Talk:Bivector/Archive 1

Refs
The whole of Gibb's Vector book can be found here - I used it to find what he thought about bivectors as I knew he was aware of them. Page 37 of the epub version. Not sure whether to replace the Google Books URL with this one. --JohnBlackburne (talk) 19:34, 22 December 2009 (UTC)

Applications
This article lacks some examples and applications, for example, as found in. Brews ohare (talk) 15:37, 23 December 2009 (UTC)

Other uses etc.
The content of this edit was misleading. The bivector described in that book is no different to the one described earlier in the article. It's just the book is using what I would consider out of date language, and the space is a 4-dimensional non-Euclidian one, so has additional properties like the pseudotensor. But the mathematics is the same.

This is something that should be added to the article — it should go up to at least four dimensions with examples and applications I think. If no-one else does it first I may have a go, probably the other side of the holiday now.--JohnBlackburne (talk) 16:55, 24 December 2009 (UTC)

Hi John: I got it. I guess the topic is the connection between tensor analysis and geometric algebra; how's this:


 * Tensors and geometric algebra
 * The term bivector also can be found in general relativity:
 * A bivector at a point p in space-time is a second order skew-symmetric tensor with components Fab = −Fba. The set of all bivectors is a six-dimensional real vector space.


 * A complex bivector is an object at point p whose real and imaginary parts are real bivectors in the sense above.

Brews ohare (talk) 17:12, 24 December 2009 (UTC)
 * This meaning is a special case of the general term bitensor. The fields of tensors and geometric algebra are closely related. For a discussion see
 * References


 * To say the term "can also be found in" is still wrong. It's not a different thing described by the same term, it's another bivector application. I think there are much better ways to do it than describe one relatively obscure mathematical object in terms of another relatively obscure mathematical object. That may have made sense in that book where the tensor is already defined but not here. And the book is pretty old; the way you'd do it now is much more in line with the article, e.g. bypass tensors entirely and do it in terms of geometric algebra fundamentals. --JohnBlackburne (talk) 17:23, 24 December 2009 (UTC)

OK, I've made an attempt at a brief connection in the article. See what you think. Brews ohare (talk) 17:36, 24 December 2009 (UTC)

General point on references
I've just changed a few things - reverted them really - as they did not make sense. The exception was just wrong. "Pseudovector" is far better. Not only is it the common name, so the name of the article, but it's the name used in GA. The last change was to take out the "preference is changing" as the book is talking about something else I think, but is certainly not WP:NPOV as it reads from just that sentence like an author preference, not a statement of the evolution of the usage. --JohnBlackburne (talk) 18:54, 24 December 2009 (UTC)
 * John: The term "pseudovector" has multiple meanings but the term "axial vector" is specific. That is why I prefer the use of axial vector, which is specific. The cited source in this connection explains a suggested restrictive meaning of pseudovector as distinct from axial vector, but this distinction is only sometimes made, leaving the term ambiguous in general usage. Have you taken a look at the Talk: Pseudovector page? Brews ohare (talk) 19:09, 24 December 2009 (UTC)

One problem seems to be there are too many references for the article, which are being used in ways that do not add to the article. Sometimes the points don't make sense outside their context, and here they just confuse. Or they overlap and contradict in subtle ways, e.g. in terms of notation. A further problem is that anything much older than ten years will be using out of date terms and making connections that now would be seen as anachronistic. Look e.g. at Geometric algebra. It is far longer than this one will ever be but only has a fraction the sources here.

It would be better to write this article as far as possible from a single good source, i.e. a modern one without any POV concerns. All of the article now could be sourced from a single source, which would make it clearer and more consistent, as well as 30% shorter. Use another source to confirm the main points, and see that there are no problems with e.g. notation particular to a source. If something is mentioned only in one place, as seems to be the case in a few things here, then it probably should not be included. --JohnBlackburne (talk) 18:54, 24 December 2009 (UTC)


 * I don't agree upon restricting the number of sources as one can use google books to consult the sources and find one that speaks to you. Also, treatment of particular topics is more complete in some sources than others, and the level of treatment varies. So more sources allows a treatment accessible to more readers, either because of sophistication or interest. I wonder whether you are considering a variety of readership? Brews ohare (talk) 19:09, 24 December 2009 (UTC)


 * I assume you refer in your last to the bivector approach of Gibbs. That methodology is widespread in some fields, and so deserves a sentence or so. Multiple meanings of terminology are a very common source of confusion, and I'd bet that sooner or later some reader will bring it up and possibly embroil everyone in some debate, as occurred in many other articles where multiple meanings came up. Its an "ounce of prevention", I think, to mention it. Brews ohare (talk) 19:09, 24 December 2009 (UTC)
 * The second rank tensor background of the bivector is a sure thing to come up. It seems that there is a bit of a war going on between the geometric algebra and the tensor analysis camps as to which is the more useful conceptually and computationally. I've tried to avoid those issues. Maybe you want to bring them forward? Brews ohare (talk) 19:19, 24 December 2009 (UTC)

Cross product
Lounesto states in so many words that the cross product is the dual of the bivector (i.e. a vector). Ławrynowicz states the wedge is dual to the "Gibbs cross product". So apparently the cross-product is dual to the bivector and therefore a vector (in some sense). So I am confused by your remarks about operators. The other confusion is about your equating cross-product to pseudovector. According to Baylis, the terms axial vector and pseudovector are often treated as synonymous, but he suggests a distinction. Baylis (see footnote) distinguishes the pseudovector from the axial vector, making the pseudovector identical with the bivector (as does Ławrynowicz, who states the pseduovector is the bivector and the pseudoscalar is the trivector (a convenient progression of terminology)), and the axial vector something different: "the real dual vector normal to the plane". Baylis says: "On the other hand, if the components are fixed and the basis vectors eℓ are inverted, then the pseudovector is invariant, but the axial vector changes sign." How do you resolve these issues? Brews ohare (talk) 15:57, 25 December 2009 (UTC)

My approach would be to accept both references at face value and state (as sourced) that the axial vector (cross product) is the dual of the bivector. According to Baylis, the axial vector is not a pseudovector; the bivector is a psudovector. However, this point can be avoided by not using the word pseudovector with its associated ambiguities of usage. Brews ohare (talk) 15:38, 25 December 2009 (UTC)


 * You are misunderstanding the Lounesto ref: it says "the cross product is dual to the exterior product". That is the two operations are dual in some sense, the sense being that the result of one is the dual of the other so the vector result is dual to the bivector result. To say as the article does that "The cross product a × b is dual to the bivector a $$\wedge$$ b" is just wrong - an operation is not dual to a bivector. The other two books are much older so I would not use them as primary sources. Also the link to pseudovector is valuable as it goes into the property of this vector in much more detail. Hence my last change.


 * More generally it is not clear you really understand the topic. You should not be pulling odd quotes like this from multiple sources and mis-interpreting them. Nor should you be trying to synthesise the article from multiple conflicting sources. You should be writing about something you understand well, referring to sources as necessary to support particular point. My recent edits have largely been fixing such things, and there are others I've not yet touched.--JohnBlackburne (talk) 15:59, 25 December 2009 (UTC)

John: Maybe you could undertake to elaborate upon the interpretation of Lounesto. You are right about his wording: The cross product is dual to the exterior product of two vectors. Then he writes:
 * $$ \boldsymbol {a \times b} = - \boldsymbol{(a \wedge b)e}_{123} \ . $$

Now, I'd identify $$ \boldsymbol {a \times b}$$ as what is normally referred to as the cross-product, wouldn't you? If so, I can replace the word "exterior product" with "cross product" in 3-D. Similarly, Schaum's outline says "the reader may recognize that the above exterior product is precisely the well-known cross product in 'R3.

So my statements do not seem to be incorrect, nor due to misconstrual of the sources (of which many more exist). Perhaps you could explain in more detail what you object to? The sources in general do not introduce the concept of operator in discussing the cross product and the dual of the bivector. However, the concept of operator is sometimes introduced in connection with the bivector itself as representing a rotation. Brews ohare (talk) 18:29, 25 December 2009 (UTC)

Rotations
I've added a subsection on rotations and bivectors referencing Baylis. Brews ohare (talk) 19:22, 25 December 2009 (UTC)


 * But you've misunderstood it again. This is one of the key applications of bivectors and it doesn't work like that: what you've written just doesn't work. e.g.


 * $$\boldsymbol{e_3\ (e_1e_2) = e_{123} = i} \, $$


 * The same is true for any vector with a non-zero z-component - the result is part trivector/pseudoscalar. Your approach to editing seems to be causing most of the problems - rather than drawing on your experience, or finding a good reference or two and drawing on those, you're pulling snippets from sources you don't understand and trying to make an article from them results in repeated mistakes.


 * Maybe you should pause adding things to this article, as beyond the basics your edits are making less and less sense, and there are issues with all sections of this article now. As you mentioned you are under a topic ban which would seem to cover much of this article, and it does seem to be a subject you don't have enough expertise in. --JohnBlackburne (talk) 19:42, 25 December 2009 (UTC)

John: I've not written:


 * $$\boldsymbol{e_3\ (e_1e_2) = e_{123} = i} \, $$

which is accurate of course. What I have written is directly from Baylis


 * $$\boldsymbol{e_1\ (e_1e_2) = e_2} \, $$

which also is perfectly correct. Your anxiety in this case is due to your not reading carefully and not consulting the sources themselves. Brews ohare (talk) 20:33, 25 December 2009 (UTC)

It's a bit of a low blow to use the fact that I am under sanction as an excuse for avoiding proper discussion of sourced material and to level general claims that I am out of my depth and misquoting sources without backup. As I have invited you, please simply engage instead of making tendentious claims. Brews ohare (talk) 20:36, 25 December 2009 (UTC)


 * You wrote "The fact is used to show that rotations in a vector space can be represented by bivector" which is wrong, or at least it's not that simple. The maths above shows how your bivector when applied to the 3-vector $e_{3}$ does not rotate it: the result is not even a vector. A rotation would transform it to another vector. This is a simple and obvious counterexample, if you are familiar with how rotations work in geometric algebra.--JohnBlackburne (talk) 21:17, 25 December 2009 (UTC)

John: I take it that the rotation here is in the xy-plane (the e12 plane), so it doesn't apply to e3 which is parallel to the rotation axis. Maybe a more general statement about rotations is needed. I've tried that change following Abłamowicz, who has a more general discussion of rotations through an arbitrary angle θ in the e12 plane. See if you think that is more appropriate. Brews ohare (talk) 01:13, 26 December 2009 (UTC) I decided the complete discussion of arbitrary rotations through arbitrary Euler angles was overkill. Do you agree? Brews ohare (talk) 11:24, 26 December 2009 (UTC)

New intro
Rather than keep moaning about it here I've decided to try and fix the article. The intro had a number of errors, not least it assumed all bivectors were associated with planes, that all bivectors were associated with exterior products, and it did not make it clear when it was discussing properties of only real bivectors or of more general ones. The new intro tries to fix that. Re-writing it was the only way as bivectors exist independent of their geometric properties, although they are important, so the very first sentence was wrong. It needs to start with algebra after which everything else needed changing. --JohnBlackburne (talk) 22:05, 28 December 2009 (UTC)

Mr Blackburn. I read the arbitration proceeding and I have to say that you were certainly misguided in your attempt there. I am also not the least bit impressed by your arguments here. I have read this article as you wrote it and it isnt very helpful to the understanding. Your objective should be that and not a purity of mathematics or didactic exposition. The main point is to give the reader an understanding of the subject and not write for experts who already know the topic. i think you are writting for the experts. So please think about the suer and what he needs to know. If that means a simplification, then that may be what is needed to get the understanding of the reader. I dont think the article in its present form is very useful. so you have a lot of work to do.72.64.45.65 (talk) 14:56, 29 December 2009 (UTC)


 * If you have any particular concerns with what I've written so far let me know and I'll do what I can to address them. But the numerous errors in the previous text meant the only way to improve it was to replace it, with something based on sound mathematics. It might now seem terse and unfriendly to new users but I hope this will get better as more of the article is improved. --JohnBlackburne (talk) 15:03, 29 December 2009 (UTC)

The introduction of terminology without explanation is confusing. I've made an attempt to provide some guidance. Brews ohare (talk) 15:07, 29 December 2009 (UTC)
 * Your change is factually wrong, so I will have to remove it, sorry. Please do not keep adding incorrect information to this article.--JohnBlackburne (talk) 15:15, 29 December 2009 (UTC)

This "factually wrong" material you removed explaining some terminology is sourced to Hestenes, who is certainly an authority. Would you care to explain what specifically is "wrong"? See &.

The object here is to present matters without undefined jargon like "elements of grade 2". That is why I added the paraphrase of Hestenes: In geometric algebra, the algebra should include the graded elements 0-vector, 1-vector, 2-vector, 3-vector..., r-vector to represent the directional properties of points, lines, planes, space... The terms scalar, vector, bivector, trivector ... are often used as alternatives. Brews ohare (talk) 16:26, 29 December 2009 (UTC)


 * The factually wrong in this case is not supplying the context for the statements you are making. That's the problem with lifting the odd line or phrase from a source: somewhere earlier in the book it will say "this only applies in xxx circumstances", or words to those effect. It is wrong to take that and make a general statement about something which only applies in a particular context. If you had read and understood the sources you claim to be basing your contributions on you would know this. --JohnBlackburne (talk) 16:51, 29 December 2009 (UTC)

The statements of Hastenes do apply only to a restricted situation, but they do explain what the meaning of the jargon you use is for such cases. Your wording just leaves the reader in the air, without any explanation. Can you fix this? Brews ohare (talk) 17:15, 29 December 2009 (UTC)


 * As I wrote at the top of this section I hope to address all the issues in this article, including adding proper explanations for everything, but not all in the intro and it won't all be done in one go. --JohnBlackburne (talk) 17:20, 29 December 2009 (UTC)

a ^ b can be associated with a bivector in the same space as the vectors
This statement is unclear. In the simple 3-D case, I believe what is meant to be said in the WP article is that the bivector a ^ b can be associated with an axial vector a × b in the same space as the vectors. See Doran & Lasenby. Even for this case, the statement needs to be reworded because the wedge product is a bivector and is not in the same space as a vector. See Baylis. In the general case I do not know what is meant. Brews ohare (talk) 18:04, 29 December 2009 (UTC)


 * I've clarified it, and removed a reference. There are far too many references in this article, as I've already noted, and pulling odd facts from here and there without context causes more problems than it fixes. Unless facts are especially contentious we don't need a reference per sentence, as the citation needed tags imply. This is very uncontroversial science not a BLP. It's better to base the article as a whole, or individual sections where appropriate, on single sources. So don't worry, everything in here will be properly sourced.--JohnBlackburne (talk) 19:04, 29 December 2009 (UTC)


 * This matter has not yet been clarified. It still seems to say that a vector space is the same as its dual, which in turn suggests that there is no point in distinguishing the space from its dual. See linked references above. Brews ohare (talk) 19:52, 2 January 2010 (UTC)

2-vectors vs. 2-multivectors
The article states: In Clifford algebra and geometric algebra they are the elements of grade 2 of the algebra, or 2-vectors, while in Exterior algebra bivectors are 2-multivectors. This sentence is replete with distinctions without a difference. The literature doesn't seem to care which term is used: 2-vector or 2-multivector. For example, see Doran for the term grade-2 multivector in geometric algebra. Exterior algebra also uses the term "2-vector", and so is not distinguished by a different terminology. See Cnops.

Inasmuch as the differences between the three algebras are a fine point in this context, a suitable replacement sentence is:

In Clifford algebras, bivectors are the elements of grade 2 of the algebra, also called 2-vectors or 2-multivectors.''

Of course, that still leaves the problem that "elements of grade 2 of the algebra" is gobbledygook for most of us. Brews ohare (talk) 19:19, 29 December 2009 (UTC)


 * The particular point of that sentence is to reference both the usage in exterior algebra and in Clifford/Geometric algebra. with links to all three. Clifford algebra and geometric algebra largely agree on terms but those used in exterior algebra are somewhat different so it's best to mention this. See e.g. Exterior algebra.--JohnBlackburne (talk) 19:37, 29 December 2009 (UTC)

If there is a significant difference in terminology or usage between the three algebras, it should be pointed out. The significant difference is clearly not a difference in usage of 2-vector vs. multivector. Maybe what is meant is:

In Clifford algebras, bivectors are the elements of grade 2 of the algebra, also called 2-vectors or 2-multivectors in these algebras and in exterior algebras. Brews ohare (talk) 19:53, 29 December 2009 (UTC)


 * No, see WP:BOLDTITLE. And there's no need to elaborate on what they're called in Clifford algebra or GA - "bivector" is the standard term.--JohnBlackburne (talk) 20:10, 29 December 2009 (UTC)

John: The first sentence: “A bivector is an algebraic object from various areas of mathematics.” is less informative than saying “Sara Palin is a woman from North America.” Can't we do better? Brews ohare (talk) 20:39, 29 December 2009 (UTC)


 * Have a look now. I thought the names didn't really belong too, so moved them to the next section as it keeps all the exterior algebra stuff together. So the first paragraph is more compact & clearer, and it says a bit more in the first sentence. Not sure you can say more than this: the stuff about planes and the geometric interpretation only is generally true in 3D, which makes for a good second paragraph with your picture. The third paragraph then rounds it off with a list of uses - this might be expanded as e.g. particular applications in physics are added. --JohnBlackburne (talk) 23:01, 30 December 2009 (UTC)

given two vectors a and b their exterior product a ^ b is a bivector in the same space
This statement is at variance with Baylis, who says the bivector space is dual to the vector space. Maybe this means the bivector and the vectors are in the space that is the union of both subspaces?? What does "same space" mean??? Is some elaboration missing here? Brews ohare (talk) 19:27, 29 December 2009 (UTC)


 * Space as in e.g. ℝn for some n. You don't have vectors in ℝ4 and bivectors in ℝ3 for example. This will be clearer when I've added some more.--JohnBlackburne (talk) 20:14, 29 December 2009 (UTC)

So the statement is: given two vectors a and b their exterior product a ^ b is a bivector in some space ℝn

Isn't that a bit silly? Brews ohare (talk) 20:34, 29 December 2009 (UTC)


 * Why ? --JohnBlackburne (talk) 20:36, 29 December 2009 (UTC)

First, so far we really have no idea what a bivector is, so we have identified it as something else we don't know. Second, saying it is in some space ℝn isn't terribly illuminating, although it is clearer than the present language, which is open to misinterpretation, e.g. as contradicting Baylis, who says the bivector space is dual to the vector space, i.e. not the same space. Brews ohare (talk) 20:42, 29 December 2009 (UTC)
 * We do now as I've just added a formal definition, as well as a couple of paragraphs on the choice of algebra. It's fairly algebraic but is is hopefully as maths intensive as the article needs to get. The main reason I've added it is the relationships between the geometric product and the interior and exterior products are important, and it's not much extra math to derive them. They're needed as the geometric product is the most fundamental product of all, so is needed to derive almost anything else. And as you noted we didn't really know what a bivector was.--JohnBlackburne (talk) 21:08, 29 December 2009 (UTC)


 * This matter has not yet been clarified. It still seems to say that a vector space is the same as its dual, which in turn suggests that there is no point in distinguishing the space from its dual. See linked references above: Doran & Lasenby & Baylis for the three-space example. Brews ohare (talk) 19:52, 2 January 2010 (UTC)

What else needs to be done
Now I've started work on the article I've a clearer idea what needs to be done, and I think it falls into three main areas.


 * A general tidy up. It currently has no real layout or structure, with sections in no particular order, odd facts in odd places and occasional duplication and repetition. It needs putting in some sort of order so it's easier to follow as a whole and easier to find particular facts and sections.
 * Supply context. Too much of the article is incorrect or misleading as it only applies to certain spaces, in particular ℝ2 and ℝ3. This needs to be made clear, and can be done as part of the sorting as facts e.g. relevant to 2D or 3D are grouped together. The second section is part of this - I think it's safe to establish that this article is primarily about real geometric algebra.
 * Expand. It right now says little about how bivectors work in dimensions higher than 3, in particular spacetime but also Euclidian 4D space and maybe 5D. This needs adding as it's one of the main reasons for using bivectors. If you stick to 2D and 3D you may as well use complex numbers, vector algebra and quaternions. --JohnBlackburne (talk) 22:21, 29 December 2009 (UTC)


 * A bit more work editing and expanding. I've added a new 2D section that better establishes the properties of bivectors in 2D and their relation to complex numbers. There's not much as you could write a lot more but it ends up being a rewrite of complex numbers using bivectors/the even sub-algebra. Most readers should be familiar with complex numbers so better just to point out the relationship and highlight a couple of key properties. Lots of wikilinks to help establish the relationship to complex numbers and other familiar topics.


 * I removed the properties section, as it consisted of some facts like antisymmetry and the relationship to sine established in the previous section and some on the wedge product that don't say much about the bivector. Some of it might be put back in with e.g. the geometric interpretation of bivectors.


 * I 'merged' the two history sections by copying the lower down one into the upper one. I think this needs trimming to properly focus on bivectors, with 'see also' links for the histories of other things.--JohnBlackburne (talk) 17:17, 31 December 2009 (UTC)

After reading the current article, I still don't know...
 * What domain a bivector lives in--a vector space's field of scalars? The vector space's dual space? The vector space itself? The intro says it's in the vector space, but then scalars and the result of exterior products are added later.
 * The formal definition of a bivector. The intro implies that it's a (finite? It may not matter--I can't tell from the current article) sum of exterior products. The "formal definition" section implies that it's the result of an exterior product, but concludes very vaguely, saying it can't be either a scalar or a vector.
 * The relationship between bivectors and (hyper?) planes/parallelograms.

I'd need to understand these to be able to understand the rest of the article on any level higher than "those algebraic operations appear to follow from the rules you've listed or implicitly used". I'm sure most of these issues are caused by my inexperience with geometric algebra. However, if I was experienced enough with it to fill in these holes, I probably wouldn't need to read this article. 67.158.43.41 (talk) 11:21, 1 January 2010 (UTC)


 * OK, yes I agree that none of that is especially clear.
 * * Geometrically they are in the same space as the vectors, e.g. in ℝ3 if the space is ℝ3, but this could be clearer. Maybe it will be clearer after more is added on the geometric interpretation.
 * * Yes, a finite sum. This again will make more sense with examples in 3D and especially 4D, as 2D is too trivial to get much across.
 * * My intention is to update the 3D section, similar to the 2D section, so it includes something on the geometric interpretation. It's best done in 3D as it's trivial in 2D and too complex and non-intuitive in 4D.
 * So overall I hope this will become clearer as more is added and what's there is sorted out so it makes more sense. Interestingly this makes me think of a reply I gave earlier, that maybe an Introduction to geometric algebra article would be useful. But first it makes sense to finish the main tasks here. -- John Blackburne (words ‡ deeds) 11:49, 1 January 2010 (UTC)

In reply to...
In reply to this:


 * This statement is unintelligible, ambiguous, and wrong in some interpretations. For example the bivector space is dual to the vector space, so in some sense the two are not the same space. It would be best to delete this claim, or to provide a footnote separating this statement from its incorrect interpretations.

I've changed the 'space' link to a more appropriate one which is maybe clearer. It's not just any old space but its a vector space, as given by the dimension of the space and the metric/signature. But it also has an additional structure, which is as well as vectors there are bivectors, trivectors etc. all in the same space. That's what that statement means. It's not as technical as the above as it's in the lede where it's not appropriate to go into too many details. Hopefully the rest of the article will expand and clarify this at some point. -- John Blackburne (words ‡ deeds) 19:59, 2 January 2010 (UTC)


 * OK, I've removed the contentious bit after adding something about the spaces further down. I'm still not happy with it but it's difficult to address properly until probably more has been added. About the dual that's only true in 3D, but one key point about geometric algebra is that all these things exist in the same space. So e.g. bivectors and vectors are dual in 3D but you can do algebra with them in the same formula as they're all in the same geometric algebra, which is also a linear space and a vector space. -- John Blackburne (words ‡ deeds) 21:39, 2 January 2010 (UTC)

formatting etc.
I just reverted the last change as the formatting was incorrect for vectors. "mathbf" is the right thing to use, i.e. unitalicised bold for both general vectors and the basis vectors. See e.g. Euclidean vector, or any article that uses vectors.

The other change just duplicated what's said in the next section about the relationship with the complex numbers. In this context also it's not clear to write √-1 as many things square to -1. You at least need to define your square root, you cannot just give it as a value. But as this is explained in the next subsection it's not needed.-- John Blackburne (words ‡ deeds) 15:55, 3 January 2010 (UTC)


 * Your comments about notation are simply a preference of some editors (apparently yourself being one), and have no substantial basis. The original choice in this article was $$\boldsymbol e $$ throughout, and I see no reason to change it.


 * The material on i is indeed repeated later in a different specific context, namely 3-D instead ot 2-D. However, it seems unreasonable to require the reader to read a later section on a somewhat different example to discover the material relevant to an earlier example and reinterpret it for the different context. Thus, I have reverted your changes. Brews ohare (talk) 16:01, 3 January 2010 (UTC)

Revision of second paragraph
Some changes made in this this edit actually make the paragraph less clear and less related to the cited source. In particular, the connection of the sign of the bivector to the sense of the bounding curve of the plane segment it represents is a very clear picture and one used by Hestenes in the citation. In contrast, the wording "the ratio of the components of the bivector determine determine the plane's rotation in space" is not clear inasmuch as the plane is not rotating or rotated and neither is any subarea on the plane that might correspond to the bivector. The point about having no specific location also is important to emphasize, as done by Hestenes as well. Various improvements you have made in the text were retained. Brews ohare (talk) 06:29, 4 January 2010 (UTC)


 * But the formulation I gave is correct. E.g. to say "the ratio of the components of the bivector determine determine the plane's rotation in space" is to say what happens. It is the ratio of the components, rather than their values, that determines which abstract plane it is. E.g. the bivectors (in 3 dimensions) e12 + 2e13 and 2e12 + 4e13 lie in the same plane. Their components are different but the ratios are the same, and that is the clearest way to describe their relationship algebraically.


 * The problems with the text as it is are many. First "orientation" is an ambiguous term. If the reader checks oriented, orientation or orientation (mathematics) they will find the text confusing. Rotation in mathematical context, i.e. Rotation (mathematics) is correct. Second what is the "direction" of a plane ? If the reader checks e.g. Direction (geometry) they will find it only applies to vectors. "Clockwise" and "counter-clockwise" are also unclear - they make sense in 2D but not in general. The same is true of "bounding curve", as a plane region is only bounded in any sense in 2D. The paragraph mentions  "in three dimensions" which is also too specific. Lastly the text is too long. The point of the introduction is to introduce the topic, not fully explain it.


 * The link to Hestenes is to a chapter on 2D geometry, where words like "clockwise" "bounded", etc. make sense. But the text does not make sense in a general context. Even if we were only dealing with 2D it is too long and unclear as noted above. Hestenes's book is very good but you have to use it with care as it's not just a general text on geometric algebra it's also a general physics text book.-- John Blackburne (words ‡ deeds) 09:36, 4 January 2010 (UTC)

John: The second paragraph originally used the example of bivectors in 3-D and applied their property of characterizing an area in a plane. So the restriction to 2-D was explicit. The notions of bounding curve and clockwise and counterclockwise are all therefore perfectly clear. The notion that a plane has orientation is obvious (and not the wildly more complicated notion of orientability you link above, more like plane (geometry)), and it is not necessary to go into how to describe it using a vector. I'd suggest that the formulation you have removed, which parallels Hestenes, is very clear and preferable to yours. There is no cause in this context for treating Hestenes with caution for the reasons you suggest. He is after all the authority on this topic. Brews ohare (talk) 13:58, 4 January 2010 (UTC)

I'd add to this that your description of how to describe the orientation of a plane as a "rotation" using e12 + 2e13 and 2e12 + 4e13 is nonobvious for the average person, and for the neophyte that doesn't know about bivectors (this article is an introduction to bivectors, after all), and not described in the article. Without this description, the wording about "rotation" (of what?) and "ratio of components" (of what?) in your paragraph is obtuse at best, and certainly not as clear as the paragraph that you have removed, which last parallels Hestenes' discussion. Also, and again, you fail to point out the significant lack of place in a bivector, present in the removed paragraph and in Hestenes. Brews ohare (talk) 13:58, 4 January 2010 (UTC)

I'd suggest your paragraph be made more like the one you have removed. If you wish to generalize to n-dimensions, I'm sure topology can characterize the analog of clockwise and counterclockwise for surfaces in n-1 dimensions without introducing bivectors and ratios of components (maybe Courant?) Or, perhaps you wish to create a subsection on the use of bivectors for such generalization of clockwise and counterclockwise? For the intro, the 3-D example is easier and clearer: a more complete n-dimensional version really requires a separate section for any clarity, intelligible once some properties of bivectors already have been established. Brews ohare (talk) 13:58, 4 January 2010 (UTC)

If you wish some further opinions on the merits, an RfC could be posted. Brews ohare (talk) 14:27, 4 January 2010 (UTC)


 * OK, have a look now. It's another re-write but is I think much better in it focusses on the geometry much more, and relates better to the diagram. It's based on Lounesto's description, page 33 & 34. He uses the word 'attitude' which is a bit archaic but as there's a page for it I've put it in too. It more directly describes the orientation in terms of the exterior product, rather than relying on e.g. clockwise and counter-clockwise or other terms that make sense only in 3D 2D. I added simple which is in Lounesto later, and it clarifies e.g. that not every bivector represents a plane.


 * We still need a section on this later on, as part of "bivectors in 3D", where it can be related to other things in 3D that readers might have a better grasp of like the cross product. It needs to be covered again in 4D as there it gets more complex but also more interesting. -- John Blackburne (words ‡ deeds) 15:46, 4 January 2010 (UTC)

Hi John: A big improvement. I look forward to your additions to the article that you mention.

In the second paragraph: The word attitude is a plus. The reference to rotation from a to b is clear enough, but the notion of "rotation" as a synonym for "attitude" is mathspeak (i.e., a technical terminology that connects obscurely to ordinary English usage). The last sentence about the bivector "inducing a rotation" is premature: no basis laid up to this point for understanding. The sentence "The bivector has no location and should not be regarded as having a place." should be restored, along with the reference to Hastenes. Brews ohare (talk) 16:06, 4 January 2010 (UTC)

The example: "In 3-D two bivectors are equal if the plane segments they represent have the same area, their planes are parallel, and their bounding curves are traversed in the same direction." is helpful to me in understanding the lack of location of the bivector. Brews ohare (talk) 16:09, 4 January 2010 (UTC)

I tried another version of your paragraph that seems to meet these points. Brews ohare (talk) 16:32, 4 January 2010 (UTC)


 * I was updating it at the same time so I've put that in. Your version had a few problems:
 * * you removed the mention of the attitude or rotation. the latter is the correct mathematical term. See e.g. Rotation (mathematics) or Rotation matrix or quaternion rotation or Rotation group, so should be understood by any mathematician. Attitude is what Lounesto uses and it also has a page, though it's not used in maths.
 * * the bivector does not have a sign, it's not a real number. So it's better to say it's negated. Similarly the sense of the rotation is not a signed number.
 * * it's not clear what this means, in that it's not clear English: "with the sense that of the rotation that aligns a along b". -- John Blackburne (words ‡ deeds) 16:53, 4 January 2010 (UTC)

Hi John: I believe we are almost converged. I'd be happy with the present text if the second sentence was removed, this one: "That is the bivector describes not only the rotation or attitude of the whole plane, but also the area of the plane segment and the orientation, i.e. the sense of rotation in the plane."

Here's what I don't like about this sentence:
 * I am unpersuaded that the sentence conveys anything useful that is not stated better in the paragraph without it.
 * Maybe "rotation" is the technically correct math term, but it is not intelligible outside those specialized in the area. Personally, I have never used "rotation" in this way, as an attribute of a plane, and I'm probably better versed than many readers. For example, Rotation (mathematics) really is about rotations as mappings, not about using "rotation" as an attribute. Likewise for Rotation matrix. I didn't check the rest of your suggested links.

Sense of rotation: this usage is common: see this or this or this.

How about removing the second sentence? Brews ohare (talk) 17:11, 4 January 2010 (UTC)


 * Rotation is a mathematical term but hardly a specialised one and this is a mathematical article, while attitude is a more general term from the source. Providing two links should be enough for any reader. Rotation is even used later in the article, especially in Bivector that you added. I can't see how you never used it.


 * I've changed "orientation" back to "rotation". It's a different (in the plane between the vectors, rather than in space) but the same sort of thing, and is only a problem if you think about it too hard as I did.-- John Blackburne (words ‡ deeds) 17:38, 4 January 2010 (UTC)

John: Maybe our disagreement is a matter of usage: the sentence says: the bivector describes not only the rotation ... of the whole plane, but also the area of the plane segment which to me means "rotation" is an attribute of the plane, as in "fashion describes not only the color of the dress, but also the choice of buttons". The usage of "rotation" as described later in this article, and in the other articles you have linked, uses "rotation" differently, as a mapping, as in Active and passive transformation. It is not a property of a plane but an action to be taken upon the points of the plane. Do you recognize this difference in usage? Brews ohare (talk) 18:30, 4 January 2010 (UTC) Maybe you wish to say the bivector governs a specific rotation of the plane? That idea is premature here as this connection is not established yet. Brews ohare (talk) 18:32, 4 January 2010 (UTC)

I have found the term rotation plane, where "rotation" is an adjective separating the plane in which a particular rotation occurs from other planes. That usage does not apply here, where a connection between bivectors and rotations has yet to be established. I find the use of "rotation" as an attribute of a plane to be unusual ( I haven't found an example of this usage), and so should not be employed in the article. Brews ohare (talk) 18:59, 4 January 2010 (UTC)


 * It's not established but it's not talking about how bivectors can be used describe rotations, just how the plane is rotated in the mathematical sense. It is an attribute of the plane, in that you can say the plane (on any object) has a rotation (or attitude) like it has a position. It is also used in the more abstract sense, to talk about rotations as an abstract thing, but they are the same thing, and it is how maths works all the time: things that have real world uses are also studied in the abstract. Number for example - we talk about the number of something (number of days in a week), and we also study numbers abstractly as natural, rational or real numbers.


 * The only other term would be orientation, but that is ambiguous as it also describes the "sense of rotation", which is how both Lounesto and Hestenes use it, in the mathematical sense, i.e. Orientation (mathematics), and what it means in "oriented plane segment". Or just leave it at 'attitude', but I think that term is slightly archaic, if more precise, and would be lost on most readers.-- John Blackburne (words ‡ deeds) 19:09, 4 January 2010 (UTC)

Can you supply some examples using google books that show this usage of "rotation" as an attribute of a plane without specifying some rotation specifically? It strikes me as arcane. If this usage appears only in rather specialized literature, it would be best avoided. Brews ohare (talk) 21:10, 4 January 2010 (UTC)

n-dimensions
The discussion of planes and parallelograms is clear in 3-D, but we need some links or some subsections explaining the connection in more dimensions. The intro implies the 3-D ideas extrapolate without difficulty to higher dimensions. Brews ohare (talk) 21:19, 4 January 2010 (UTC)


 * There's nothing there that depends on 3D: adding simple limits the geometric interpretation to those bivectors that can be associated with planes. Planes and parallelograms exist in all dimensions ≥ 2. Avoiding words like "clockwise" help as they suggest you can e.g. look at the plane from one side, something that breaks down in higher dimensions.


 * As noted above what's needed is:
 * * a further elaboration of this in 3D where it's most intuitively obvious and can be linked to other things in 3D the reader might be familiar with.
 * * a discussion on how it dies and doesn't generalise, as where it does we can say things about e.g. planes, rotations and the properties of space that are difficult to do with other tools.
 * * Some applications in 4D and 5D.-- John Blackburne (words ‡ deeds) 21:29, 4 January 2010 (UTC)

John, I think that you have been too swift to assume that matters relating to vectors in 3D geometry can automatically be extrapolated to higher dimensions. Unfortunately, I don't have my old notes handy right now. But I do remember that the expression,

$$\mathbf{a} \times \mathbf{b} = a b \sin \theta \ \mathbf{\hat{n}}$$

follows from the basic axioms of the 3D cross product. Some of those axioms do not hold in the 7D case. The vector triple product axiom and the Jacobi identity axiom do not hold in the 7D case. I'm pretty sure that will have a bearing on the proof which links the cross product to the expression,

$$\mathbf{a} \times \mathbf{b} = a b \sin \theta \ \mathbf{\hat{n}}$$

but until I see my old notes again, I can't be totally sure. Something tells me that if the vector triple product doesn't work, then the Lagrange identity won't work either, and the sinθ equation is all tied up with the Lagrange identity. My memory tells me that everything to do with the above sinθ expression is linked exclusively to 3D geometry. And as we already know, apart from 1, 3, and 7, we can't have any others. The 7D cross product does not map to the hypothetical 7D space in the same way that the 3D cross product maps to 3D space. A truly 7D cross product would involve all seven components at once in an operation. We have already had this discussion over on the Seven dimensional cross product talk page and I see that you may be pushing this same error here too. David Tombe (talk) 08:27, 5 January 2010 (UTC)


 * This has little to do with bivectors. Please raise any issues related to Seven dimensional cross product on its talk page, where I or another editor can help you with them. -- John Blackburne (words ‡ deeds) 08:41, 5 January 2010 (UTC)

John, It's got everything to do with wrongly extrapolating issues in 3D geometry to n-dimensions. You are the one that is badly in need of help. Your whole problem lies in assuming that the equation,


 * x &times; y|2 = |x|2 |y|2 &minus; (x &middot; y)2

is general for all dimensions. It is not. The Lagrange identity is a complex expression, and this is only the special 3D version.

Not even the Jacobi identity or the vector triple product hold in 7D. And 7D is the only D apart from 1 and 3 that anything holds in. And yet you seem to think that all this 3D stuff can nicely be extrapolated to all other dimensions? David Tombe (talk) 08:47, 5 January 2010 (UTC)

Usage of "rotation"
JohnBlackburne: The article says: "That is, the bivector describes not only the rotation ... of the whole plane, but also the area of the plane segment and the orientation, i.e. the sense of rotation of the plane ."

Of course, no actual rotation is envisioned here in the sense of a mapping or in the sense of Rotation (mathematics) (a transformation in a plane or in space that describes the motion of a rigid body around a fixed point). I have tried to google search for as a descriptor of a property of a plane. The returned links all refer to an actual rotation in the sense of Active and passive transformation. They do not refer to a "rotation property" of the plane. So I cannot see how your support on this Talk page for this sentence in the article, viz: "things that have real world uses are also studied in the abstract" has any bearing upon the matter.

Can you supply some examples using google books that show published literature using "rotation" as an attribute of a plane ? Failure to find some such sources means this usage is arcane, and therefore should be deleted from the article.

Will you please provide some support for this usage of "rotation"? Brews ohare (talk) 05:00, 5 January 2010 (UTC)


 * Hestenes NFfCM, page 420. He lists a table of properties, lists the relevant property as "Attitude", but in the text above says "the spinor-valued function R = R(t) determines the time dependent rotation". Lounesto also uses "Attitude". It is perhaps the more precise term as "rotation" has other uses, but "rotation", as in Rotation (mathematics), is the usual modern term in mathematics. -- John Blackburne (words ‡ deeds) 11:06, 5 January 2010 (UTC)


 * John: The source Hestenes NFfCM, page 420 discusses the position and attitude of a rigid body, locating its center of mass by a vector and the relative position of each particle in the body relative to a fixed reference by a time dependent rotation. For a rigid body that is standard practice. So I gather you wish to suggest that specification of the "rotation of a plane", referring to a property of an arbitrary stationary plane, implies (i) setting up reference axes, and (ii) locating three points in the plane using vectors, and (iii) specifying the orientation of these vectors as a rotation (or combination of rotations) relative to the chosen coordinate reference axes. That is all terribly complex and unnecessary in general, although it might be helpful in some specific situations. I can find no source where such a process is implied or employed as a general property designated by the term: "the rotation of the plane". Can you?


 * Another significant point: any such "property" of the plane is altered by the choice of the reference axes, and so is a property of both the plane and the axes, established by their relative orientation. That is not a property of the plane itself.


 * The article Rotation (mathematics) does not support this usage either: it discusses rotation in the transformational sense: "a transformation in a plane or in space that describes the motion of a rigid body around a fixed point."


 * Bottom line: nobody I can find refers to "rotation" as a "property" of a plane. The sources you suggest don't either. Brews ohare (talk) 14:24, 5 January 2010 (UTC)


 * Rotation is a property of anything that can be rotated in space. There's no special "rotation of a plane" that is different from everything else, there's just rotation, as described in the article, used in Hestenes, and in many other places. It's the common term in mathematics, which is why e.g. Hestenes uses it without elaboration, as do have in this article.-- John Blackburne (words ‡ deeds) 14:38, 5 January 2010 (UTC)

John: You are inventing your own terminology. Rotation is not the property of anything that can be rotated. Rotation is an action that can be performed upon anything that can be rotated. Hence, rotation can be used to describe the relative orientation of two coordinate systems, that is, it describes the action that must be applied to map one into the other. But the rotation mapping one into the other is a property of neither, but a property of the two of them relative to each other. Your sources say exactly this, and do not support your usage. Brews ohare (talk) 14:45, 5 January 2010 (UTC)

As an example: in the vector space R2, a rotation through angle θ is given by the matrix:
 * $$R=

\begin{pmatrix} \cos \theta & \sin \theta\\ -\sin \theta & \cos \theta \end{pmatrix}, $$ where angle θ is a property of the two systems: the one rotated and the one that is used as reference. The angle θ is not a property of either system alone. Brews ohare (talk) 14:51, 5 January 2010 (UTC)

Usages cont'd
I see the issue as one of two different usages: the vague general term "rotation" that might be used in common English where a reference system is implied but not specified (a synonym for "orientation"), and the technical usage of rotation that can be made quantitative in the sense of transformation. In the article defining Bivector a technical terminology is likely to be understood, when you mean to advance only the vague general usage. Yes? Brews ohare (talk) 15:03, 5 January 2010 (UTC)


 * The rotation is relative to the base coordinate system or frame of reference, as are all these things. When you write "the position of the object is x" you don't need to give the coordinate system, you usually just assume there is one. The same for "the rotation of the object is R". Just as a position can be absolute or relative, so can a rotation be. But you don't need to say so if it's clear, or if they mean the same thing.-- John Blackburne (words ‡ deeds) 15:07, 5 January 2010 (UTC)

I am happy that the need for a reference system is agreed to. The sentence in the article is, to repeat: :"That is, the bivector describes not only the rotation or attitude of the whole plane, but also the area of the plane segment and the orientation, i.e. the sense of rotation of the plane."

I am stuck trying to define the "sense of rotation of the plane". In my mind, the plane exists independent of any reference system and its properties are also independent. Without some reference system, the "sense of rotation of the plane" is impossible to determine. I'd suggest that this phrase can be dropped entirely. From the standpoint of the bivector, whatever this "sense of rotation of the plane" may be, it is irrelevant. What is relevant is the sense of the bivector, which is decided by the rotation taking a into b as described by the subsequent sentence. This rotation is opposite to that taking b into a, also described by the following sentence. That leaves us with:
 * That is the bivector describes not only the attitude of the whole plane, but also the area of the plane segment and its orientation. If the bivector is a^b then the area, which is the magnitude of the bivector, is the same as the parallelogram with edges a and b, and the orientation is given by the rotation that would align a with b. The exterior product is antisymmetric, so b^a negates the bivector, producing a rotation with the opposite sense.

I'd settle for this, although the first sentence seems unnecessary to me. What do you think? Brews ohare (talk) 15:34, 5 January 2010 (UTC)

I made this change and added a note and sources for the use of the term "attitude". The source Jancewicz provides a pretty thorough discussion. Brews ohare (talk) 20:12, 5 January 2010 (UTC)

Source of font ℝ
John: What font is ℝ and where did you find it? Brews ohare (talk) 15:42, 5 January 2010 (UTC)

It's not a font it's a unicode character, in many fonts. I'm on a Mac so can get it through the Character Palette, or you can copy and paste it from anywhere you see it.-- John Blackburne (words ‡ deeds) 15:53, 5 January 2010 (UTC)

Thanks, John. It isn't on my PC, but I found it following your suggestion at Letterlike symbols. Brews ohare (talk) 16:38, 5 January 2010 (UTC)

Problems with second paragraph
Compared to the version here the second paragraph now has numerous problems. In particular


 * Bivectors are no more associated with a family of planes than vectors are associated with a family of lines. The source is not about bivectors.
 * Rotation removed again, even though it's the usual mathematical term and is sourced.
 * Curious emphasis of "family of planes (italicised, moved to top, repeated)
 * attitude does not "identify"
 * Spaces removed around operators, more difficult to read and inconsistent with the rest of the article where TeX does it properly. I can't find anywhere on Wikipedia that says this is wrong, probably as no-ones ever thought to do it like this.
 * It's just badly written - apart from the reordering and removals there have been a lot of small grammatical and punctuation changes which make it more difficult to follow.

As linked to above there was a better (clearer, more correct, better written) version two days ago. Not perfect but the changes since then have made it far worse. So should I rewind it back to that version? -- John Blackburne (words ‡ deeds) 16:47, 6 January 2010 (UTC)

From Brews_ohare:

 * The basic issue is clarity.


 * The version suggested by JohnBlackburne has a few deficiencies:
 * The simple bivector is not an oriented plane segment: this language may suggest the segment is a part of some particular plane, while the bivector is not attached to a particular plane but to an ensemble of parallel planes, commonly called a family. The independence of the bivector from any position is inadequately conveyed in this paragraph, and only introduced as an afterthought at the end.
 * The use of the word "rotation" (the rotation of the whole plane) as advocated by JohnBlackburne is inadvisable, IMO, as outlined in earlier back-and-forth on this Talk page. The word "rotation" in this version is used as though it were a technical term, when its only valid interpretation is as a vague synonym for the common English usage of "orientation". Technically, the word "rotation" cannot be made quantitative without explicit introduction of a reference frame, and a major goal of geometric algebra is to be frame independent.


 * The present version follows Jancewicz pretty closely in identifying three attributes of the bivector: attitude, magnitude, and orientation. See Table 28.1, p. 403 in Jancewicz available on line at this link, and Jancewicz' detailed discussion leading up to this table. These attributes are all frame-independent.
 * The present version of the paragraph has two parts: a general description, followed by an example a^b. It is accurate, but could be simplified IMO by avoiding the general verbal statement of the first few lines and focusing upon the description in terms of the example a^b. Brews ohare (talk) 17:40, 6 January 2010 (UTC)

Discussion on "family of planes"
The problem on the "family of planes", which accounts for many of the changes, is it's unsourced. Whereas Hestenes in NFfCM writes: I.e. there is only one plane, through the origin. Rotation, as in Rotation (mathematics), is a standard mathematical term, and again is sourced in NFfCM, as gone over above. Attitude is more precise, not used much in mathematics today but common in sources. So best to use both. -- John Blackburne (words ‡ deeds) 18:28, 6 January 2010 (UTC)
 * "Given a (nonxero) bivector B, the set of all vectors x which satisy the equation x ^ B = 0 is said to be a 2-dimensional vector space and may be referred to as the B-plane. ... So every bivector which is a non-scalar multiple of of i determines the same plane as i. Such a vector is a called a pseudovector of the plane" (page 49)


 * The "plane" of Hestenes above is not a single plane, but any of a set of parallel planes. Of course, one is free to define a plane as a set of planes, but that is not common usage. For example, crystallography identifies individual planes as well as sets of parallel planes; a common capacitor is between two parallel planes; interplanar separation is a common concept; and so forth. The preponderance of readers will not think of a plane in Hestenes' sense of a set of planes. Two sources for the usage "family of planes" are provided. Brews ohare (talk) 18:44, 6 January 2010 (UTC)

This article is about the bivector. Neither source says anything about bivectors. The source I gave does precisely describe how a bivector relates to a single plane through the origin, given by the equation x ^ B = 0-- John Blackburne (words ‡ deeds) 19:01, 6 January 2010 (UTC)


 * The objective is to explain bivectors. I don't see that usage of the word "plane" has to be taken only from a particular source that also mentions bivectors, suggesting that somehow a "plane" is a new item in this context, with a new meaning. Do you? If we have to introduce the "new" B-plane, it cannot be used to define the properties of the bivector, as that would be circuitous. Brews ohare (talk) 19:06, 6 January 2010 (UTC)


 * You have not addressed Jancewicz' exposition, which seems to me to be a good compromise. He uses "attitude" in the sense needed here, although that word has ambiguity in common usage, occurring in the "attitude" of a rigid body, the "attitude" of an airplane, which are related but not exactly what is wanted. Brews ohare (talk) 19:06, 6 January 2010 (UTC)


 * No, the point is to correctly describe the relationship of a bivector to a plane, as Hestenes does with precise mathematics. If you have a source for the "bivector == family of planes" interpretation you prefer please give it. On the other point I do not object to attitude, best use both it and rotation.-- John Blackburne (words ‡ deeds) 19:16, 6 January 2010 (UTC)

John, I'd say that in three-dimensions there is no doubt that the bivector determines a family of parallel planes. Do you agree? That is also well sourced. From the notion from Hestenes above that the plane is two-dimensional, and from the stated objective of this source to describe classical mechanics, I'd take it that Hestenes also has only 3--D in mind in the above quotation.

More generally, Jancewicz says that the "attitude of the bivector is a plane". Cartan says "two bivectors are said to be equal if their two associated parallelograms are in the same plane (or in parallel planes), have the same area and sense of direction". That clearly recognizes a set of distinct planes that are parallel. Hazewinkel says: "A non-zero bivector generates a unique two-dimensional space in an affine space A, its carrier. Two bivectors are said to be parallel if their carrier planes are parallel." Here there is a distinction drawn between the "carrier" and its constituent "carrier planes".

Introduction of a "plane" that uses the bivector to define it is circular in a discussion of "What is a bivector?", eh?

In general, one can easily argue that the present paragraph is not perfect, but it is clear, and it is accurate. Brews ohare (talk) 19:35, 6 January 2010 (UTC)


 * "attitude of the bivector is a plane" agrees with what I wrote. The other source is French (the link it so a translation), almost 60 years old, and rather odd (no vector derivations, just coordinates). Hestenes writing 30 years later largely established modern geometric algebra. Anything pre-dating his work has to be read vary carefully and should not be used if there's a more modern source that is clearer.-- John Blackburne (words ‡ deeds) 20:09, 6 January 2010 (UTC)


 * As per WP:REDACT you should not, as you have just done, change your comment after I've replied to it. Please undo this change and reply properly to my points.-- John Blackburne (words ‡ deeds) 20:22, 6 January 2010 (UTC)

Hi John: In my edit I added Hazewinkel as another source. Your remark that Hestenes is modern and other sources less so really does not establish that Hestenes has the clearest or even the most commonly used view, and does not address the issue of circularity in the second paragraph. Brews ohare (talk) 20:42, 6 January 2010 (UTC)

Some other sources referring to parallel planes in connection with the bivector are: Hestenes "bivectors with the same direction can be represented by plane segments in parallel planes" (I take that as inclusive of the statement that two identical bivectors can be so represented) Fordor "We may accordingly represent the bivector [(b-a)(c-a)] by any area in one of its parallel planes..." (I take that as meaning the bivector refers to many such parallel planes) Lounesto "Two bivectors A and B in parallel planes have the same attitude..." (I take that as inclusive of two identical bivectors A = B.) There seems to be lots of literature referring to multiple parallel planes.

Is it fair to say that you understand perfectly that a bivector characterizes a family of parallel planes, any one of which serves as well as any other, and that the converse does not hold because bivectors can differ in other properties, even though they may have the same attitude?? Brews ohare (talk) 20:42, 6 January 2010 (UTC)


 * Again, please put your replies in order so I can reply to them properly. -- John Blackburne (words ‡ deeds) 20:52, 6 January 2010 (UTC)

Just reply to them as they appear immediately above your remark. Brews ohare (talk) 21:02, 6 January 2010 (UTC) I'll repeat them below for you:

Further discussion on "family of planes"
Hi John: In my edit I added Hazewinkel as another source. Hazewinkel says: "A non-zero bivector generates a unique two-dimensional space in an affine space A, its carrier. Two bivectors are said to be parallel if their carrier planes are parallel." Here there is a distinction drawn between the "carrier" and its constituent "carrier planes".

Your remark that Hestenes is modern and other sources less so really does not establish that Hestenes has the clearest or even the most commonly used view, and does not address the issue of circularity in the second paragraph.

Some other sources referring to parallel planes in connection with the bivector are: Hestenes "bivectors with the same direction can be represented by plane segments in parallel planes" (I take that as inclusive of the statement that two identical bivectors can be so represented) Fordor "We may accordingly represent the bivector [(b-a)(c-a)] by any area in one of its parallel planes..." (I take that as meaning the bivector refers to many such parallel planes) Lounesto "Two bivectors A and B in parallel planes have the same attitude..." (I take that as inclusive of two identical bivectors A = B.) There seems to be lots of literature referring to multiple parallel planes.

Is it fair to say that you understand perfectly that a bivector characterizes a family of parallel planes, any one of which serves as well as any other, and that the converse does not hold because bivectors can differ in other properties, even though they may have the same attitude?? Brews ohare (talk) 21:06, 6 January 2010 (UTC)


 * Hazelwinkel is a tertiary source and isn't even talking about bivectors. Otherwise there's no contadiction to say "bivectors are the same if their planes segments are parallel, have the same area and orientation". It is clearer in some ways as everyone knows what "parallel" means but other words like "attitude" are less common. But this does not mean a bivector generates a family of planes. It is associated with a single plane, given mathematically by Hestenes.


 * There's a reason I keep referring to Hestenes: he invented modern geometric algebra and is probably the most referenced author in geometric algebra today - everything depends on him or is at best out of date.


 * But more importantly I'm happy to use one or two sources, i.e. Hestenes and Lounesto, as they're the books I have in front of me and cover pretty much everything there is to know. there's no need to search Google books as you seem to be for fragmentary references, which you frequently misunderstand based on your comments here and contributions. It's good to look at more than one source to get a well rounded view of the topic, but to repeatedly search until you get something you think backs you up but actually is about something else weakens not strengthens your point. -- John Blackburne (words ‡ deeds) 21:27, 6 January 2010 (UTC)

So, to be more succinct about it, your answer to:
 * "Is it fair to say that you agree that a bivector characterizes a family of parallel planes, any one of which serves as well as any other, and that the converse does not hold because bivectors can differ in other properties, even though they may have the same attitude??"

is :
 * No. There is no such thing as a family of parallel planes, even in 3-D, that has the same attitude as a particular bivector a^b. Therefore, the attitude of a bivector cannot be construed as characterizing such a family. Moreover, all the sources you cite (including Hestenes and Lounesto) that appear to invoke parallel planes are either (i) outdated, or (ii) refer to subjects other than bivectors, or (iii) have been misconstrued in ways I don't wish to pursue in detail.

Is that what you would say? Brews ohare (talk) 22:07, 6 January 2010 (UTC)

If the above is not a statement of your views, I'd be obliged if you would point out where it diverges. Brews ohare (talk) 22:31, 6 January 2010 (UTC)


 * roughly yes. Though there's also the point that even when they say "parallel planes" it does not mean distinct. They might say a precondition for equality is the planes are parallel, as parallelism is easy to visualise.


 * E.g. vectors. We associate a vector ae1 + be2 +  ce3 with the point (a, b, c). The set of vectors parallel to this generate points on a line, through the origin, parametrically given by (at, bt, ct) providing the vector is non-zero. All the vectors (ae1 +  be2 +  ce3, 2ae1 +  2be2 +  2ce3, etc.) are parallel, and you would say this. But they lie in the same line.


 * The same for bivectors: they are parallel but lie in the same plane, through the origin. Otherwise all sorts of things don't work well, e.g. you can't treat bivectors as linear subspaces, where products have interpretations in terms of subspace operations.


 * Also there is a way to associate objects in GA precisely with points and lines not through the origin, given e.g. in this paper, by Hestenes, and built on since by others. -- John Blackburne (words ‡ deeds) 22:34, 6 January 2010 (UTC)

OK John. Let's compare viewpoints. We'd agree that all bivectors u^v, α u^v have the same attitude. Likewise, in fields like geology, aviation, astronomy, crystallography etc. all "planes" in a "family of planes" have the same attitude. That is, in these contexts, all bivectors u^v, α u^v belong to the same family of parallel planes. Making a comparison, your definition of "plane" has properties similar to those of the common definition of a "family of parallel planes". Your definition of "plane" is understandable in a narrow context with a special definition; mine is understandable in many fields, such as crystallography, high school geometry, etc. etc. Right? Brews ohare (talk) 23:53, 6 January 2010 (UTC)


 * I've given my views, repeatedly, with sources and algebraic definitions. Please do not try to put words into my mouth. Read what I've written already it makes perfect sense.-- John Blackburne (words ‡ deeds) 00:02, 7 January 2010 (UTC)
 * I am putting words in my own mouth, John, not yours. There is no doubt that in fields like geology, aviation, astronomy, crystallography, high school geometry etc., all "planes" in a "family of planes" have the same attitude. They can also all be characterized (since this is 3-D) by the attitude of the same normal vector that is expressible as a × b where a and b lie in any one of these planes, and this axial vector is directly related to a^b. So these words I have put in my mouth are sourced. I understood you to say that the words "plane" and "family of planes" as employed here do not apply in geometric algebra à la Hestenes, at least in some of his formulations? Brews ohare (talk) 07:12, 7 January 2010 (UTC)


 * Plane ≠ family of planes. One is a single object the other is a usually infinite set, not necessarily parallel. See e.g. Pencil (mathematics). It's standard mathematical usage but is also plain English. So "family of planes" has no place here. As for crystallography, here is a geometric algebra treatment of it. As you can see it uses a very different approach to specify points, lines etc. in space. -- John Blackburne (words ‡ deeds) 09:29, 7 January 2010 (UTC)

Hi John: I am happy to see you recognize individual planes, as I had thought you wished to use the word "plane" to refer to an "equivalence class" of planes that share parallel normals. I have, as you know, persistently distinguished individual planes from a set of planes called a "family of planes". You have objected to the word "family" as being more inclusive than the definition used in crystallography and geology. However, these sources are pointed out in the paragraph in the article to make perfectly clear that a family is a set that share the same normal. That is not an invention or conceit of my own. Your link to the geometric treatment of space groups by Hestenes is interesting, but I have not found a passage there that is pertinent to this discussion. Perhaps you could point something out? Brews ohare (talk) 15:58, 7 January 2010 (UTC)


 * Please stop trying to second-guess or interpret what I mean. I do not "recognise individual planes", I'm not even sure what it means.


 * No, a family of planes does not necessarily share the same normal - again see Pencil (mathematics). The link was to a paper on GA applied to crystallography, and which deals with general points, lines and planes, which is far more relevant to the article than this that you provided (which anyway links to so many things that it shows nothing). You still have not provided a source for the "bivector == family of planes" interpretation, as so far the lniks you've provided have been as irrelevant as that one. -- John Blackburne (words ‡ deeds) 16:20, 7 January 2010 (UTC)

From RDBury:
You can characterize vectors in several ways, one of them is as a magnitude and a direction and another is as a magnitude, an equivalence class of parallel lines and an orientation. Here orientation means which of two ways the vector is pointing. Bivectors can be characterized analogously in two ways, so the question isn't which one is correct but which is more commonly used in the mathematical community and which gives a better understanding of the concept. In this sense, it looks like the original version of the paragraph meets both criteria. The references given in the current version support an inference, but that inference is still on the part of an editor; it's better to have a source that supports the statement in article directly. Also, bringing crystallography here is explaining a concept in terms of an equally unfamiliar (to most people) concept; it's better to stick with ideas that the reader will be familiar with.--RDBury (talk) 13:26, 7 January 2010 (UTC)


 * RDBury: Thanks for joining this discussion. To fill in your discussion of the bivector, to replace the word "similarly" above, I believe it would go like this:
 * You can characterize bivectors in several ways, one is as a magnitude, attitude, and orientation; and another is as a magnitude, equivalence class of planes, and orientation.


 * The "equivalence class of planes" is a synonym for a "family of planes", that is, it refers to all planes with parallel normals. The normal to a plane, of course, determines its "attitude". Planes with the same attitude are in the same equivalence class. Does that seem like a correct version, as applied to bivectors?


 * Thanks for your reply. Brews ohare (talk) 15:29, 7 January 2010 (UTC)
 * This source provides the relevant discussion for 3-D. Brews ohare (talk) 17:17, 7 January 2010 (UTC)
 * This source and this provide generalization of "normal" to higher dimensions. Brews ohare (talk) 18:08, 7 January 2010 (UTC)


 * OK ,we seem to have finished with this debate so I'm removing the RfC tag -- JohnBlackburne wordsdeeds 22:46, 10 January 2010 (UTC)

Fallacies
Again I have had to revert a version with numerous problems (formatting and simple mathematical errors) -- John Blackburne (words ‡ deeds) 16:42, 7 January 2010 (UTC)
 * A plane normal only exists in 3D
 * Clockwise and counterclockwise are only defined in 2D
 * Overuse of italics
 * Re-introducing unclear usage of attitude

JohnBlackburne: Your reversion here was made with a one-line edit referring to "fallacies". The reverted lead sentence is:

Do you take exception to this sourced introduction of the three properties of a bivector? Brews ohare (talk) 16:43, 7 January 2010 (UTC)
 * Geometrically, a simple bivector has three properties: magnitude, attitude and orientation.


 * See here -- John Blackburne (words ‡ deeds) 16:44, 7 January 2010 (UTC)

I will abandon my attempts to engage in a discussion with JohnBlackburne. Maybe after a few weeks things will go better. Brews ohare (talk) 16:47, 7 January 2010 (UTC)

New 3D intro
I've added an intro to the 3D section which establishes the properties of bivectors in 3D: it replaces a previous subsection as there's no need to introduce the Kronecker delta or re-introduce the geometric product. Lots of maths but it skips over the details: it just looks a lot because of the extra indices whenever you write general bivectors. In some ways it's just restating earlier results but it's good to do so as it's only in 3D that a lot of these things are non-trivial. -- John Blackburne (words ‡ deeds) 23:06, 8 January 2010 (UTC)


 * The 3D section is pretty much done now, i.e. I've added everything I wanted to. I'll probably come back and copyedit it in the next few days, as I've done with previous sections, especially this last one as its so long and wordy. -- JohnBlackburne wordsdeeds 00:53, 15 January 2010 (UTC)

Biquaternions
I've just reverted this change as I had a number of issues with it. First it removed the mention of quaternions which every history agrees were an important part of the development of Clifford's algebra. See e.g. Hestenes, NFfCM, page 59.

More importantly I don't think the changes made sense on their own. The 'bivector' is mentioned by Hamilton in footnotes: he does not discuss them at length. And from their description they seem just to be the same as Gibbs's bivector, which is already mentioned in the text, with references which cover it in more detail. Gibbs was developing his vector algebra at this time so it would not be surprising if Hamiton was aware of them. They are definitely not the same a biquaternions, which are quite different in many ways, and the source does not support any such assertion.-- JohnBlackburne wordsdeeds 13:49, 14 January 2010 (UTC)


 * The loss of the quaternion link is regretable. Restoring it was right. However, "Gibbs was developing his vector algebra at this time so it would not be surprising if Hamilton was aware of them" shows you have not noted the years involved. My contribution clearly gives 1866 has Hamilton's use of the term, long before Gibbs was writing on vectors. The Hamilton source has bivector in common text, not a footnote, and clearly related to biquaternion, as he says, the ratio of a bivector to a vector. Hamilton's priority over Gibbs is clear, as is the relation of bivector to biquaternion. Please continue to discuss your issues so that this article can be appreciated appropriately.Rgdboer (talk) 03:36, 16 January 2010 (UTC)


 * I have it in front of me now: I was looking at the second volume, while you are referring to the first. But I'm still not convinced it's a notable use of bivector or a notable part of the history of them. His bivectors are something other than what we are discussing here, being six-dimensional but clearly not bivectors in 4D. He does not develop them, simply stating what they are, as part of a program of classifying complexified objects (BIVECTOR, BISCALAR, BIQUATERNION), and noting that dividing a bivector by a vector yields a biquaternion. They are essentially Gibbs's bivectors, though you are right Hamilton cannot have got them from Gibbs, so it was likely the other way around. Gibbs does find a use for them and it's this application of them which finds adherents so Gibbs's version were still being used very recently. See e.g. here.


 * The reason they are mentioned at all is because of this, Gibbs's usage persisting for almost a century. i.e. they are not part of the history of bivectors as we know them, but are part of the diversion that was Gibb's approach to the subject. See e.g. here. That Gibbs' bivectors were based on Hamilton is interesting but I don't think notable, especially as Hamilton does not seem to have a clear idea what they are, or at least he does not do anything with them.-- JohnBlackburne wordsdeeds 12:30, 16 January 2010 (UTC)

Relationship with vector algebra
This section is unnecessarily confusing because it uses the term "pseudovector" initially as a synonym for "axial vector", and later restricts this usage to three dimensions. For all but three dimensions, we are told the pseudovector is the (n-1) grade element.

Support for this approach can be found in the literature. In realms where geometric algebra is not employed, the equating of "pseudovector" with "axial vector" is common, as the article Pseudovector attests. Inasmuch as cross-products don't generalize, this terminology is ipso facto limited to three dimensions.

However, support also can be found for a different approach that uses the (n-1) grade element for all dimensionalities, so in three dimensions the bivector is the pseudovector. That approach has the merit of consistency.

The various books like Hastenes, and Baylis, and Doran & Lasenby, and DeSabbata & Datta, all use the identification of pseudovector with bivector, and claim that advantages in the interpretation of the applications attach to this approach. See this, for example. I'd suggest that within the context of geometric algebra, taking the pseudovector as the bivector in 3D is the more consistent, the more useful, and the predominant terminology within geometric algebra.

I'd like to see this section of Bivector changed to follow this approach, and to relegate the pseudovector = cross-product terminology to an aside. That would correspond to the approach used in Pseudovector. Brews ohare (talk) 15:34, 15 January 2010 (UTC)


 * Thanks for the comments. That was a difficult section to organise as I wanted to establish the relationship with the vector algebra without getting tied up in the "what is a pseudovector?" issue. In GA a pseudovector is a bivector in 3D, so at the start of that section the vector is an axial vector (which is the term I prefer for it anyway) first, a pseudovector second. Later I've added a note, with pseudovector in bold, which gives the usual GA definition.


 * But as I said here "pseudovector" is not as much used in GA. The book you link to is a good example. a quick search turns up exactly two uses of the word in several hundred pages. Which is two more than in either Hestenes NFfCM or Lounesto. And these are fairly typical It's not a term generally used to describe a bivector, or much used by any author in GA. At most at some point they say "the bivector is a pseudovector in 3D", then leave it at that, after which it's discussed as a bivector. So that's the usage I've followed here.


 * It is restricted to 3D as it's part of the 3D section, and I've tried to get as much in that section as possible as there are more links to topics/articles on topics readers might be familiar with and more opportunities to describe things intuitively. It will become clearer when there's something on 4D, which will largely consist of some algebra and a lot of exceptions to the things established in 3D.-- JohnBlackburne wordsdeeds 16:21, 15 January 2010 (UTC)

Hi John: Once the pseudovector is established as equivalent to a bivector (or a trivector, etc.) there is not a lot of need for the term, I suppose. Nonetheless, it is clear that the usage of pseudovector is not that of axial vector and is so stated in so many words by about half a dozen texts in the arena of geometric algebra. It seems out of place to me to drag up the extraneous usage in this article on bivector. The use of bivector to replace the use of axial vector is more important than the usage of pseudovector. I don't find the present discussion really hits the right emphasis. Brews ohare (talk) 18:40, 15 January 2010 (UTC)


 * The usage in vector algebra, which precedes that in GA, is that the pseudovector is a vector which transforms differently. And this section is about vector algebra as well as GA, so it has to reflect both usages. As noted above I've deliberately referred to the axial vector first, as it's the term I'm used to and it is a common name for it. But the article is called pseudovector, also a common name, so that name should be used to. As far as I can tell axial vector has always been a redirect to pseudovector, for over six years. i.e. the two are different names for the same things, at least in vector algebra.


 * I just noticed that axial vector was removed, I'm sure accidentally, from the pseudovector article so I've put it back in. As noted it's the name I'd use for it, has been in the article since its creation, and it's supported by sources below.-- JohnBlackburne wordsdeeds 19:17, 15 January 2010 (UTC)

John: As you know the lead-in sentence for this article is In mathematics, a bivector is a quantity in geometric algebra that generalizes the idea of a vector. That is, this article in WP is about the bivector in geometric algebra (although it could be redesigned to deal with tensor algebra, I suppose, which topic is presently an afterthought at the end). As such, the section on Vector algebra is a subheading of the main topic, the bivector in geometric algebra. Consequently, the reference to the pseudovector in this subsection should make note of the predominant usage in texts on geometric algebra, namely that the bivector in 3D is the pseudovector, just another example of the (n-1) grade multivector being the pseudovector. No? Do you disagree about the common usage? I've pointed out Hastenes, and Baylis, and Doran & Lasenby, and DeSabbata & Datta.

On that basis it is not exactly a mis-statement but certainly an easily mis-read statement in the article that says: This is the same behavior as bivectors and in geometric algebra axial vectors are usually interpreted as bivectors. In fact, what is usual and less open to misconception is Ławrynowicz "Hence in 3D we associate the alternate terms of pseudovector for bivector, and pseudoscalar for the trivector". Baylis goes to some length to separate the transformation properties of bivectors from axial vectors under inversion. I will produce five more quotes from other sources if you wish. These authors go on to point out that the axial vector is the dual of the bivector, and therefore the dual of the pseudovector, as is of course readily established by simple algebra. Brews ohare (talk) 05:31, 16 January 2010 (UTC)


 * Looking at the two sources Ławrynowicz seems to agree with what we have here: "Hence in 3D we associate the alternate terms pseudovector for bivector", and seems to treat them as interchangeable. But it's not a significant source: he only has a few pages on geometric algebra and the book covers many other things. so its difficult to draw any general conclusions. Bayliss is more interesting, as he does seem to be saying that a pseudovector and axial vector are different, but the distinction he draws is quite subtle, as he says. But it's in an aside, towards the end of the book: it again isn't significant coverage.


 * It might be worth looking at other sources but if they are as different and varied as the above two I'm not sure it will show anything. It also relates to this which I raised a while ago. If you use e.g. Google book search you can find many mentions of a phrase, but they will often be fragmentary, contradictory and drawn from sources which are primarily on something else. Which is why I prefer to use one or two main sources, referring to others only for more obscure points or confirmation. It makes the article more consistent and makes it more likely all main points are covered. Trying to use many sometimes contradictory sources is bound to lead to a confusing article.-- JohnBlackburne wordsdeeds 12:51, 16 January 2010 (UTC)

Axial vectors and bivectors behave similarly under inversion
The article states:
 * Axial vectors or pseudovectors are vectors that have an opposite behaviour to normal or polar vectors under inversion, that is when the frame of reference is reflected or otherwise inverted. ... This is the same behaviour as bivectors and in geometric algebra axial vectors are usually interpreted as bivectors;...

JohnBlackburne: As you know, and as the article also states A bivector is the dual of its associated axial vector, and the axial vector is dual of the bivector. Now this one-to-one correspondence is not an identification, nor does it mean the two are not meaningfully different. In particular, the relation in 3D that:
 * $$\boldsymbol {a \times b} = (a^2b^3 - a^3b^2) \boldsymbol e_1 + (a^3b^1 - a^1b^3) \boldsymbol e_2 + (a^1b^2 - a^2b^1) \boldsymbol e_3, \, $$

while
 * $$ \boldsymbol{a \wedge b} = (a^2b^3 - a^3b^2) \boldsymbol {e_{23}} + (a^3b^1 - a^1b^3) \boldsymbol {e_{31}} + (a^1b^2 - a^2b^1) \boldsymbol {e_{12}} \, $$

implies by simple substitution that under the inversion that replaces all the unit vectors eℓ by −eℓ, the bivector does not change sign, while the cross-product does. This difference in transformation behavior is a mathematical fact, and is pointed out by Baylis. Now, whether or not this fact plays a prominent place in geometric algebra, or is a backwater, one doesn't wish the article to make statements that may be construed to contradict this fact.

Therefore, some rewording of the article is indicated. Brews ohare (talk) 15:27, 16 January 2010 (UTC)


 * The Hodge dual is an identification, i.e. an isomorphism, which its article say is natural and geometrical, and where the first example is vectors and bivectors. But this article doesn't stress that, it instead says they are interpreted or represented, which is what I was trying to get across (to e.g. a reader used to using vectors for things like torque): some quantities can be represented by, or interpreted as, bivectors, with various benefits.


 * I'm still not convinced by Baliss. He seems to be the only one trying to draw a distinction between bivectors and pseudovectors. But he does so in a footnote, in a book largely on something else, and does not develop it further, i.e. to say why its interesting or useful. Generally what he's describing - changing the sign of the basis vectors - doesn't happen in real world applications. Even he admits its a subtle point, which he seems to be the only one making. -- JohnBlackburne wordsdeeds 15:52, 16 January 2010 (UTC)

John: It appears that you intend to insist that the behavior under inversion {eℓ} → {−eℓ} of the axial vector and the bivector is the same, when it is not. It is a matter of simple fact, not of whether Baylis points it out and others don't. The article cannot claim the same behavior under inversion when it is not the same. Brews ohare (talk) 16:24, 16 January 2010 (UTC)

Incidentally, Baylis does not distinguish between bivectors and pseudovectors; he distinguishes between axial vectors and pseudovectors. Brews ohare (talk) 16:33, 16 January 2010 (UTC)


 * My point was the circumstance that Bayliss describes, where the basis vectors are inverted, is not one that's recognised in physics. You rotate and reflect objects, i.e. vectors, bivectors, and the things they represent. But you don't invert the frame. Some of the fundamental laws of physics are tied into the chirality of the frame of reference, so inverting it is meaningless. which perhaps why Baliss describes it as a subtle point in a footnote, and no-one else even mentions it.


 * The other possibility is that Baliss is wrong. If no other source, including those that go into much more detail on bivectors, choses to highlight it, then maybe it's simply incorrect. As Baliss does not elaborate on it, e.g. say why it's useful or interesting, or prove anything from it, it's difficult to say either way. Whether he's wrong or describing something that never happens, it's from a footnote of a book that's not a primary source so we should not rely on it too much. -- JohnBlackburne wordsdeeds 16:36, 16 January 2010 (UTC)

John, you raise several points:
 * (i) frame inversion never occurs. Of course, this is simply a math transformation. Mathematicians exploit symmetry all the time. One is free to do it if one wants. It shows up a difference in behavior between axial vectors and bivectors, and so is a fact of these entities that separates them.


 * (ii)It's not useful. Whether this is useful depends upon the problem, of course. I'd say this is a point that need not be settled here in a discussion of behaviors, not their usefulness.


 * (iii) Baylis is 'wrong'. Clearly the mathematical transformation property is accurately stated and does exist. So Baylis may be making a point that need not be made, but he isn't wrong except (possibly) to suggest it may be useful. Brews ohare (talk) 17:08, 16 January 2010 (UTC)

In any event, the paragraph quoted makes incorrect statements that should be fixed. It says axial vectors and bivectors have the same properties. They don't. Doran makes this a central point of his work. Brews ohare (talk) 17:40, 16 January 2010 (UTC)


 * Just replying in one place: what incorrect statements ?-- JohnBlackburne wordsdeeds 18:01, 16 January 2010 (UTC)
 * This paragraph is incorrect. In particular, the behavior under inversion {eℓ}  → {−eℓ} of the axial vector and the bivector is not the same. Brews ohare (talk) 18:22, 16 January 2010 (UTC)

You have not commented upon the major purpose of introducing geometric algebra to replace axial vectors with bivectors. By minimizing the differences, the article seems to play down this activity. Brews ohare (talk) 18:16, 16 January 2010 (UTC)

Axial vectors are pseudovectors
This identification is common in tensor and vector algebra, but is not common in geometric algebra. The sentence:
 * "axial vector is dual of the bivector. Mathematically they have the same properties and are given the same name pseudovector"

is incorrect in the context of geometric algebra.

The bivector and the pseudovector are the same thing, but the bivector and the axial vector are different. As noted in the above section of this talk page, the axial vector has transformation properties different from the bivector.

JohnBlackburne: I have raised this issue above with references:
 * Pezzaglia says: "Hence, in 3D we associate the alternate terms of pseudovector for bivector, and pseudoscalar for the trivector." (page 131, end of first paragraph of §2.3) You have interpreted this statement as in agreement with the article, while I interpret as in disagreement. Our differences hinge on the various ways "associate" can be used. You see "associate" as a mapping, I see an "alternate term" as an identification of terminology.
 * Rivas " a spatial bivector or pseudovector"
 * Baylis "It is useful to be able to distinguish a bivector (an imaginary vector, the pseudovector) from its dual (a real vector, the axial vector)." You regard this statement as "insignificant coverage". I regard the difference as an incontrovertible fact, whatever its importance, and the terminology is consistent with the n-1 grade multivector.
 * Baylis  "i times a vector is a pseudovector (bivector) and represents a plane"

My observation is that a great many authors in geometric algebra never use the term pseudovector at all. They universally agree that in n dimensions the pseudovector is the (n-1) grade multivector, which of course, for n = 3 means the bivector is the pseudovector.

So, here is my proposal: I see no reason to insist that the pseudovector is something different in 3D. It is more straightforward to say that in tensor algebra and vector algebra the term pseudovector means axial vector, and geometric algebra it does not. That makes the article consistent for all dimensions, and in agreement with almost everybody. What do you say? Brews ohare (talk) 16:55, 16 January 2010 (UTC)


 * how is it incorrect ? It doesn't say they are the same thing, just that they have the same name. They are isomorphic via the Hodge dual, so have the same properties as vector spaces. They also have the same properties when rotated and inverted. But it's also a relatively minor point, as pseudovector is not a widely used term in geometric algebra. It's interesting as it's helps tie bivectors to (axial) vectors, but it's not worth overemphasising.-- JohnBlackburne wordsdeeds 17:15, 16 January 2010 (UTC)

John:


 * 1. Axial vectors are not the same as bivectors, regardless of the fact that they are Hodge duals. In particular, they have different transformation properties under inversion of basis vectors, and exist in dual spaces. These differences should not be glossed over in the article.
 * 2. In 3-D, bivectors are called pseudovectors in geometric algebra by Baylis, by Rivas and by Pezzaglia. Many authors in geometric algebra don't use the term in 3D, but its usage should not be muddied in the article.
 * 3. In n-D, grade (n-1) multivectors are called pseudovectors in geometric algebra. Consistent usage requires bivector = pseudovector in 3D.

Can you comment directly upon these points and their bearing upon the article? I believe they require revisions. Brews ohare (talk) 17:36, 16 January 2010 (UTC)

Proposed wording
Rather than get into a long semantic exchange, I've posted a revised version that differs from the previous by rather little. I hope it proves satisfactory. If not, perhaps comments could be made here. Brews ohare (talk) 19:02, 16 January 2010 (UTC)


 * Rather than by edit from the top of the section, the problems I can see looking at the changes as a whole are:


 * * Change of UK to US spelling, contrary to WP:RETAINJohnBlackburne wordsdeeds 00:59, 17 January 2010 (UTC)
 * As I originated this article, the shoe is on your foot. Brews ohare (talk) 06:07, 17 January 2010 (UTC)
 * * Loss of the reason why bivectors are associated with axial vectors, that they transform the same way JohnBlackburne wordsdeeds 00:59, 17 January 2010 (UTC)
 * Of course, the bivector and the axial vector do not transform the same way under the inversion operation spelled out. There is no loss of this info so far as I can see. There is hardly any rewording on this point other than the rewording of phrases that suggested the two were identical, not just duals. Brews ohare (talk) 06:07, 17 January 2010 (UTC)
 * * Otherwise the wording is worse: "are usually" is better English, and nothing is "replaced" JohnBlackburne wordsdeeds 00:59, 17 January 2010 (UTC)
 * Wording changes like this are minor; I won't argue about most of them. However, Doran & Lasenby in fact do suggest replacement of certain axial vectors by bivectors. Brews ohare (talk) 06:07, 17 January 2010 (UTC)
 * * vector cross product is just a redirect to cross product, the usual name, so should not have been changed JohnBlackburne wordsdeeds 00:59, 17 January 2010 (UTC)
 * Not a worrisome thing; change it back if you like. Brews ohare (talk) 02:04, 17 January 2010 (UTC)
 * * Confusing reordering. It suggests the reason why bivectors and axial vectors are related is due to the cross product, which is wrong.JohnBlackburne wordsdeeds 00:59, 17 January 2010 (UTC)
 * The cross product is stated to be an axial vector, which is hardly arguable. It is also clear that the bivector is the dual of the cross-product (the determinant relation you included shows that), and therefore related to it, and by implication to this axial vector. Perhaps you have a more general connection to axial vectors in mind other than the cross product. If so, please suggest what. If something else is on your mind, please clarify. Brews ohare (talk) 06:07, 17 January 2010 (UTC)
 * With the re-ordering it's difficult to compare the two versions, but there are other changes that seem needless: it's an advanced algebra topic, there's no need to say what a determinant is for example. JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 00:59, 17 January 2010 (UTC)
 * There is no reason not to indicate what the notation means and link to what a determinant is; it is not certain that only "advanced" readers will read the article, and even the advanced can use a little guidance. Brews ohare (talk) 06:07, 17 January 2010 (UTC)
 * But overall the changes have done a lot of damage to the section so the reason why bivectors are called pseudovectors is no longer clear.JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 00:59, 17 January 2010 (UTC)
 * What is that reason, John? It lies in the general n-dimensional definition related to n-1 grade multivectors, rather than to the exceptional case of 3D, yes? Brews ohare (talk) 06:07, 17 January 2010 (UTC)
 * I'm also unhappy with your other changes since then, in particular replacing ∧ with ʌ. Wikipedia is Unicode based, and MOS:MATH says "symbols that correspond to named entities are very likely to be displayed correctly", which includes ∧ (it's &and;). It's also used on other pages like exterior algebra. Far more obscure things are used on many pages without anyone feeling the need to change them, including things like ℝ and ℂ. Making them bold is also wrong: it's inconsistent with the TeX rendering, with other operators in HTML on this page, and other pages JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 00:59, 17 January 2010 (UTC)
 * John, on my PC, which is a very standard product, running Windows XP, a very standard system, and a Firefox and Explorer browsers, again, very standard items, the wedge symbol ∧ in main text shows up in the editor, but not on the published page, where it shows up as ∘ . Therefore the change in symbol is warranted as a service to readers like myself, which must be legion, even if Mac users would like to ignore us. If you don't like the wedge ʌ in bold face, that's easily fixed. Brews ohare (talk) 06:07, 17 January 2010 (UTC)
 * The other two changes do no harm but there's a better link for attitude and the other change is I think redundant. -- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 00:59, 17 January 2010 (UTC)
 * John: your link to attitude is to a disambiguation page, which makes little sense to the reader. The link to attitude (geometry) is pertinent. Brews ohare (talk) 06:07, 17 January 2010 (UTC)

John, please elaborate on your complaints and clarify your objections if I have missed their point. Brews ohare (talk) 06:07, 17 January 2010 (UTC)


 * WP:RETAIN says "If an article has evolved using predominantly one variety, the whole article should conform to that variety,", and that variety is British English. And MOS:MATH says "...what is most important is to consistency within an article, with deference to previous editors. It is considered inappropriate for an editor to go through articles doing mass changes from one style to another." XP is perfectly capable of displaying Unicode. See Help:Special_characters if you need help. Changing the article to use an incorrect symbol, and the formatting so it's inconsistent with the rest of the article, is wrong and I have to revert it.JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 11:16, 17 January 2010 (UTC)
 * In view of your intransigence, I have made an RfC below Brews ohare (talk) 16:36, 17 January 2010 (UTC)


 * On the general point the article as you've changed it does not say why the bivector and axial vector/pseudovector are related, which is the whole point of that section to draw a parallel between them. Though badly worded so it's difficult to be sure it now suggests the that it's because of the cross product and exterior product, which is wrong. So I have to revert that too. Sorry, I hoped you would either fix it yourself or explain better what you are trying to do. But now the section is wrong and I should fix it. -- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 11:16, 17 January 2010 (UTC)

Pythagorean trigonometric identity
The Pythagorean trigonometric identity is appealed to here in the section Bivector to establish the cosine rule for inner product, and applies to two vectors in three-space. But is this the appropriate link for the general case? Maybe the Cauchy–Schwarz inequality is more general? It can be used to point out that a·b ≤ |a||b| and that can be used to define the cosine of the angle as (see Use):


 * $$ \cos \theta = \frac{\boldsymbol{a \cdot b}}{|\boldsymbol{a}||\boldsymbol{b}|} \ ? $$ Brews ohare (talk) 05:42, 17 January 2010 (UTC)

The Pythagorean identity is fine: that section as a whole deliberately uses very elementary maths where possible, with links so the user can see this for themselves, and if they want work through the algebra and in the gaps. Bivectors are conceptually hard but once you are used to them the maths is very easy. -- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 11:00, 17 January 2010 (UTC)


 * John: The use of "elementary maths" when they are inappropriate is, well, inappropriate. The Pythagorean identity is applicable only in limited cases. If you wish to formally state its use in a limited context and then say "it can be shown...", that would be a correct usage. However, I see no need to avoid the Cauchy-Schwarz inequality, which is general. The proof of the inequality may be a challenge, but its use in this context is very straightforward. Brews ohare (talk) 15:35, 17 January 2010 (UTC)

RfC: Unicode wedge symbol
On my PC, which is a very standard product, running Windows XP, a very standard system, and a Firefox and Explorer browsers, again, very standard items, the wedge symbol ∧ in main text shows up in the editor, but not on the published page, where it shows up as ∘. Therefore the change in symbol from &and; to any of the readable alternatives ʌ, Ʌ, or $$\wedge$$ is warranted as a service to readers like myself, which must be legion, even if Mac users would like to ignore us. See alsoTurned v. Brews ohare (talk) 15:23, 17 January 2010 (UTC)
 * Use of Unicode symbol &and; for the wedge product ʌ, Ʌ, or $$\wedge$$ in inline text


 * Comments
 * From JohnBlackburne: XP is perfectly capable of displaying Unicode. See Help:Special_characters if you need help. JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 11:16, 17 January 2010 (UTC)
 * I believe it is ludicrous to require the general user of WP to be sophisticated enough to change their browser settings to read this symbol. Instructions at Help:Special_characters are likely to be unknown to the user (they were to me) and do not apply to all browsers, Firefox in particular. Brews ohare (talk) 15:23, 17 January 2010 (UTC)
 * Moreover, if all the reader sees as wedge products is a ∘ b, they will be unaware that the common notation is a Ʌ b, and so will not be motivated to search for a browser setting change. Brews ohare (talk) 16:13, 17 January 2010 (UTC)

The guidelines here say that symbols that correspond to named entities are very likely to display correctly, and this seems to make sense as 60 of the fonts on my computer have glyphs for ∧, compared to 5 for "LATIN CAPITAL LETTER TURNED V". Interestingly the Turned v, the middle of your three alternatives, appears as normal when viewing but as a box when editing. So for me that character is no good. So of the various non-Ascii characters used on maths pages there are those of far more concern as far as users being able to see them. Another is ℝ which has been used here and which Brews ohare has used at Active and passive transformation, but is also far rarer than ∧. But the guidelines on that are clear.

WP is a Unicode environment. If a standard HTML entity like ∧ cannot be used then pretty much everything non-Ascii will have to be avoided, and removed from articles, or replaced with images, so they are all in Ascii. Or perhaps, as written here, "This isn't the internet of 1995". -- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 21:09, 17 January 2010 (UTC)


 * Hi John: So your inability to see one of the alternatives rules that one out, while my inability to see ∧ simply means I have to suck it in? Why on earth wouldn't one use a universally viewable symbol instead of one that is viewable just to some readers? Because of a policy that says explicitly that $$\wedge$$ is a better alternative than ∧ if "a significant number of viewers will have problems seeing the character" ? Brews ohare (talk) 00:02, 18 January 2010 (UTC)


 * No, I simply wrote that the turned v doesn't work for me. If I were editing a WP article that used it (though I know of no purpose for that character) I would need to fix it by e.g. installing a font or changing my settings so the edit window used a different font. I.e. it's a user issue if they cannot see even an obscure Unicode char like a turned v. I especially don't see "significant numbers of viewers" having problems with a common HTML entity.


 * MOS:MATH actually 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", with a link to MOS:MATH which says "This should usually only be done for formulae displayed on their own line". A little above this it says "Having LaTeX-based formulae in-line which render as PNG under the default user settings, as above, is generally discouraged" followed by reasons. i.e. it's a bad idea. It's used for only one thing in the article currently, $$\mathcal{G}_n$$ which I could not find a Unicode equivalent for.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 00:24, 18 January 2010 (UTC)
 * John, you digress. The question is: Why not use a universally viewable symbol instead of one that is viewable just to some readers? WP policy is not opposed to such use. Brews ohare (talk) 00:54, 18 January 2010 (UTC)

If by "universally viewable" you mean an image see my previous reply. If an alternative more obscure Unicode char like turned v see that and my reply before that.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 13:32, 18 January 2010 (UTC)
 * JohnBlackburne: If I understand you correctly, your position is that no symbol other than the Unicode wedge itself, &and;, is remotely acceptable, even though there are other versions like $$\wedge$$ and ʌ, Ʌ, that look equivalent and have wider accessibility, in particular on PC's such as mine, that constitute a very large fraction of the population of equipment used by readers and do not display &and;. Yes?


 * I find this position peculiar when WP aims to be readable to as large a population as possible. Suggesting that the large and unversed population monkey with their browsers to try to register &and; is not a solution.


 * You claim further that WP policy guidelines make any other choice unacceptable to WP policy policy. Yes?


 * I find this interpretation of policy to be invalid, and to ignore provisions that state a preference for the use of more readily displayed options, that include explicitly $$\wedge$$ and indeed, any choice that is widely viewable. There is no requirement that an image be used where other options are available. Brews ohare (talk) 15:30, 18 January 2010 (UTC)


 * You say turned v has "wider accessibility" ? By what measure ?-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 16:28, 18 January 2010 (UTC)

Hi John: If you read the RfC you will see the configuration of my computer setup. This configuration is a typical PC set-up running Microsoft applications. I believe it is readily accepted that such systems are very common. On my system I can read any of the symbols for wedge product except &and;. I'd assume that any other user with this plain vanilla set-up can do likewise. That is to say, ʌ, Ʌ, or $$\wedge$$ are immediately visible on such set-ups while ∧ shows up as ∘, except in the browser editor. That experience will be that of all readers with similar set-ups, and as these set-ups are common, will be common to many readers, yes? Brews ohare (talk) 18:10, 18 January 2010 (UTC)

The obvious advantage to many readers (and disadvantage to none) of replacing &and; with $$\land$$ or $$\wedge$$, thereby allowing proper display of wedge products to almost all, if not all, WP readers has led me to make this replacement throughout the article. Brews ohare (talk) 18:43, 18 January 2010 (UTC)


 * I've just reverted it. See e.g. MOS:MATH — "Either form is acceptable, but do not change one form to the other in other people's writing — as well as a number of other places.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 18:47, 18 January 2010 (UTC)


 * John, you have no support for your actions; for example,MOS:MATH says: "to maximize the size of the audience who can read an article, it is better to be conservative in using symbols.". Frankly, I can see absolutely no purpose in your stance that clearly inconveniences WP readership. Brews ohare (talk) 18:53, 18 January 2010 (UTC)

There's this, from MOS:MATH: and this, from MOS:MATH: as you seem to want more reasons, and have obviously not read them yourself MOS:MATH gives the following: (apologies to other editors for repeating myself and the excessive copy and paste)
 * but do not change one form to the other in other people's writing
 * Having LaTeX-based formulae in-line which render as PNG under the default user settings, as above, is generally discouraged
 * The font size is larger than that of the surrounding text on some browsers, making text containing in-line formulae hard to read.
 * Misalignment can result. For example, instead of ex, with "e" at the same level as the surrounding text and the x in superscript, one may see the e lowered to put the vertical center of the whole "ex" at the same level as the center of the surrounding text.
 * The download speed of a page is negatively affected if it contains many images.
 * HTML (as described below) is adequate for most simple in-line formulae and better for text-only browsers.

The only reason you've given for changing it is your PC has problems. To extrapolate from that to all other PC users makes no sense at all. If any number of readers and editors on WP had problems with ∧, a standard HTML entity, it would have been well documented by now. It is simply a problem with your setup, which you can easily fix.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 19:29, 18 January 2010 (UTC)


 * I made some browser changes that aren't great but enable the wedge to be seen. These changes are here. I suspect that better advice could be provided, but it is an improvement over no advice. Brews ohare (talk) 20:24, 18 January 2010 (UTC)

Signs in Hodge dual equation
The signs are chosen to agree with the source. See footnote 5 in Baylis. It is simply a matter of convenience for the reader if the equations agree with the source. The footnote has been edited to note the possibility of different signs. Brews ohare (talk) 15:26, 19 January 2010 (UTC)


 * JohnBlackburne: You reversed the signs in the dual equations despite the above note, and without comment. Please read Baylis and if you don't wish to use his sign conventions, replace the references to Baylis with something else. At the moment we have inconsistency. Brews ohare (talk) 15:42, 19 January 2010 (UTC)


 * Bayliss is writing about something different, the Clifford Dual, the same as the Clifford dual given by Lounesto - page 39, the same as in the reference at the end of the section. The Hodge dual though is as given. The whole point of it being a dual is it's symmetric, i.e. do it twice and you get back where you where. This is not generally true for the Clifford dual.


 * I was not aware that Baliss does the Clifford dual in a different way to Lounesto - but if that is the case rather than introduce it and deal with this discrepancy just using the Hodge dual is much clearer. -- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 15:45, 19 January 2010 (UTC)


 * As you suggested I've removed the Baliss ref, as all the theory is given in Lonesto ref.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 15:48, 19 January 2010 (UTC)

Thanks, John. You know, despite the fact we argue like cats and dogs, I think this article is the better for it. Brews ohare (talk) 16:18, 19 January 2010 (UTC)


 * I've just had to fix this again. The version I've just put back is correct - see the source page 39 - and has all the relations, i.e. the Hodge dual and how it can be got from the product with the pseudoscalar. It does not need changing, certainly not when it seems introduce errors.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 17:38, 19 January 2010 (UTC)

John: We now have:


 * $$\mathbf{A} = \star \mathbf{a} = \mathbf{a}i\ ;\ \ \  \mathbf{a} = \star \mathbf{A} = - \mathbf{A} i $$

However, it would seem:


 * $$ \star \mathbf{A} = \star (\mathbf{a} i ) = (\star \mathbf{a}) i =\mathbf{a} i^2 = -\mathbf{a}  $$

and not


 * $$\mathbf{a} = \star \mathbf{A} \ . $$

Is the associative rule inoperable? Brews ohare (talk) 17:49, 19 January 2010 (UTC)

Baylis suggests there is an ambiguity as $$e_{T}^2 = \pm 1 \. $$ I don't understand what that is about. Dimensions of the space, maybe? That is, $$e_{123}^2 = - 1 \ ? $$ The dual may or may not be associative?? Should something be said about this?? Brews ohare (talk) 18:12, 19 January 2010 (UTC)


 * I see (I think) where you are misunderstanding this. The dual is not an associative operator: it's derivation is rather more complex, see e.g. Lounesto p 38. So it's wrong to try and calculate with it, like you seem to be trying to do. The version that I wrote, and have put back twice, states it and the relations to the products with i completely.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 18:13, 19 January 2010 (UTC)


 * Is the non associativity of the dual peculiar to odd-dimensional spaces, maybe? I think some explicit statement about dual of the dual should be included along with the math so my incorrect assumption will be cautioned against. Maybe it isn't associativity at fault, but is as simple as $$e_{12}^2 = - 1 $$ while $$e_{123}^2 =  1 \  ? $$ Brews ohare (talk) 18:21, 19 January 2010 (UTC)

It's just not associative, like a lot of things. E.g. cos is not associative - you don't have cos(2θ) = cos(2) * θ. But what's there clearly shows how they are related via both the Hodge dual and product with the pseudoscalar. In 2 and 3 but not 4 dimensions the pseudoscalar squares to -1; check it yourself. It's tied into the reversion operator. but is not important in itself: in places where you use the pseudoscalar you just introduce a -1 where its needed, as is there in the article.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 19:15, 19 January 2010 (UTC)
 * To repeat myself, I think some explicit statement about dual of the dual should be included along with the math so my incorrect assumption will be cautioned against. I know you don't need that, but some readers will find that helpful. Brews ohare (talk) 19:55, 19 January 2010 (UTC)


 * This article is not about the mechanics of the dual, and WP in particular is not a maths text book, i.e. a how to do problems. See WP:NOTHOW.


 * I you want to learn how to do GA there's a complete course here. It's the one I learned from by downloading, after which I scoured the web for many more papers and bought two books. It's much better than what you're trying to do, learn from snippets, as you never get the whole picture and so the background. The course and most books have exercises you can do to which are especially helpful. -- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 20:08, 19 January 2010 (UTC)

Thanks for the link. I understand WP:NOTHOW. However, a line or so to avoid a likely pitfall is within purview, I think. Or do you feel that a very long explanation is all that will suffice? Brews ohare (talk) 20:36, 19 January 2010 (UTC) Maybe the Hodge dual page would lighten the burden? Brews ohare (talk) 20:44, 19 January 2010 (UTC) Perhaps the following:
 * Note that:
 * $$ \star \mathbf {A} = \star (\mathbf {a} i ) \ne (\star \mathbf {a}) i \ ,$$
 * that is, the dual operation for a vector is simply a multiplication by the unit pseudoscalar, but the dual operation for a bivector involves an additional sign reversal. Brews ohare (talk) 21:13, 19 January 2010 (UTC)

Have a look at it now: i've changed it, largely based on the source where it's laid out clearly in one place, splitting one line into two and adding a note as to the differences between them from Lounesto's footnote on page 39, tweaked the descriptions and made the relations similar for the cross and exterior products also to match Lounesto.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 21:50, 19 January 2010 (UTC)

I'll look at that in an hour or so. In the meantime, just as a note, here's some details:

This last relation can be understood by expressing the bivector A in terms of its components:


 * $$ \mathbf {A} = A_1 \mathbf {e_2e_3} + A_2 \mathbf {e_3e_1} + A_3 \mathbf {e_1e_2} \ . $$

One finds:
 * $$i \mathbf A = \mathbf{e_1e_2e_3 A} = A_1 \mathbf{e_1} (\mathbf {e_2e_3})^2 + A_2 \mathbf{e_2} (\mathbf{e_3e_1})^2 + A_3 \mathbf{e_3}(\mathbf{e_1e_2})^2 \ $$
 * $$= - \left( A_1 \mathbf{e_1} + A_2 \mathbf{e_2} + A_3 \mathbf{e_3} \right) = - \star \mathbf A \, $$

which is the relation between A and its dual ⋆A. Brews ohare (talk) 22:11, 19 January 2010 (UTC)


 * There's no need to work it out in the article - large parts of this article miss out derivation, and this section already has far more than most, more than I would have added. If people want to learn how to do geometric algebra there are many other places, including the main article.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 22:24, 19 January 2010 (UTC)

John: I built a bit upon your new copy, which is nice. I'd like to keep these few lines in here because they answer questions that bothered me, and I expect I am not alone. These few lines make clear what is involved and add to the usefulness of the article by clearing up some natural questions about the signs. Hope you can indulge me on this one. Brews ohare (talk) 23:22, 19 January 2010 (UTC)

John, you may be a bit contemptuous on this, but the extra lines also make it plainer what the dual is by exhibiting explicitly the use of the unit bivector basis and comparing with the unit vector basis. Brews ohare (talk) 23:30, 19 January 2010 (UTC)


 * No, see WP:NOTHOW. That section already had too much working things out for me - giving the dual for the exterior and cross product just duplicates the duals earlier, and those have been expanded. The products are trivial, but if a reader has problems doing basic GA there are better places for them than the middle of this article. It was clear and explicit before, it now looks like badly laid out working out. -- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 23:32, 19 January 2010 (UTC)

John: I put the detail in a footnote where it will not distract from the main text. Brews ohare (talk) 02:24, 20 January 2010 (UTC) I don't think of this as material for an WP:NOTHOW instruction manual or a textbook. It is more in the line of WP:NOTHOW A Wikipedia article should not be presented on the assumption that the reader is well versed in the topic's field. Brews ohare (talk) 02:34, 20 January 2010 (UTC) Taking myself as an example of an interested reader without a lot of sophistication, I find these details useful in understanding not just where the results come from but as an aid to understanding the vocabulary. Brews ohare (talk) 02:36, 20 January 2010 (UTC)

I moved this material to Hodge dual. As a result, the Bivector: Axial vector section could be considerably shortened, if you wish. Brews ohare (talk) 18:28, 20 January 2010 (UTC)

Axial vectors section
I just undid a number of changes to this section - there were collectively far too many change for me to go through them one by one and too many problems to fix individually. Mostly it was changing words and phrasing for the worse, sometimes so it was grammatically incorrect, sometimes so it was just a poor choice of words (it's not "easy to use" - you were making mistakes with this maths a day ago Brews so were not finding it easy yourself), with things pointlessly repeated.

If you have some good (new and correct) content to add on the topic then please add it. If you spot mistakes I've made (and you have) then fix them. But don't just re-word or reformat things, or repeat things already clearly expressed. Maybe not every edit has problems but the cumulative effect is to make the article much less clear and much more difficult to follow. -- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 21:00, 20 January 2010 (UTC)


 * Reply to JohnBlackburne: Suit yourself: revert without reading or looking at the purpose of the changes. I find your English unintelligible in places, and tried to fix it. For example "This is easier to use as the product is just the associative and generally easier to work with geometric product" - what does that mean? Now you say I came up with this "easier to use" concept, which was yours from the beginning! "Alternately" should be "Alternatively": a traffic light changes alternately between red and green; given a green light, one may alternatively cross or not. And so on: you even reverted the replacement of a hyphen with a subtraction symbol in −1. Tsk, tsk. Your logic doesn't flow: it skips around; I tried to fix that too. Good luck. Brews ohare (talk) 00:44, 21 January 2010 (UTC)


 * I know you do fix things, and I leave them in if I spot them. But what I saw was a paragraph in much worse state than before, with the problems far more of an issue than minor fixes, and the way you had done it made it impossible to untangle. The edit summary issue is something I raised on your talk page which has improved a little but still if you compare yours to e.g. mine there's still a big problem. The too many edit issue is something that you need to fix by taking more time and using 'preview' not 'save' between changes. If I or another editor sees a long list of changes done at the same to the same section by the same editor without good summaries it's natural to treat them as a block, i.e. a single edit which if it introduces too many problems is often easier to revert, especially if the fixes are much smaller (and so difficult to spot) than the problems introduced.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 00:55, 21 January 2010 (UTC)

A plane has two normals
The present text contains the statement "The plane has two normals, one on each side". Usage of the word normal is ambiguous.

One common meaning takes the normal to a plane as a line with attitude but no direction attached to it (see David Hilbert); i.e. normal refers to a "normal line", not a vector. If the vector is to be designated, the term normal vector is used unambiguously. The term "unit normal" also refers to a vector.

I have made "normal" an adjective to avoid ambiguity. Brews ohare (talk) 16:06, 23 January 2010 (UTC)


 * "normal" is a surface normal. Given in the previous sentence, with link in case anyone does not know what it means (as you seem not to). So it's a noun, not a verb, no ambiguity. The rest of the text you added did not make sense. A bivector does not "point" - maybe a vector does. And I'm not sure what "pointing oppositely along" means. The "one on each side" is clear: readers will hopefully know a plane has two sides. See e.g. the 1st picture at surface normal, which readers will see if they follow the link for further explanation. -- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 16:25, 23 January 2010 (UTC)

OK John:

1. The word "normal" is commonly an adjective, as in "normal vector", "normal line". The links normal line to 1900 texts & normal vector to 2200 texts are abundant illustration of this usage, yes?

2. I agree that surface normal makes the point, but using the words normal vector in place of making "normal" an ambiguous noun is not an outrageous gift to the reader, yes? The word "normal" as a noun is used in many ways, and to me its most common usage is to refer to a "normal line". Hilbert does the same.

3. What is the point of raising the issue of their being two normal vectors here without any connection to the subject of bivector?? Brews ohare (talk) 16:44, 23 January 2010 (UTC)


 * 1&2. Here it's used as a noun, the common usage as reflected in the article name, surface normal. That name is given in full in the previous sentence, with a link. If readers are confused they can follow the link, the usual way it works in WP.


 * 3. the connection is "giving the two possible orientations for the plane", and orientations are discussed in detail two paragraphs before.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 16:51, 23 January 2010 (UTC)

I am a bit amazed that you would take the view that forcing the reader to use a link to consult another article is preferable to adding one word to form a very common and unambiguous phrase "normal vector".

I understand, of course, that there is a connection between different normal vectors and a property of bivectors, but the paragraph doesn't connect the dots. I agree my wording was incorrect. The notion of sense or circulation has to be connected to the two vector directions, I guess, as in Hestenes' discussion. Brews ohare (talk) 16:59, 23 January 2010 (UTC)


 * It is not just I but the writers of surface normal. I.e. it's the usage here, so is preferred, unless there's some compelling reason to use a different name. "Surface normal" is also common and unambiguous, I don't see why you have a problem with it.-- JohnBlackburne words<sub style="margin-left:-5.0ex;">deeds 17:20, 23 January 2010 (UTC)

It's funny to see you adopting the usage of one article in WP (of all places) as setting the appropriate terminology. It is very evident that the word "normal" is used all over the place as an adjective, and using it in "normal vector", as I have pointed out is very common, as is "normal line". Making "normal" an adjective in place of using the adjective as a noun makes the wording unambiguous at no cost, and in keeping with common usage. You are simply exerting your own whims here. Brews ohare (talk) 01:07, 24 January 2010 (UTC)

Axial vectors are pseudovectors, a tautology
John: This one-line edit summary of yours doesn't make sense. As you know, in n-dimensions the pseudovector is is the n-1 blade. For n = 3, this makes the bivector the pseudovector, which is a meaning of "pseudovector" incompatible with "axial vector" because the bivector is the dual of the axial vector. This usage of pseudovector is supported by Baylis, among others. This division in usages of "pseudovector" is a point that should be made clear in the article. Brews ohare (talk) 20:23, 30 January 2010 (UTC)


 * The bivector in 3D is sometimes called the pseudovector, so it mentions that. It's not an important point though: many sources do not use the term pseudovector, and even those that do often refer to it only in passing. I.e. it's just an alternate name for the bivector, only in 3D and not often even there.


 * "axial vectors, commonly called pseudovectors in three dimensions," was the tautology, but the article's not about axial vectors/pseudovectors - a link to that article's all that's needed. There's a whole section on the relationship with the axial vector, so it links to that too. The more general interpretation of a pseudovector as an n - 1 blade is even less important, at least here where we're only concerned with bivectors not the general properties of geometric algebra. -- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 20:47, 30 January 2010 (UTC)