which can be proved. It is not a definition. And until we know what angle means in 4D, I can't see how we can extrapolate the theorem to higher dimensions. That is why I think that the article should segregate the two issues. David Tombe (talk) 17:47, 18 May 2010 (UTC)


 * David, it appears that you are trolling. Please stop. DVdm (talk) 17:51, 18 May 2010 (UTC)


 * The classical Pythagoras' theorem is in 2D, not in 3D. And Pythagoras' theorem in general inner product spaces is not a "definition". Even if we consider the most simple case of Rn with the standard inner product, the definition only consists of the special case of the identity where the number of vectors equals the dimension of the space, and the ith vector is of the form (0,...,0,ui,0,...,0) with the only nonzero entry at the ith coordinate. Other instances of Pythagoras' theorem are not included in the definition, they have to be proved.—Emil J. 18:02, 18 May 2010 (UTC)

Emil, The classical Pythagoras's theorem is in 3D. The third dimension is the rotation axis associated with the angle between the two sides. The Lagrange identity only leads to Pythagoras's theorem when in 3D. Try getting Pythagoras's theorem from a Lagrange identity in 2D and see how you get on. David Tombe (talk) 18:24, 18 May 2010 (UTC)
 * Please stop the trolling. It's simply inconceivable that a physicist would believe that there needs to be a third dimension for the Pythagoras theorem to be stated or proved, or a one-dimensional axis for an angle to make sense. I don't care whether this is a problem with WP:POINT or one with WP:Competence. In either case, once a discussion has firmly reached pseudo-mathematics territory it needs to stop. Hans Adler 18:54, 18 May 2010 (UTC)
 * David, Re "The classical Pythagoras's theorem is in 3D. The third dimension is the rotation axis associated with the angle between the two sides." - I didn't understand that. Would you care to explain in more detail? Thanks. --Bob K31416 (talk) 20:28, 18 May 2010 (UTC)
 * Bob, do we really need this here? This is the talk page of Pythagoras' theorem, not the talk page to entertain David Tombe. Please do this on his or on your talk page? Thanks. DVdm (talk) 20:33, 18 May 2010 (UTC)
 * David, I think I just realized what you are talking about. You recognize that an angle corresponds to rotation about an axis. In the case of a triangle, one can think of an axis of rotation at each vertex that can be considered as extending into the third dimension. However, one doesn't have to look at it that way. Triangles can be constructed simply by the joining of three line segments in a plane, without any consideration of an axis of rotation that is in the third dimension.


 * Also, note that the triangle itself, i.e. three joined line segments, lies completely in a two dimensional plane. You can talk about axes of rotation in a third dimension but they are not part of the triangle, which is the three line segments.


 * Here's a dictionary definiton of triangle,
 * The plane figure formed by connecting three points not in a straight line by straight line segments.
 * --Bob K31416 (talk) 20:40, 18 May 2010 (UTC)

The axis of rotation in two dimensions is a point. But rather than defining it that way, one could define rotations as happening only around 12-dimensional flats in 14-dimensional space, and based on that definition declare that the Pythagorean theorem is "really about" 14-dimensional space. I don't see any difference between that point of view and Tombe's claims that rotations can only be defined around lines in 3d and that the theorem is 3-dimensional. It's crankery and it needs to stay out of the article. —David Eppstein (talk) 21:15, 18 May 2010 (UTC)


 * ... and out of this talk page. DVdm (talk) 21:25, 18 May 2010 (UTC)

Bob, The equation,


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

makes it clear that Pythagoras's theorem involves both an inner product and an outer product. The section in the main article which extrapolates Pythagoras's theorem to 'n' dimensions focuses on the inner product and totally ignores the outer product. In fact, the title of that section has just been changed today to 'inner product spaces'. But if we consider the outer product as well, which we should do, we will then be restricted to 3D. There is no such thing as a 2D Pythagorean identity. I'll cite Lounesto to back me up on that point. David Tombe (talk) 22:45, 18 May 2010 (UTC)
 * Please stop. See WP:DEADHORSE. Justin W Smith talk/stalk 22:56, 18 May 2010 (UTC)


 * The classical Pythagorean theorem has neither inner nor outer products. It only speaks about lengths of line segments and about right angles, all of which are well defined on planes. The more general form of the theorem replaces "length" with "norm", but also needs to replace "perpendicular" with "orthogonal". This is why the theorem generalizes to inner product spaces rather than to normed spaces. On the other hand, the Pythagorean theorem has very little to do with cross products in general (particularly because cross products exist in very few of the spaces for which the Pythagorean theorem holds). &mdash; Carl (CBM · talk) 00:22, 19 May 2010 (UTC)
 * The classical Pythagorean theorem is not about lengths; it is about areas of squares. Michael Hardy (talk) 01:09, 19 May 2010 (UTC)
 * Chacun à son goût. &mdash; Carl (CBM · talk) 01:17, 19 May 2010 (UTC)

Carl, You keep saying that the outer product has got nothing to do with Pythagoras's theorem. You will however agree that,


 * $$ {\cos}^2 \theta + {\sin}^2 \theta = \frac{a^2 + b^2}{c^2} = 1,$$

and that therefore the sine term is clearly an integral part of Pythagoras's theorem. It then follows that,


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

which is Lagrange's identity in 3 or 7 dimensions. We can forget about the 7 dimensional case because the vector triple product and the Jacobi identity do not hold in the 7D case, and so that rules out its connection with angles and geometry. This suggests that Pythagoras's theorem is strictly a 3D affair.

The section entitled 'inner product spaces' would certainly seem to be in line with the modern literature. But the bit were it says that the Pythagorean identity can be generalised to inner product spaces seems to put an entirely different interpretation on Pythagoras's theorem which totally ignores the outer product connection. As you have correctly pointed out, they even change the terminologies from 'perpendicular' to 'orthogonal', and 'length' to 'norm'.

Although I didn't write the two sections entitled 'areal interpretion' and 'metric interpretation', which caused the controversy, I am pretty sure that the underlying purpose of those two sections was to highlight this important distinction. David Tombe (talk) 09:17, 19 May 2010 (UTC)


 * "... therefore the sine term is clearly an integral part of Pythagoras's theorem." => By that reasoning the number 1 is also clearly an integral part of Pythagoras's theorem. Per WP:NOCLUE and WP:IDIDNTHEARTHAT, you are disrupring this talk page. This is the 4th time you are kindly asked to stop this. Take this to your own talk page please - thank you. DVdm (talk) 09:27, 19 May 2010 (UTC)

DVdm: It is inappropriate to squelch discussion on a talk page. Several authors (Hardy, Tombe, BobK, EmilJ) are trying to sort out just what constitutes the Pythagorean theorem in n-dimensions, and this discussion may lead to clarification in the main article. Many readers of the article may benefit. It is appropriate for you to help draw water, but not to poison the well. Brews ohare (talk) 14:39, 19 May 2010 (UTC)
 * It is not the proper role for editors of Wikipedia to "sort out just what constitutes the Pythagorean theorem in n-dimensions". That sounds like WP:OR (or at least WP:SYNTH) to me. This talk page has apparently become a forum for all sorts of fringe ideas on how *we* (the editors) should generalize the Pythagorean theorem to arbitrary dimension. The article is too long/complex already.  Keep it simple. Justin W Smith talk/stalk 14:28, 20 May 2010 (UTC)

Areas vs. squared lengths
Response to Michael Hardy and Carl: As cited in the article, Tobias Dantzig has pointed out that Pythagoras' theorem in 3D is both statement about the sides of the right triangle, and about the areas of the squares on the sides. The proofs of the theorem are various, with the similarity proof based upon ratios of sides and the Euclidean proof directly related to areas. That difference may seem a hair split, but when the theorem is generalized, one has a choice of which aspect to emphasize, and it may be that both properties do not hold simultaneously.

Generalization of the theorem can take a variety of forms, and the dimensionality of the space is only one of them. So it is claimed in an older version of this article that the similarity proof is preferable, based upon lengths, because in general set theory the notion of areas becomes much more complex, leading, for example, to the Banach–Tarski paradox.

When it comes to generalization in dimensionality, there is no doubt that the sum of squares form is widely referred to as Pythogoras' theorem. A number of references to that effect were deleted without paying any attention to them, for example, Folland Also in any number of dimensions, angle is defined by:
 * $$\cos \theta = \frac { \mathbf{a \cdot b}} {\|\mathbf a \| \| \mathbf b \| } \, $$

which obviously also defines the sine of the angle. However, the cross product requires a x b to have two properties. One is that its magnitude be ab sin θ, and the other that it be perpendicular to a and b for arbitrary choices of a and b. It is the orthogonality requirement that cannot be met in every dimensionality, and restricts the Pythagorean theorem involving the outer product to only 3D and 7D. Brews ohare (talk) 15:09, 19 May 2010 (UTC)


 * The "areal" interpretation works just as well in higher-dimensional spaces as in 2D. The point, as always, is that both the triangle and the squares determined by its sides can all be found in a single 2D plane, and so any counterexample in an ambient space of higher dimension is actually a planar counterexample. Any such counterexample would work both for the "lengths of sides" and "areas of squares determined by the sides" versions of the 2D Pythagorean theorem. (As Maor's reference points out, the reason the Greeks phrased it in terms of area is that they had no other way to refer to the square of a number. This is no longer a problem after the introduction of coordinates and the interpretation of the plane as a vector space.)


 * One issue here is that neither the "areal" interpretation (about areas) nor the "metric" one (about lengths) is about cross products in the first place. The existence of a cross product is a very limited phenomenon, and when a cross product exists it will satisfy some formulas. But these formulas are not "the Pythagorean theorem" as such, they are just special formulas that happen to hold for spaces with a cross product. I have no idea where the idea that the Pythagorean theorem is fundamentally about cross products could have come from. The original theorem is a theorem of two-dimensional geometry, where there is no cross product. The extension of the theorem to three dimensions is no different than the extension to 4, 8, or 15 dimensions: in each case the definition of "right triangle" relies only on the plane in which the triangle sits.


 * In any case, the article seems to be cleaned up now. &mdash; Carl (CBM · talk) 15:30, 19 May 2010 (UTC)


 * Carl: We may be at cross purposes here. There are three things that seem to have become blurred together, when I wished to keep them separate:


 * 1. The generalization of Pythagoras' theorem based upon areas, versus the generalization based upon lengths, regardless of dimensionality. In this respect, my (weak) understanding of the Banach–Tarski paradox is that Pythagoras' theorem expressed in terms of areas won't work in some cases, while the other will. This is not a dimensionality argument.


 * 2. The extension to n-dimensional spaces, which refers to "lengths" or "norms" and has no restrictions on dimensionality of the space. This extension does not refer to areas, but to norms. Area doesn't come up.


 * 3. The Pythagorean theorem as used by Lounesto, and maybe only by him, as involving the cross product. I agree that this appears to be a limited usage. I understand cross-product in terms of areas.


 * So my question would be: do you agree that the extension of Pythogoras' theorem based upon lengths or norms is generalizable in different ways than the theorem based upon areas? Brews ohare (talk) 16:08, 19 May 2010 (UTC)


 * No, I don't agree. First, the Banach–Tarski paradox is about measures ("areas") of extremely complex sets. There is no difficulty with the areas of simple sets such as squares. Moreover, as I pointed out, any counterexample to the "length" form of the Pythagorean theorem immediately gives a counterexample to the "area" form, and vice versa, for spaces over the reals. For spaces over the complex numbers, one usually doesn't talk about "areas of squares", but presumably the area of a square would still be the square of the norm of its side, and so again the length and area versions of the Pythagorean theorem would coincide.


 * I think the difficulty may be that you are reading Lounesto's use of the phrase "Pythagorean theorem" as if he is defining the Pythagorean theorem. What he is actually doing is referring to a particular property of the cross product which follows from the Pythagorean theorem. This does not mean that the Pythagorean theorem itself is dependent on cross products in any way. Similarly, when Lounesto uses the phrase "orthogonality" on the previous line on p. 96, he is not defining orthogonality, but rather using the word to describe a property of the cross product. &mdash; Carl (CBM · talk) 16:14, 19 May 2010 (UTC)

Hi Carl: Thanks for the response. If I may, I'll reword your answer to be sure I got it straight: (i) Banach–Tarski paradox is irrelevant in this context.

(ii) As for Lounesto: He footnotes WS Massey, if v x w is orthogonal to both v and w, and
 * $$ \| \mathbf {v \times  w}  \| ^2= \| \mathbf v \|^2 \|\mathbf w \|^2 - (\mathbf v \cdot \mathbf w )^2 \, $$

then dimensions of space are 3 or 7. So the restriction is upon the relation to the cross-product, and Pythagoras' theorem always refers to sums of squares. How's that? Brews ohare (talk) 17:52, 19 May 2010 (UTC)


 * (i) Yes. (ii) Massey is interested in the question: "for which n can you define a binary operation on $$\mathbb{R}^n$$ that acts somewhat like the cross product?" That is very different than: "for which n does the Pythagorean theorem hold in $$\mathbb{R}^n$$?"  The condition on $$ \| \mathbf {v  \times  w}  \| ^2$$ is just one property of the actual cross product that Massey includes in his requirements on a putative higher-dimensional cross product. (By the way, Massey also requires the cross product operation to be bilinear. I am able to view Massey's paper on JSTOR at .) &mdash; Carl (CBM · talk) 18:38, 19 May 2010 (UTC)

Complex/real
Yes, you seem to be misunderstanding something very basic, Bob K31416. For example, $$\mathbb R^n$$ with the usual inner product is a real inner product space, but it is not a complex inner product space. Neither is more general than the other. Just read the definition.—Emil J. 11:23, 19 May 2010 (UTC)


 * EmilJ is right here. In order to be a complex inner product space, a vector space has to be a vector space over the complex numbers first.


 * When I wrote the section yesterday, I did it in a hurry, mostly copying text that was elsewhere. I fully expected that this issue would be corrected in time. But I was waiting for comments before doing to much more editing myself. &mdash; Carl (CBM · talk) 11:32, 19 May 2010 (UTC)


 * Thanks. Looks like my mistake since a real inner vector product space excludes vectors that are complex and not real. Is that the point?  --Bob K31416 (talk) 11:55, 19 May 2010 (UTC)


 * Pretty much. &mdash; Carl (CBM · talk) 12:00, 19 May 2010 (UTC)

the theorem generalizes to inner product spaces rather than to normed spaces.
Carl: Will you kindly elaborate upon this statement of yours and consider how that distinction might be placed in the article? Brews ohare (talk) 15:38, 19 May 2010 (UTC)


 * I don't think it should be placed in the article. However, I can expand on it. In order to state the Pythagorean theorem, one has to have both a concept of length and a concept of orthogonality. If you start with just a normed vector space then the notion of length is given by the norm, but without an inner product there is no naive way to define orthogonality. Moreover, there are some norms that do not come from inner products (e.g. $$\ell_\infty$$) and so it is not always possible to "cook up" an inner product on a normed vector space to turn it into an inner product space with the same norm. Without a notion of orthogonality, one cannot even state the Pythagorean theorem, much less prove it. &mdash; Carl (CBM · talk) 16:00, 19 May 2010 (UTC)

A norm comes from an inner product if and only if it satisfies the parallelogram law. Only in that case does the Pythagorean theorem make sense. Michael Hardy (talk) 18:43, 19 May 2010 (UTC)

Deletion of side of pentagon
I feel that Dick Lyon's deletion of this section is simply a petty assertion of his individual opinion, and that all his objections were adequately discussed above, and never responded to. Brews ohare (talk) 06:42, 20 May 2010 (UTC)


 * Surely it's easier to prove directly using similar isosceles triangles that the side and diagonal are in extreme and mean ratio, avoiding any mention of Pythagoras? The demonstration DickLyon deleted seems to me to be more mathematics made difficult than something illuminating. Your text here claims that the use of Pythagoras to prove the ratio of side to diagonal in a pentagon can be sourced to Brodie's cut-the-knot page, but that is a gross misreading of that source. What Brodie actually does is use similar isosceles triangles to prove the ratio (the paragraph beginning "The Greek geometers realized" prior to mention of Pythagoras), and then uses Pythagoras to compute the length of a line segment in a particular compass-and-straightedge construction and compare to the known lengths in a pentagon to show that the construction is correct. So this derivation would be relevant only in an article about compass-and-straightedge constructions of pentagons, not in more general articles about the Pythagorean theorem and the golden ratio. —David Eppstein (talk) 07:07, 20 May 2010 (UTC)


 * Eppstein: There is no "gross misreading" of the source; come on! The presentation in the deleted material is the same as that in the source. The source has other objectives, namely to compute the ratio of the diagonal to the side (the golden ratio), but the omission of that goal already is explicitly stated from the outset. Go ahead and say "Me too, Yea Dick!!", but, please, avoid fabrication and annoying nonsense. This example illustrates a more complex use of Pythagoras' theorem than the preceding 3-sided, 4-sided examples, and follows the sequence with a 5-sided example. And unlike some other determinations of the side, it relies only upon Pythagoras' theorem to do so. Brews ohare (talk) 14:04, 20 May 2010 (UTC)


 * I agree with David. The Brodie source was grossly mischaracterized, or misapplied, if not misread.  And the point of the rewrite remains unclear.  And not every place where people use the Pythagorean theorem is a good addition to the article.  And as before, if several editors push back on what you're doing, the burden should be on you to justify it.  Asserting your feelings about my motivations doesn't go very far in that direction; see WP:NPA; and no, you never did address my objections, so after a long enough wait I took out the offending material that now at least three of us have objected to.  But I'll stop now; I need to get back to not letting you get on my nerves. Dicklyon (talk) 08:07, 21 May 2010 (UTC)

Dick: The points of the rewrite are as follows.
 * 1. To state the side of the pentagon is an irrational number, give its value and source it.
 * 2. To show a method of construction, and source it.
 * 3. To show that Pythagoras' theorem by itself can verify the construction, and source it.
 * 4. To add a sentence about the historical origins of the idea of the irrational and its relation to the pentagon, and source it.

Inasmuch as this subsection is about connection of Pythagoras' theorem to the irrational numbers, and because of the pentagon's historical role in this topic, and because Pythagoras' theorem is able single-handedly to establish this result, this example seems on topic to me. The Brodie source is used to source the construction, and Ludlow is added as yet another source. Please state explicitly just how this misapplies these sources (it does not), and state explicitly how the points above have not been clearly explained (they are). It is difficult for me to be more clear when only vague charges are made without examples to support them, and no clear statement of the directions for remedy are pointed out. Brews ohare (talk) 14:34, 21 May 2010 (UTC)


 * I've already stated explicitly how it misapplies the sources and you just blew it off. I don't see why it would be more likely to get through to you a second time. —David Eppstein (talk) 15:25, 21 May 2010 (UTC)


 * David. Here is what I heard:
 * 1. The side and diagonal golden ratio can be established without Pythagoras. The golden ratio is not what this sub-subsection is about. An irrelevant remark, therefore.
 * 2. The determination of the side of the pentagon is a "gross misreading" of Brodie. No claim is made that determination of the side is the main objective of Brodie's work, but only that he determined the side. He did. That is not a misreading.
 * So, David, what are you talking about? Brews ohare (talk) 15:52, 21 May 2010 (UTC)


 * I agree with David: you've been told, you don't get it, and telling you again or in more detail has proven historically to be a waste of effort.  Dicklyon (talk) 01:38, 22 May 2010 (UTC)

An easy out, eh Dick and David: don't attempt to get your point across, don't answer to responses, speak in vague generalities, and blame the poor communication upon me. Great collaboration, eh? I work at writing, making diagrams, and finding sources, and you two yawn in your easy chairs and click the remote. Brews ohare (talk) 02:53, 22 May 2010 (UTC)

Root rectangles
Say, how about using Root rectangles instead. Figure 10 in that section looks like a simple and sound way to construct lengths that are the square root of any integer, including lengths that are irrational numbers such as the square roots of 3, 5, etc. It's very elegant and very Pythagorean, IMO. Regards, --Bob K31416 (talk) 08:11, 20 May 2010 (UTC)


 * It would be better to stick to math sources. Dicklyon (talk) 08:07, 21 May 2010 (UTC)


 * Shouldn't the idea be evaluated independently of what one personally defines as a "math source"? It seems like it is as sound an idea as anything else in the article, and it is a purely mathematical construction based closely on the Pythagorean Theorem. --Bob K31416 (talk) 21:13, 21 May 2010 (UTC)


 * Yes, as I said, there's nothing wrong with the construction or the idea. But to attribute it to Hambidge and his "root rectangles" seems odd.  There are plenty of more appropriate sources for math relevant to the Pythagorean theorem without resorting to its appications in obscure pyramidology-like rantings. Dicklyon (talk) 01:27, 22 May 2010 (UTC)
 * So you feel "there's nothing wrong with the construction or the idea" yet you want to keep it out of the article because "Hambidge and his 'root rectangles' seems odd". It appears that excluding it would be on the basis of your own feelings against Hambidge and wouldn't that be a violation of WP:NPOV? --Bob K31416 (talk) 02:19, 22 May 2010 (UTC)


 * Dick, I just noticed in the next section that you found a reliable source for a simpler geometric construction of roots of integers. That would be fine with me.  --Bob K31416 (talk) 02:37, 22 May 2010 (UTC)

It's hard to object to this construction as it follows directly from Pythagoras' theorem. I'd like it to be added to the article rather than replacing the simple examples. Brews ohare (talk) 23:03, 21 May 2010 (UTC)

Other version of Figure 10

 * Brews ohare, I moved the image you just created to your message, since it is not the figure in the section Root rectangles that I was referring to. --Bob K31416 (talk) 00:13, 22 May 2010 (UTC)

This figure is constructed the same way as Hambidge's Figure 10 in the article Root rectangles. Brews ohare (talk) 00:25, 22 May 2010 (UTC)


 * And I created this subsection and moved the image here, so as not to digress from the discussion in the above section.
 * Re your remark, not to me it isn't. --Bob K31416 (talk) 00:56, 22 May 2010 (UTC)


 * There's an even simpler and clearer construction, equivalent but without the distraction of the rectangles (and the colored rectangles) in this version of Euclid, as simple consequency of theorem I-47, the Pythagorean theorem. Use that instead of some wack job maybe?  Dicklyon (talk) 01:36, 22 May 2010 (UTC)

Does this make sense?
Brews wrote that "This construction ... has the merit of requiring only Pythagoras' theorem for verification." What does that mean? Can it be true? It's hard to see how. If you look at Brodie, it's clear that he has to derive a value for the golden ratio using geometric arguments, and he uses that verifying the construction. I guess Brews means to verify not the construction, but that the length constructed is the same as the well-known result. If you rely on what's known, I admit, the problem is a lot simpler that way. But that's not what's meant by verifying the construction, is it?

And a question: Brodie says "since the ray PR bisects angle CPQ, ..." My geometry is a bit rusty, so I don't see what this is deduced from. Can someone help me out? Dicklyon (talk) 02:05, 22 May 2010 (UTC)


 * Dick: Let me repeat myself for the third time: There is no attempt in the present sub-subsection to derive the golden ratio. Yes, I am verifying the construction, using Pythagoras only. That is also done by Brodie, and is why he is sourced. Yes, it verifies the construction: because the correct result is known, all you have to do is show that the construction produces the correct result. I do not use the bisection at all. If you want a proof of bisection, look at the construction in Pentagon. Brews ohare (talk) 02:59, 22 May 2010 (UTC).


 * Yes, I realize you don't rely on the bisection, but do rely on the answer being known; so maybe you can say "This construction ... has the merit of requiring only Pythagoras' theorem for verification, when the length of the side of the pentagon is already known by some other method." or "This construction ... has the merit of requiring only Pythagoras' theorem and knowledge of the answer for verification.".  But I was just asking for help in understanding the Brodie page, which claims to be a proof but omits an explanation of a key step.  I don't see what you mean in Pentagon; the only "bisect" there is a construction, not a deduction about angles in the pentagram.  Dicklyon (talk) 20:17, 22 May 2010 (UTC)


 * I think the "Consequences and uses of the theorem" section has become too diffuse.


 * This article is not a catalog of methods to answer all questions of the form, "How can I construct a length that is a specific irrational number?" If Wikipedia were to have such a catalog, it should probably be in a separate article.


 * I think the previous, shorter subsection "Distance in Cartesian coordinates" was adequate. The newly renamed section "Distance in various coordinate systems" does not offer results as general as advertised. I don't count each higher-dimensional Cartesian coordinate system as a separate system. The expanded version only extends to polar coordinates. Again, this article is not a how-to repository. People looking for how to calculate a distance in a particular coordinate system might do better to start in the article on that coordinate system. - Ac44ck (talk) 03:36, 22 May 2010 (UTC)
 * The section on distance does explain how the general case may be treated, and shows only one example. If more examples were provided it would be a digression indeed. The example of polar coordinates happens to connect with the generalization of Pythagoras' theorem to the Law of cosines which is linked to the discussion of that topic. Brews ohare (talk) 03:58, 22 May 2010 (UTC)

Spacing
Brews ohare, if your browser displays Δc/Δa so that the c/ looks like d, it is seriously broken. You should fix your setup, not insert any blanks there, which make it look like c followed by a large empty space followed by / for the rest of us.—Emil J. 14:56, 20 May 2010 (UTC)
 * My personal opinion is that formulas that require minute spacing adjustments should just use the &lt;math> tag and be done with it. There is too much variability between browsers to hope that micro adjustments will display consistently. For simple formulas such as "2x+1" plain HTML is usually better. In the case of inline fractions, I have no idea whether they usually display correctly in HTML. &mdash; Carl (CBM · talk) 15:31, 20 May 2010 (UTC)
 * Probably that is the simple solution, so I'd use $$c/ \,$$ rather than c&thinsp;/ to avoid c/ looking like d. However, I think that Emil is being very picky to object to c&thinsp;/ as leaving a horrible space. Brews ohare (talk) 15:56, 20 May 2010 (UTC)
 * If one version looks too narrow to you, it's not surprising if another version looks to wide to him. Using math mode seems like an easy compromise. &mdash; Carl (CBM · talk) 16:05, 20 May 2010 (UTC)

Carl: A sensible observation. What I don't like about the math approach is that alignment with the surrounding text is lost: while $$ c / d \,$$ looks pretty good, $$a^2 + b^2 = c^2 \, $$ drops below the surrounding text in Firefox and $$a^2 + b^2 = c^2 $$ aligns, but looks too big. The form a&thinsp;2 +b&thinsp;2 = c&thinsp;2 aligns and doesn't crowd the superscript into the letter. In IE this leads to a slightly larger spacing, but nothing jarring. Maybe Emil could tell me what the appearance is in his browser? Brews ohare (talk) 17:19, 20 May 2010 (UTC)
 * It looks like this: [[File:Pythagoras snapshot.png]]. The superscripts are dangling in the void halfway between their actual base symbol and the next character. Anyway, this kind of adjustments to fit a particular browser setup are fundamentally wrong, you should take seriously what Mr.98 told you on RD/C.—Emil J. 13:10, 21 May 2010 (UTC)
 * Emil: It is unfortunate that a reader has to tinker with browser settings to read the wedge-product symbol and then find that italicized symbols are poorly spaced. I would guess that my problems are not peculiar to myself, and the resulting very poor display is far worse than the minor issue your screen shot displays. Lucida Sans Unicode works better than Arial Unicode MS, but it's not great either. There is no solution, it seems. Brews ohare (talk) 16:13, 21 May 2010 (UTC)

Irrational numbers
I've rewritten this section and provided the figure from Euclid. Brews ohare (talk) 03:59, 22 May 2010 (UTC)
 * Good move. Regards, --Bob K31416 (talk) 04:14, 22 May 2010 (UTC)


 * Yes, much better. Dicklyon (talk) 20:13, 22 May 2010 (UTC)

The Lagrange identity and Pythagoras's theorem
Brews, regarding your latest edits to the section entitled "The Lagrange identity and Pythagoras's theorem", a linkage between the Lagrange identity and Pythagoras's theorem has been demonstrated for the cases of 3 and 7 dimensions. Should we now add any comments regarding the possible elimination of the 7D case on the basis that the Jacobi identity only holds in 3D, and based on the linkage between the sine of the angle and the Jacobi identity in the proof which I supplied on your talk page? As you point out, the proof shows that the sine of the angle follows from the Jacobi identity. You then ask, "does the logic work in reverse?". I'm not sure, but I would think that it does work in reverse. David Tombe (talk) 14:14, 23 May 2010 (UTC)


 * David:
 * The relation
 * $$ \| \mathbf{a \times b} \| = \|\mathbf a \|^2 \|\mathbf b \|^2 -(\mathbf{a \cdot b}) ^2 \ ,$$


 * is a defining property of the cross product, so existence of a cross product implies:


 * $$ \| \mathbf{a \times b} \| = \| \mathbf a \|^2 \| \mathbf b \|^2  \left( 1  -\cos ^2 \theta \right) \  .$$


 * Assuming the definition of sin and cos imply:


 * $$ \sin^2 \theta + \cos^2 \theta = 1 \, $$


 * then
 * $$ \| \mathbf{a \times b} \| = ab \ \sin \theta \ . $$


 * This line of logic shows the correct formulation of this section should be to derive the above relation assuming the Pythagorean trigonometric identity is fundamental to the meaning of the trig functions, and the present formulation in the article has this turned around. Accordingly, I have rewritten this section. Please tell me if you approve. Brews ohare (talk) 16:47, 23 May 2010 (UTC)


 * Brews: since David Tombe is as far as I know still indefinitely restricted from comments concerning you, it seems inappropriate for you to provoke him into breaking that restriction by replying directly to him here. —David Eppstein (talk) 17:48, 23 May 2010 (UTC)
 * “Advocacy for or commenting upon Brews_ohare” within that context does not refer to comparing notes on a particular topic. It refers to engaging in any discussion or petition regarding the bans imposed upon me about contributing to physics articles. Brews ohare (talk) 20:53, 23 May 2010 (UTC)
 * You have a very strange idea of what "broadly construed" means. —David Eppstein (talk) 21:32, 23 May 2010 (UTC)

Brews, I've reworded it slightly so as to add a proviso. The proviso caters for any doubt surrounding the issue of whether or not the sine relationship applies in the case of the 7D cross product. We all know that the Jacobi identity only holds in 3D, and the proof which I supplied on your talk page is somewhat persuasive of the fact that the sine of the angle in the cross product is dependent on the applicability of the Jacobi identity. David Tombe (talk) 18:05, 23 May 2010 (UTC)
 * Your version is better. Brews ohare (talk) 20:55, 23 May 2010 (UTC)

Angle brackets
In the section Inner product spaces there is a problem with the angle brackets on some computers where they appear as question marks or boxes. Currently the angle brackets are implemented with the html codes &amp;lang; and &amp;rang;. Unfortunately, on some computers these don't appear as angle brackets but rather as boxes or question marks.

The following are possible ways of displaying angle brackets.

1) article page:       of a vector v is  &lang;v,v&rang;1/2.     (appears as boxes on some computers)


 * edit page:       of a vector v is  &amp;lang;v,v&amp;rang;1/2.

2) article page:      of a vector v is  〈v,v〉1/2.


 * edit page:       of a vector v is  〈v,v〉1/2.

3) article page:      of a vector v is  $$\langle$$v,v$$\rangle$$1/2.


 * edit page:        of a vector v is  $$\langle$$v,v$$\rangle$$1/2.

4) article page:     of a vector v is  1/2.


 * edit page:        of a vector v is  1/2.


 * or:  of a vector v is  &amp;lt;v,v&amp;gt;1/2.

5) article page:    of a vector v is $$\langle \mathbf{v},\mathbf{v} \rangle ^{1/2}$$.


 * edit page:        of a vector v is $$\langle \mathbf{v},\mathbf{v} \rangle ^{1/2}$$.

6) article page:    of a vector v is $$\scriptstyle \langle \mathbf{v},\mathbf{v} \rangle ^{1/2}$$.


 * edit page:    of a vector v is $$\scriptstyle \langle \mathbf{v},\mathbf{v} \rangle ^{1/2}$$.

7) article page:    of a vector v is $$\scriptstyle \langle$$v,v$$\scriptstyle \rangle $$1/2.


 * edit page:    of a vector v is $$\scriptstyle \langle$$v,v$$\scriptstyle \rangle $$1/2.

--Bob K31416 (talk) 14:28, 23 May 2010 (UTC)

Update added 4, 5, 6, and 7. --Bob K31416 (talk) 06:31, 27 May 2010 (UTC)
 * I'm generally opposed to anything that mixes inline LaTeX code with html typesetting. I don't know if there is a restriction against this in the WP:MOSMATH, but there probably should be one since it will display quite unpredictably depending on fontsizes, etc.  Of the first two options, I think that it is generally preferable to use named html entities rather than ad hoc unicode glyphs.  There was some discussion about this somewhere in the past, that there are several possible ways of achieving &lang;&rang;, but not all of which are fully standards-compliant.   Sławomir Biały  (talk) 18:47, 23 May 2010 (UTC)


 * Option 1 is best. Option 2 has too much space between the angle brackets and the exponent, and mixing html and tex (as in option 3) is almost always a bad idea. Something like option 2 but without the extra space between the angle brackets and the exponent would probably be even better than option 1, due to the compatibility problems you mention. —David Eppstein (talk) 03:27, 24 May 2010 (UTC)


 * Option 1. I agree with David.  #1 is clearly the best in my opinion. I find it strange how wide those two angle-bracket characters ("chevrons"?) used for option 2, i.e., note the excess(?) space before/after: '〈' and '〉'. (I'm using google's chrome browser.) Justin W Smith talk/stalk 03:38, 24 May 2010 (UTC)


 * Option 3 looks perfect on my system. Option 1 produces boxes in stead of brackets. DVdm (talk) 09:42, 24 May 2010 (UTC)


 * @DVdm: I'm just curious... which browser/OS are you using? Justin W Smith talk/stalk 23:26, 24 May 2010 (UTC)


 * Nothing fancy: standard Win XP Pro with standard Internet Explorer 8. DVdm (talk) 06:35, 25 May 2010 (UTC)


 * This is so strange; Option 1 is showing up as boxes for me now. I'm on my home PC: Win 7/Chrome. My work PC (XP/chrome) was ok with it. Justin W Smith talk/stalk 13:09, 25 May 2010 (UTC)
 * My home PC with firefox or IE8 shows the brackets fine. Justin W Smith talk/stalk 13:12, 25 May 2010 (UTC)


 * I have tested a number of other XP Pro systems: Option 1 is OK with Firefox and Google Chrome. IE8 shows boxes on all of them. Opera refuses to load the talk page (only tried it on one system). DVdm (talk) 13:31, 25 May 2010 (UTC)


 * The recent version of Safari/TigerOS shows Option 1 as boxes.
 * So far, have all the browser/OS displayed Option 3 in an acceptable manner? --Bob K31416 (talk) 14:10, 25 May 2010 (UTC)
 * On my system (OS X) the angle brackets on option 3 are about twice as big as the surrounding text. I don't think that's acceptable. —David Eppstein (talk) 14:24, 25 May 2010 (UTC)


 * Option 3 has the disadvantage of mixing LaTeX code with html. Better would be Option 4:
 * article page:     $$\langle \mathbf{v},\mathbf{v} \rangle ^{1/2} $$
 * edit page:        $$\langle \mathbf{v},\mathbf{v} \rangle ^{1/2}$$
 * Tricky, and perhaps not ok for OS X.
 * If we could only get the exponent 1/2 closer to the brackets in option 2, that one would be ideal. DVdm (talk) 14:32, 25 May 2010 (UTC)


 * David, If you had to choose between boxes and what you saw for Option 3 (or DVdm's option 4), which would you prefer? --Bob K31416 (talk) 14:37, 25 May 2010 (UTC)
 * If those were my choices, I'd prefer (4) left than / greater than in place of real angle brackets, or (5) do it all in rather than trying to mix math and html. —David Eppstein (talk) 14:46, 25 May 2010 (UTC)
 * I'm not exactly sure what you were saying, since option 4 has real angle brackets and is all in . Perhaps you mean to use 1/2 as a new option 5? That seems reasonable.
 * In any case, it seems like we need to choose whether or not to have Option 1, which is best on some computers but is catastrophically unreadable on some other computers. This is in comparison to the other options which are readable on all computers, so far, but don't look the nicest on all computers. --Bob K31416 (talk) 14:56, 25 May 2010 (UTC)
 * I meant 1/2 as option 4, $$\langle\mathbf{v},\mathbf{w}\rangle^{1/2}$$ as option 5, and maybe with scriptstyle $$\scriptstyle\langle\mathbf{v},\mathbf{w}\rangle^{1/2}$$ as option 6. All of these are preferable to option 3 to me but I think 4 looks a lot better than 5 or 6. —Preceding unsigned comment added by David Eppstein   (talk • contribs)  15:12, 25 May 2010 (UTC)


 * Ah... didn't know this was possible (see the original Option 2). Let's just take the simplest form for inline text: 1/2. DVdm (talk) 15:32, 25 May 2010 (UTC)
 * That would be fine with me, although some might object to having < > instead of angle brackets 〈 〉. It seems like < > is the least of various evils. (Damned software!) Any objections? --Bob K31416 (talk) 15:54, 25 May 2010 (UTC)


 * I agree with Bob, using &lt; and &gt; is "the least of various evils". Justin W Smith talk/stalk 16:23, 25 May 2010 (UTC)


 * It appears that consensus has been reached, and if there are no objections, I'll replace the angle brackets with < > in the article section "Inner product spaces". But first I'll wait a day to give anyone following this discussion time to raise any objections. If the consensus holds up, it will be stronger that way and we can refer any editor who wants to change it back to angle brackets to this discussion when their edit is reverted. --Bob K31416 (talk) 01:51, 26 May 2010 (UTC)


 * Whoa, hold on. How is it that a discussion with five or six editors on the talk page of Pythagorean theorem has anything to do with the article Inner product spaces?  If there is to be a blanket decision to make such a change across many articles, then it should be brought up at WT:WPM for others to comment.  By the way, there was consensus in the past against using the less-than greater-than symbols for inner products.  I, for one, am opposed, at least until many others have commented.   Sławomir Biały  (talk) 09:54, 26 May 2010 (UTC)


 * Oh sorry, you said "article section 'Inner product spaces'". That makes a difference, my mistake.  Still, this discussion probably could benefit from more input before implementing the change.  Sławomir Biały  (talk) 10:00, 26 May 2010 (UTC)


 * Re "Still, this discussion probably could benefit from more input before implementing the change." - Then please give your input. Which form would you prefer for the inline text in the section "Inner product spaces" of this article and why? Thanks.


 * BTW I looked at the article Inner product space and it has a mixture of various and non-  symbols and variables in the text, including an instance of < > . So because of its inconsistency and the use of  for inline text, perhaps it isn't a good model for the use of math symbols and variables in the inline text of the Pythagorean theorem article. --Bob K31416 (talk) 11:29, 26 May 2010 (UTC)
 * See my comment way above, where I advocated the first option (using named html entities). Any way: &amp;lt; and &amp;gt; are not used in professional typesetting for this purpose.  They should not be used on Wikipedia either.   Sławomir Biały  (talk) 12:49, 26 May 2010 (UTC)

I too don't see the problem with using proper angle brackets. They are standard Unicode characters, and common ones not obscure ones. On my Mac (OS 10.6) there are over 60 fonts with them in, not all fonts but probably all which have sets of Unicode punctuation and symbols. WP is unicode based, and has been for a while. If anyone has problem viewing Unicode the best solution is to help them with it, not replace standard Unicode with ASCII.-- JohnBlackburne wordsdeeds 10:23, 26 May 2010 (UTC)


 * See the ending comments at here, and indeed Help:Multilingual support. Will we actually be able to "help them" by telling them to buy MS Office or a standalone font, or by telling them to use another browser? Wouldn't using simple ASCII brackets, which work for everyone, be a lot easier? DVdm (talk) 10:44, 26 May 2010 (UTC)
 * There are very good free fonts, but it seems it's a problem other than this as the glyphs are very common in the fonts I have. So yes, if a browser has broken Unicode support switching browser might be best, if only for WP use.-- JohnBlackburne wordsdeeds 11:52, 26 May 2010 (UTC)
 * John, I'm not sure what you are proposing.
 * 1) Are you suggesting using in the article another Unicode for angle brackets that doesn't have the problems of the present one? If so, what is the unicode that you propose?
 * 2) Or are you suggesting that there is some practical way of helping with the angle bracket problem, each person who reads the section "Inner product spaces" of the article and has trouble with the angle bracket code displaying as boxes? How would you do that? Also, I noticed in Help:Multilingual support that you linked to in your message, that it discusses some browsers, but not the browser Safari, which is at least one of the browsers where the angle bracket in the article comes up as boxes. BTW on your Mac, I presume you don't get boxes for the angle brackets in this article? If you don't have the problem, what browser do you use?  Thanks. --Bob K31416 (talk) 11:18, 26 May 2010 (UTC)
 * Safari, the latest version. Can you see the angle brackets in the "Math and logic" section of the edit tools (the popup below the edit window), just before the frac ?-- JohnBlackburne wordsdeeds 11:46, 26 May 2010 (UTC)

As a general principle, we should not use greater-than and less-than signs for inner products. This is universally shunned in the mathematics literature; authors who use these signs come off as naive about mathematics. If there are issues with the symbols, we should use math mode. (As a separate issue, we should not use scriptstyle for running text.) &mdash; Carl (CBM · talk) 11:40, 26 May 2010 (UTC)


 * Carl, Oh it definitely would be better to use angle brackets if we didn't have problems with displaying angle brackets in Wikipedia. I suspect the mathematics literature that you are referring to has the capability of typesetting that displays angle brackets without these problems. So far, < > seems to be the "least of various evils" for displaying inner products in the inline text of Wikipedia. Note that angle brackets would still be used in stand alone equations. Also, for readers who aren't focusing on this detail like we are, I don't think they will even care about any difference between  and 〈v,v〉in the inline text, but I think they would definitely care if it came up as boxes. Regarding using  for inline text, that would be fine with me but others here don't seem to like it and one editor suggested that it is discouraged by WP:MOSMATH. So what do we do? We try to find the "least of various evils" in order to reach consensus.  BTW if angle brackets were used for inline text, I thought that option 3 above would be the best of those available for angle brackets, but of course not perfect since they are larger than desirable. In fact, I'm OK with practically all the possibilities, except option 1 because it can come up as unreadable boxes. Ironically I was the one who introduced option 1 into the article before I knew that it came up as boxes on a Mac! --Bob K31416 (talk) 12:11, 26 May 2010 (UTC)


 * Using less-than and greater-than signs for inner products makes us look foolish; it's not a solution to anything. WP:MOSMATH certainly does not advocate using less-than and greater-than for inner products simply to avoid using inline math. Presumably the section you are referring to is that "very simple formulas" section. If there are difficulties getting the symbols to display, then the formula is not "very simple". In the section on special symbols, MOSMATH says,
 * "One way to guarantee that an uncommon symbol is rendered correctly for all readers is to force the symbol to display as an image, using the &lt;math> environment."
 * Inline images may not be ideal, but they are at least guaranteed to work and avoid any issue with fonts.


 * The normal solution for this type of thing, though, is to just tell people that they need to install proper fonts in order to view the article. Wikipedia as a whole uses Unicode characters, but the symbols under discussion here are named entities that have been around since HTML 4.0, which means there is very little reason to be shy about using them. &mdash; Carl (CBM · talk) 12:42, 26 May 2010 (UTC)
 * Re "there is very little reason to be shy about using them" - Of course there is. They come up as boxes! Sorry but LOL. --Bob K31416 (talk) 12:51, 26 May 2010 (UTC)


 * These entities were introduced in HTML 4.0 in 1997. Every browser supports them. If a particular person sees boxes for &amp;lang ("&lang;") and &amp;rang; ("&rang;"), it means that person needs to adjust their font settings (there is some guidance at Help:Special characters). Even if we try to "work around" the issue here, such a person has a seriously broken browser configuration. &mdash; Carl (CBM · talk) 12:56, 26 May 2010 (UTC)


 * Right, and unfortunately these persons represent more than half of the browsing population. DVdm (talk) 13:53, 26 May 2010 (UTC)


 * No, because Internet Explorer does support these characters. The problem is just bad font settings on individuals' computers, not their browser. However, if we don't want to use the HTML entities, then the other option is math mode. &mdash; Carl (CBM · talk) 14:08, 26 May 2010 (UTC)


 * Yes, the problem is that the number of individuals who stick with the default font, or dont have ARIALUNI.TTF, or don't know how to (illegally) copy it from another system, is rather large. I personally don't mind the boxes, nor do I mind selecting Arial Unicode MS, or even copying it from whereever... DVdm (talk) 14:40, 26 May 2010 (UTC)

I updated the option list above with 4, 5, and 6, and added a new option 7. Since there doesn't seem to be an agreement on the best option, one way we can come to a consensus is for each interested editor to list all that are acceptable, in their order of preference. We could reach a consensus by considering the option that is acceptable to most editors and break any ties using the order of preferences. Would that be an acceptable way to proceed? --Bob K31416 (talk) 06:36, 27 May 2010 (UTC)


 * Note that in option 4, in order to avoid errors with HTML-tags, we really must type "&amp;lt;,&amp;gt;", showing as "&lt;,&gt;". So option 4 should be


 * 4') article page:     of a vector v is  &lt;v,v&gt;1/2.


 * edit page:        of a vector v is  &amp;lt;v,v&amp;gt;1/2.


 * DVdm (talk) 08:38, 27 May 2010 (UTC)
 * Re "in order to avoid errors with HTML-tags" - I'm not familiar with that error. Could you explain what it is for < > and whether such an error is likely to occur for < >? Thanks. --Bob K31416 (talk) 13:51, 27 May 2010 (UTC)


 * Hard to explain in HTML. For instance, suppose you want to put the string "br" (without the quotes) between angle brackets.
 * This is how to do it: &lt;br&gt; shows what you want it to show.
 * This is not how to do it: generates a line break.
 * You see? See and enjoy also, for instance, this nice little HTML-intro. DVdm (talk) 14:50, 27 May 2010 (UTC)
 * I think I understand your point now, that < > can have an undesirable effect if the text between the < > also happens to be an executable HTML code, such as &lt;br>. In the case we're considering, that seems highly unlikely, if at all possible, IMO. However, I added that alternate method to option 4. Thanks. --Bob K31416 (talk) 15:01, 27 May 2010 (UTC)

Please note that the recently added option 7 above displays nearly the same as option 1 which is currently in the article, except that option 7 does not have the problem that option 1 has of displaying as boxes on some computers.


 * Option 1:      of a vector v is  &lang;v,v&rang;1/2.     (appears as boxes on some computers)
 * Option 7:    of a vector v is $$\scriptstyle \langle$$v,v$$\scriptstyle \rangle $$1/2.

--Bob K31416 (talk) 17:42, 27 May 2010 (UTC)

putting it together
About option 7 and such... mixing LaTEX and HTML typesetting is a very bad idea. Option 6 is much better (and more efficient). Remember that on some systems every $$...$$ construct generates a separate picture (i.e. png- or gif-file). That's why pure HTML-solutions (without this math-pair) are the most efficient. Examples are Options 1, 2 and 4. O1 is best but doesn't work for everyone. O2 is hard to type and there is this ugly space between the right angle and the exponent. O4 always works for everyone, and is safe if we use the &amp;lt; and &amp;gt; construct. If we use a LaTEX construct, we must at least make sure we put the entire thing in one packet. Examples are O5 (ugly when used in-line) and O6, which looks perfect. O7, which puts the exponent outside the Math is bad, and O3, doing the same, plus having two math-constructs is evil ;-) DVdm (talk) 13:18, 28 May 2010 (UTC)


 * Re "mixing LaTEX and HTML typesetting is a very bad idea." - In this specific case of option 7, it is only a bad idea if it does not display well. Does it display poorly on your computer? And I put that question to everyone who is reading this message. Please report here whether or not option 7  displays well on your computer. Here it is again.


 * Option 7:    of a vector v is $$\scriptstyle \langle$$v,v$$\scriptstyle \rangle $$1/2.


 * Thanks. --Bob K31416 (talk) 16:49, 28 May 2010 (UTC)

How about this?


 * {| class="wikitable" style="text-align: center;"

! |Option1 ! |Option2 ! |Option3 ! |Option4 ! |Option5 ! |Option6 ! |Option7 ! |User |style="background: #66EE22;" |HTML |style="background: #66EE22;" |HTML |style="background: #EE4422;" |Math+HTML+Math+HTML |style="background: #66EE22;" |HTML |style="background: #EECC22;" |Math |style="background: #EECC22;" |Math |style="background: #EE4422;" |Math+HTML+Math+HTML ! | |inline &lang;v,v&rang;1/2 test |inline 〈v,v〉1/2 test |inline $$\langle$$v,v$$\rangle$$1/2 test |inline 1/2 test |inline $$\langle \mathbf{v},\mathbf{v} \rangle ^{1/2}$$ test |inline $$\scriptstyle \langle \mathbf{v},\mathbf{v} \rangle ^{1/2}$$ test |inline $$\scriptstyle \langle$$v,v$$\scriptstyle \rangle $$1/2 test | |OK |very ugly |ugly |OK |very ugly |OK |OK |DVdm (talk) |unreadable (boxes) |poor |fair |fair |poor |fair |good |Bob K31416 Mac |good |poor |fair |fair |poor |fair |good |Bob K31416 Windows |poor (can't read) |poor (extra space) |fair (looks fine, but Math/HTML mix) |poor (wrong use of character) |fair (too big for inline) |good |fair (looks fine, but Math/HTML mix) |Blue Moonlet Mac
 * }

Bob, the "unreadable (boxes)" version on your Mac works fine on my Mac (Intel Macbook Pro, 10.5.8) on Camino, Chrome, Firefox, and Safari. Maybe you need to reset your font cache to clear the font caching bug? Or something else? I wouldn't call it good, though, as there's an extra space built into the glyph (looks same as option 2, which was poor on your mac).

As for "it is only a bad idea if it does not display well," I find that idea abhorrent. Understandability and maintainability of the source is at least as important. Dicklyon (talk) 05:39, 29 May 2010 (UTC)


 * Boxes aren't something that is peculiar to just my computer. See e.g. Justin's remark and the edit summary here.


 * Re "Understandability" for editors - this can be satisfied with a hidden comment


 * Re "I wouldn't call it [option 7] good, though, as there's an extra space built into the glyph (looks same as option 2, which was poor on your mac)." - There's no extra space in option 7 on my computers. Has anyone else observed this problem? But surely, if there was a display problem with some computers, wouldn't you prefer that the problem be extra space instead of the unreadable boxes of option 1 that is presently in the article?
 * Regards, --Bob K31416 (talk) 15:17, 29 May 2010 (UTC)

So, Bob, if it's only about this obscure little section in this article, just go ahead, make a choice between Option 4 (my favourite) and Option 6, since (1) they show up for everyone, (2) the quality is acceptable, and (3) they don't mix LaTEX with HTML typesetting. I don't think anyone will object. If o.t.o.h. you take Option 3 or Option 7, probably someone, if not me, will object sooner or later, if not within 10 minutes :-) I have added some colors to the table: green=cheep, yellow=math, red =evil. DVdm (talk)


 * In addition to my query of you on your talk page, I also queried Dicklyon on his talk page, and  the situation doesn't look too good to me, so I'll drop out. Maybe you will try what you suggest. I won't be reverting anything regarding angle brackets. Regards, --Bob K31416 (talk) 10:05, 31 May 2010 (UTC)


 * Sure, maybe tomorrow. There's no rush. Cheers! DVdm (talk) 16:26, 31 May 2010 (UTC)


 * Ok, I did O4. If someone prefers O6, don't hesitate. DVdm (talk) 14:08, 1 June 2010 (UTC)


 * I'm sorry I'm joining the discussion so late, but please don't do O4. There's plenty of reasons why good typesetting avoids it; think about writing an inequality with an inner product: &lt;v,v&gt;&gt;0. Unreadable! If you're shy about using O1 in inline text, just use (v,v)&gt;0. It is standard mathematical notation and it's always readable. —Preceding unsigned comment added by Tercer (talk • contribs) 03:44, 2 June 2010 (UTC)


 * It's too bad the O2 characters enforce all that space, otherwise it would be easily the best option. Please don't use O1.  My Mac can't read it, and even if there is a way to fix that, you can't expect the casual reader to go to the trouble.  I agree with Tercer that it's bad practice to use the inequality character "off-label", as O4 does, and I also agree that mixing HTML and LaTeX is probably not a good idea.  There should be no problem in principle with just using LaTeX, which is already done extensively in the article.  O6 is superior to O5 as it is the proper size for inline text.  --BlueMoonlet (t/c) 14:31, 2 June 2010 (UTC)