Talk:Seven-dimensional cross product/Archive 4

Introduction of a paragraph
The paragraph: "Notice that any one of the seven unit vectors can result from the cross product of three distinct pairs (representing three distinct planes) from among the other six. For example, from the first row of the listing above one finds e1 is given by e2 × e4, e3 × e7, and e5  × e6. Thus, unlike the 3-dimensional cross product, the same 7-dimensional cross product can result from multiplying pairs of vectors residing in different planes." is beyond controversy, a simple observation based upon the multiplication table, and should be reinserted in the discussion of the table. Brews ohare (talk) 22:24, 8 July 2010 (UTC)

So far the following objections have been raised:
 * The cross product is not unique. That is, the multiplication table is misleading in indicating that there is only one possible cross product that applies to a given plane. Every multiplication table will lead to a different conclusion. —The response here is that every possible multiplication table has this property: they all lead to a unique cross product for any two vectors. And that in any case, the paragraph is describing only this particular table, nothing more general.
 * There are actually infinitely many cross products associated with a plane. —This statement confuses "infinitely many" normals to the plane with "infinitely many" directions for cross products. In fact, there is only one direction of a cross product (within a sign) for each plane. That is clear from the given multiplication tables, and holds for every multiplication table. By definition, a plane consists of all vectors that can be made up as a linear combination of two independent vectors. Whatever two vectors one selects, they have a unique cross product, and hence the plane has a unique direction associated with this cross product.
 * One cannot make general statements based upon a particular multiplication table. —The only general statement made is that this example shows cross products from various planes point in the same direction, which is an existence proof that this phenomenon does occur.

If there are other objections not listed or if the objections have been inadequately refuted, please comment. Otherwise, the paragraph should be reinstated. Brews ohare (talk) 22:37, 8 July 2010 (UTC)


 * No, it confuses multiple things, is confusingly worded and is OR as you are using flawed arguments and not sources to justify it. This has already been explained to you, please review these previous replies as they go into it in more than enough detail.-- JohnBlackburne wordsdeeds 22:43, 8 July 2010 (UTC)


 * There is no OR here, as has been pointed out earlier. Here is that response:
 * John: There is no OR in looking at the first line of your listed multiplication rule and observing that three different cross products lead to e1. Calling this "parts of the whole" seems to suggest that one can talk only of general x × y and not about particular vectors like the unit vectors, which of course means a multiplication table and the distributive law are not useful ideas. And finally, suggesting that a concrete instance cannot be taken as proof of the possibility of existence of such an instance is simply a logical error. Brews ohare (talk) 21:54, 8 July 2010 (UTC)
 * The multiplication tables are sourced, and that is all that is used here, so no further source is required.
 * No flawed arguments are employed, although the above listed claims have been advanced, they are soundly refuted. If you continue to disagree, present valid arguments. Brews ohare (talk) 23:07, 8 July 2010 (UTC)

Distributive law and Lagrange identity
David: Can you elaborate upon the connection between the distributive law and the Lagrange identity?

I take the distributive law to be:
 * $$\mathbf{a \times}\left(\mathbf{c + d}\right) = \mathbf{a \times c} + \mathbf{a \times d} \ . $$

I'm guessing you mean by the Lagrange identity
 * $$ \sum_{1 \le i < j \le n} \left(a_ib_j-a_jb_i \right)^2 =| \mathbf a |^2 \ | \mathbf b |^2 - (\mathbf {a \cdot b } )^2 \, $$

which, in 7 dimensions, results in:
 * $$|\mathbf{a} \times \mathbf{b}|^2 = \sum_{1 \le i < j \le 7} \left(a_ib_j-a_jb_i \right)^2 $$

Plugging in b = c + d:


 * $$|\mathbf{a \times}\left( \mathbf{c+d}\right)|^2 = \sum_{1 \le i < j \le 7} \left( a_i(c_j+d_j)-a_j(c_i+d_i) \right)^2 \ ,$$

and I guess formal manipulation of the sum results in
 * $$|\mathbf{a \times}\left( \mathbf{c+d}\right)|^2 =|\left(\mathbf{a \times c}\right) +\left( \mathbf{a \times d}\right)|^2 \ ,$$

to demonstrate the distributive law holds. Is that the idea? Brews ohare (talk) 19:56, 8 July 2010 (UTC)


 * Brews, Yes, that's the basic idea. The Lagrange identity is the proof of the distributive law in the context, which is why it is a desired defining property. Let's look at it another way. Let's cast the Lagrange identity aside and define a cross product which doesn't need it. Let's make up a bilinear 5D multiplication table and assume the distributive law. What's stopping us from doing that? Here we have a perfectly good bilinear 5D cross product which seems to be perfectly functional.


 * The answer is that we cannot assume the distributive law. The very reason why the Lagrange identity is chosen as a defining property is because it brings the distributive law with it, and that limits us to 3D and 7D.


 * In a 3D cross product course, we prove the distributive law using geometry. The distributive law must be established as legitimate before we do any expansions of vectors in their unit vectors. Likewise with the 7D cross product, we need to prove that the distributive law is valid before we do any unit vector expansions. We cannot assume the distributive law just from bilinearality. That's why I said something along the lines of 'As a result of the defining properties, vectors in 3 and 7 dimensions can be multiplied out distributively in terms of their unit vectors - - - -'.


 * But as regards your specific manipulations, I was thinking more in terms of what I wrote higher up, in that if we distribute everything out within the Lagrange identity, it balances only in 3 and 7 dimensions.


 * $$\sum_{1 \le i < j \le n} \left(x_iy_j-x_jy_i \right)^2 = \| \mathbf x \|^2 \ \| \mathbf y \|^2 - (\mathbf {x \cdot y } )^2\, $$


 * In 3D the left hand side becomes,


 * (x2y3-x3y2)2 + (x3y1-x1y3)2 + (x1y2-x2yundefined)2 which happens to equate to z12 + z22 + z32 in the cross product.


 * In 7D the left hand side becomes,
 * (x2y4-x4y2)2 + (x3y7-x7y3)2 + (x6y5-x5y6)2 + (x1y4-x4y1)2 +  (x3y5-x5y3)2 + (x6y7-x7y6)2 + (x1y7-x7y1)2 +  (x2y5-x5y2)2 + (x4y6-x6y4)2 + (x1y2-x2y1)2 +  (x3y6-x6y3)2 + (x5y7-x7y5)2 + (x1y6-x6y1)2 + (x2y3-x3y2)2 + (x4y7-x7y4)2 + (x1y5-x5y1)2 +  (x3y4-x4y3)2 + (x2y7-x7y2)2 + (x1y3-x3y1)2 +  (x2y6-x6y2)2 + (x4y5-x5y4)2


 * which happens to be equal to z12 + z22 + z32 + z42 + z52 + z62 + z72 in the cross product. It's just a question of taking,


 * z1 = x2y4-x4y2+x5y6-x6y5+x3y7+x7y3


 * z2 = x3y5-x5y3+x6y7-x7y6+x4y1-x1y4


 * z3 = x4y6-x6y4+x7y1-x1y7+x5y2-x2y5


 * z4 = x5y7-x7y5+x1y2-x2y1+x6y3-x3y6


 * z5 = x6y1-x1y6+x2y3-x3y2+x7y4-x4y7


 * z6 = x7y2-x2y7+x3y4-x4y3+x1y5-x5y1


 * z7 = x1y3-x3y1+x4y5-x5y4+x2y6-x6y2


 * and squaring each of the z terms. You will get 252 terms in total. 168 of these will mutually cancel. In fact, the 168 terms will be 2x84 terms, with each group of 84 containing two groups of 42 mutually cancelling terms. The remaining uncancelled 84 terms can then be reduced to the 21 squared terms. David Tombe (talk) 22:00, 8 July 2010 (UTC)


 * Actually you can just "assume" the distributive law from bilinearity as it follows immediately: because it's a linear product and because it's over vectors which you can add. It's bilinear and there are vectors on both sides so it's left and right distributive (the two are not the same property for non-commutative products). As all the properties depend on this it makes no sense to try and prove the distributive law from the properties: you would at best have a circular argument, proving what you started with or proving something you rely on. And as noted in reply to Brews earlier, unless it's properly sourced you can't insert your own reasoning like the above into the article as that would be OR.-- JohnBlackburne wordsdeeds 22:09, 8 July 2010 (UTC)

John: You beg the question. The cross product is defined using two requirements, and it cannot be assumed that an arbitrary definition involving two arguments is bilinear. Clearly some definitions involving two arguments could be quadratic or whatever. You have to establish consistency of the definitions with the requirement of linearity. Brews ohare (talk) 22:34, 8 July 2010 (UTC)

It's interesting that Lounesto and Massey do not specify that the cross product must be linear in their definitions. Brown and Gray do make that a separate stipulation. Nobody I've found attempts to show consistency between linearity and the two fundamental properties of any cross product. So it seems this is an unreported point, although all authors I found assume its truth. Brews ohare (talk) 15:57, 9 July 2010 (UTC)

RfC: Is the following paragraph confusing or misleading?
Is the following paragraph confusing or misleading?

Based upon a sourced multiplication table displayed on the Talk page below this request, the following observations are made:
 * “Notice that any one of the seven unit vectors can result from the cross product of three distinct pairs (representing three distinct planes) from among the other six. For example, from the first row of the listing above one finds e1 is given by e2 × e4, e3 × e7, and e5 × e6. Thus, unlike the 3-dimensional cross product, the same 7-dimensional cross product can result from multiplying pairs of vectors residing in different planes.”

It would seem useful to include these observations in the article as a description of some of the consequences of this particular multiplication table. Please comment. Brews ohare (talk) 23:16, 8 July 2010 (UTC)

Below is the table in question:
 * Remark


 * $$\mathbf{e}_2 \times \mathbf{e}_4 = \mathbf{e}_1, \quad \mathbf{e}_3 \times \mathbf{e}_7 = \mathbf{e}_1, \quad \mathbf{e}_5 \times \mathbf{e}_6 = \mathbf{e}_1,$$


 * $$\mathbf{e}_3 \times \mathbf{e}_5 = \mathbf{e}_2, \quad \mathbf{e}_4 \times \mathbf{e}_1 = \mathbf{e}_2, \quad\mathbf{e}_6 \times \mathbf{e}_7 = \mathbf{e}_2,$$


 * $$\mathbf{e}_4 \times \mathbf{e}_6 = \mathbf{e}_3, \quad \mathbf{e}_5 \times \mathbf{e}_2 = \mathbf{e}_3,\quad \mathbf{e}_7 \times \mathbf{e}_1 = \mathbf{e}_3, $$


 * $$\mathbf{e}_5 \times \mathbf{e}_7 = \mathbf{e}_4, \quad \mathbf{e}_6 \times \mathbf{e}_3 = \mathbf{e}_4, \quad \mathbf{e}_1 \times \mathbf{e}_2 = \mathbf{e}_4,$$


 * $$\mathbf{e}_6 \times \mathbf{e}_1 = \mathbf{e}_5, \quad \mathbf{e}_7 \times \mathbf{e}_4 = \mathbf{e}_5,\quad\mathbf{e}_2 \times \mathbf{e}_3 = \mathbf{e}_5, $$


 * $$\mathbf{e}_7 \times \mathbf{e}_2 = \mathbf{e}_6,\quad \mathbf{e}_1 \times \mathbf{e}_5 = \mathbf{e}_6, \quad \mathbf{e}_3 \times \mathbf{e}_4 = \mathbf{e}_6,$$


 * $$\mathbf{e}_1 \times \mathbf{e}_3 = \mathbf{e}_7,\quad \mathbf{e}_2 \times \mathbf{e}_6 = \mathbf{e}_7, \quad \mathbf{e}_4 \times \mathbf{e}_5 = \mathbf{e}_7.$$

It is taken directly from Lounesto. Brews ohare (talk) 23:16, 8 July 2010 (UTC)


 * I don't think it should be reinstated. The last sentence confuses multiple things as John said. The rest is a trivial observation which obscures the fact that pairs of vectors from many planes (not just three) map to e_1, or any particular vector, under any 7D cross product. Holmansf (talk) 23:20, 8 July 2010 (UTC)

Holmansf: The last statement says “the same 7-dimensional cross product can result from multiplying pairs of vectors residing in different planes”. Now the first line of the multiplication table shows an example of exactly this:
 * $$   \mathbf{e}_2 \times \mathbf{e}_4 = \mathbf{e}_1, \quad \mathbf{e}_3 \times \mathbf{e}_7 = \mathbf{e}_1, \quad \mathbf{e}_5 \times \mathbf{e}_6 = \mathbf{e}_1 $$

So please, say what is being confused? Your second claim that many planes (not just three) map to e1, not just those listed in the first line of the table, is manifestly contradicted by the table. All vectors v = x2 e2 + x4 e4 in the plane e2 e4 have cross products in the direction e1. Likewise for the two other planes in the top row of the table. No planes other than these three have cross products in this direction, as the listing of all the other cross products shows clearly.Brews ohare (talk) 23:55, 8 July 2010 (UTC)


 * It is the various meanings of "cross product" that are being confused. From the definition in the article, a 7D cross product is a mapping that satisfies certain properties. In this sentence "7-dimensional cross product" is apparently being used to refer to a vector.


 * On the other point- Using the cross product defined by Lounesto, (1/2)*(e_2 + e_3) x (e_4 + e_7) = (1/2)*( (e_2 x e_4) + (e_2 x e_7) + (e_3 x e_4) + (e_3 x e_7) ) = 1/2 * (e_1 - e_6 + e_6 + e_1) = e_1. Apparently the plane spanned by e_2 + e_3 and e_4 + e_7 is not the same as any of the three you already know give e_1. So more than three planes map to e_1. Holmansf (talk) 00:25, 9 July 2010 (UTC)
 * Looks that way. However, the paragraph is still 100% accurate. It doesn't state (as I mistakenly did above) that the matter is limited to the three planes mentioned. Brews ohare (talk) 03:41, 9 July 2010 (UTC)
 * BTW, the seven-dimensional cross product is normally understood to be a vector. See here and here. Brews ohare (talk) 04:09, 9 July 2010 (UTC)
 * I added verbiage to this effect to the article quoting Massey, a definitive source already cited there. Brews ohare (talk) 15:10, 9 July 2010 (UTC)


 * :See already here and here for example, and also user:Holmansf's reply here. And yes, it is OR to draw conclusions from your own observations - from WP:OR (my emphasis):
 * "Wikipedia does not publish original research. The term "original research" refers to material—such as facts, allegations, ideas, and stories—not already published by reliable sources. It also refers to any analysis or synthesis by Wikipedians of published material, where the analysis or synthesis advances a position not advanced by the sources."


 * So please name the source that presents the same arguments you are making. Otherwise it is OR and has no place here (even if it made sense, which it doesn't).-- JohnBlackburne wordsdeeds 23:22, 8 July 2010 (UTC)

JohnBlackburne: If you wish to make accusations of OR, it is necessary to say exactly what constitutes the OR you are objecting to. My understanding is that you are objecting to the observation of the first row of the multiplication table, which is sourced to Lounesto. Or is it OR in your opinion to say two orthonormal vectors such as e2 e4 define a plane? What exactly is the OR? Brews ohare (talk) 23:55, 8 July 2010 (UTC)

"“...for the cross product a × b in R7 there are also other planes than the linear span of a and b giving the same direction as a × b”" Brews ohare (talk) 00:25, 9 July 2010 (UTC)
 * Perhaps you both would enjoy this quote from Lounesto (page 97)


 * I would support putting this quote, or something almost exactly the same as it somewhere in the article. 108.1.37.152 (talk) 02:31, 9 July 2010 (UTC) Whoops, that was me. Holmansf (talk) 02:32, 9 July 2010 (UTC)


 * Changing the order of the list in order to make a point that is not explicitly in the source is elementary WP:OR. DVdm (talk) 09:26, 9 July 2010 (UTC)


 * Having had some time to think this matter over, I can see now that there has been an initial problem of cross purposes, and that as a consequence it has now degenerated into a turf war. The issue of non-uniqueness which John and Holmansf want to highlight relates to the issue of convention as regards setting up a multiplication table. There are many ways of setting up a multiplication table. The issue of uniqueness/non-uniqueness which Brews and I wish to highlight is an issue that arises within the context of a given multiplication table. Both of these points can be covered within the same section. But it seems now that since the latter was not the issue which John and Holmansf originally had in mind that it now becomes necessary to obliterate all mention of it and to unsort the example multiplication table so as to hide this fact. David Tombe (talk) 09:35, 9 July 2010 (UTC)

Product rule from Lounesto
I've just restored the product rule from Lounesto to as I first entered it. Not only is this the version from the source but it flows logically to the next point, the observation of the pattern, also from the source. Rearranging it breaks this reasoning and is unnecessary as the product is expressed term by term, with e.g. the e1 term on the first line and so on, immediately below and in the matrix straight after that. There is no need to have all of them arranged the same way, as to do so loses valuable information contained in the arrangement in the source.-- JohnBlackburne wordsdeeds 23:36, 8 July 2010 (UTC)


 * I hope your rearrangement of the table will not confuse readers who look at the RfC, which uses your original tabulation. Brews ohare (talk) 00:09, 9 July 2010 (UTC)

John, I've restored the table to the version which highlights the point which you have decided that you don't want to be mentioned in the article. You cannot unsort a table that somebody else has taken the care to sort, and then try to argue that somehow there is more merit to the unsorted version. Just because Lounesto didn't sort it doesn't mean that it forever has to remain unsorted. As you yourself have stated in the section below, you do not change the format of something from one legitimate form to another without having some valis reason for doing so. David Tombe (talk) 09:27, 9 July 2010 (UTC)


 * I have undone this rather blatant piece of WP:OR. DVdm (talk) 09:28, 9 July 2010 (UTC)

DVdm has got to it already, but his point here and in the above section is correct: rearranging content differently from the source to make a point is original research. The reason for my change was to undo your invalid change of properly sourced content, without giving a reason other than one based on WP:OR.-- JohnBlackburne wordsdeeds 09:39, 9 July 2010 (UTC)


 * John, Unfortunately you seem to have forgotten that this is an encyclopaedia for general readers. It is not about copying chunks out of pure maths textbooks verbatim. The purpose of the requirement of sources is to confirm material facts. Do you have some problem with the ordered version of the table? Do you wish to unsort it because it speaks too loudly of the point about that any one unit vector can be the product of three pairs from the other six? Do you have a problem with that fact? David Tombe (talk) 09:45, 9 July 2010 (UTC)


 * And when these general readers check the source and notice that someone changed the list in order to demonstrate a point that is not mentioned in that source, they will have second thoughts about whether this is an encyclopedia or a textbook. DVdm (talk) 09:48, 9 July 2010 (UTC)

DVdm, There is no requirement that one must copy verbatim from textbooks. In fact the rules are quite the opposite, and John Blackburne has on occasions recently re-ordered wordings in order to avoid direct copying. He admitted it himself. But on this ocasion you are both insisting that the table must be copied verbatim from Lounesto right down to the ordering. And in doing so, you are concealing a point. David Tombe (talk) 10:00, 9 July 2010 (UTC)


 * David, please stop adding your original research to this encyclopedia and stop disrupting article talk pages by persistently pushing moot points. DVdm (talk) 10:10, 9 July 2010 (UTC)

DVdm, You have contributed nothing but disruption to this article. I have been trying to make it accessible to the general reader, whereas others have been trying to cloud the article up in cryptic pure maths speak. David Tombe (talk) 10:31, 9 July 2010 (UTC)

DVdm: It is amazing to learn that sorting of a sourced table is OR. You could claim that using the sorted table to make a new claim was OR, but not the sorting. And in this case (which I doubt you actually looked at, because it is such a simple matter you would never involve yourself) sorting isn't necessary anyway. What is done is to look at a sourced table and notice that a particular entry occurs in the table three times. Blackburne says that is OR, but he is a little overwrought. As you are an uninvolved editor, I would guess that you'd agree that a statement that an entry appears 3 times in a sourced table does not quite rise to the level of OR, even if the author did not point out that fact, eh? Brews ohare (talk) 13:23, 9 July 2010 (UTC) To save you a moment, here is the so-called OR reworded from the listing that Blackburne altered to refer to the table instead: "For example, from the tabulated multiplication table to the right one finds e1 occurs three times; it is given by e2 × e4, e3 × e7, and e5 × e6. Thus, unlike the 3-dimensional cross product, the same 7-dimensional cross product can result from multiplying pairs of vectors residing in different planes." Brews ohare (talk) 13:43, 9 July 2010 (UTC)


 * Brews, you asked for comments. You got comments. Together with Tombe you are trying to push your original research into an article. Please stop. My being involved or uninvolved has nothing to do with this. DVdm (talk) 13:59, 9 July 2010 (UTC)
 * DVdm: You disappoint. Rather than make a specific comment (namely, about applicability of OR to the above text) you skip to vague generalities in order to sermonize. Brews ohare (talk) 14:13, 9 July 2010 (UTC)
 * You got comments from three editors. If you find comments disappointing, then don't ask for them. DVdm (talk) 14:49, 9 July 2010 (UTC)
 * DVdm: Rhetorical trickery, eh? It is not comments in general that disappoint, but your particular evasion referred to in particular just above. Brews ohare (talk) 16:01, 9 July 2010 (UTC)
 * Brews you have comments from multiple editors now explaining how your and David's interpretations, rearrangements and extrapolations are all OR. That some of it is mathematically wrong as well is not important, except to highlight the problem with doing amateur research rather than trusting reliable sources. But it's all OR and you've had the comments. Don't then accuse others of "trickery" because you don't like what they say. It simply suggests that you yourself have run out of reasonable arguments, if you are resorting to disparaging others' replies instead of responding properly.-- JohnBlackburne wordsdeeds 16:10, 9 July 2010 (UTC)
 * "Rhetorical trickery, eh?" => Brews, I have no idea where this comes from or what it means, but it surely sounds ugly. Anyway, your referring to my ignoring your comment just above as "evasion" shows that you still have not understood the simple concept of original research in the context of Wikipedia. You should know better by now. DVdm (talk) 16:26, 9 July 2010 (UTC)
 * Ah me. DVdm, it would be constructive and instructive if you would point out to me specifically what is OR in the following sentences:
 * "For example, from the tabulated multiplication table to the right one finds e1 occurs three times; it is given by e2 × e4, e3 × e7, and e5 × e6. Thus, unlike the 3-dimensional cross product, the same 7-dimensional cross product can result from multiplying pairs of vectors residing in different planes."
 * Of course, you aren't under any obligation to instruct me, but I honestly do not see anything here that is OR by any stretch of imagination. The first sentence is simply description of the sourced table. The second involves only the notion that two vectors define a plane. Please enlighten me with some detailed comment pertinent to this text, and not generalities that I may be unable to apply. Brews ohare (talk) 16:38, 9 July 2010 (UTC)

Reply to Blackburne 16:10, 9 July 2010: Vague references to discredited earlier OR challenges as though they were now established is specious. Critique the text supplied above specifically, rather than sermonizing on unsupported generalities. Brews ohare (talk) 18:40, 9 July 2010 (UTC)

unnecessary formatting changes
I have just undone some changes that only changed the formatting of references. This is the third or fourth time I've brought this up in recent days but the manual of style is clear: editors should not change the formatting from one valid format to another without a very good reason, i.e. some other reason than their personal preferences. This is especially true of of footnotes where the differences between them are largely a matter of preference.-- JohnBlackburne wordsdeeds 07:55, 9 July 2010 (UTC)
 * Pardon me John. The formatting changes I made had the effect of leaving exactly the same footnotes as before, but adding a link to the source they refer to in the list of references. That is, you could click on the footnote and be taken to the full listing of the reference details. That obviously is an aid to the reader, and changes absolutely no content. You reversion of this assistance to the reader is purely an act of annoyance and disservice. To justify this action as a priority of your preference to mine is just silly. Brews ohare (talk) 13:15, 9 July 2010 (UTC)

Non-sequitor
I took out the two sentences below: "‘However, unlike in three dimensions, there is no unique (up to sign) cross product defined in seven dimensions. This is because in seven dimensions there are more than two unit vectors perpendicular to any given plane.’" My reasoning is as follows:
 * It is not clear what is meant by "no unique cross product". A probable interpretation is that two vectors x and y can be associated with several different vectors x × y, making x × y not unique. Based upon any multiplication table, that interpretation is mistaken, as every specific pair of vectors (x, y) is mapped by the operation x × y to one and only one vector in the space.
 * The second sentence is meant to serve as justification for the first. It is obvious that a plane is described by two non-parallel vectors, and as there are seven dimensions, that means there are 5 normals for any plane. However, that fact bears no apparent connection to the statement it is intended to support, as the multiplicity of normals to a plane does not clearly and simply imply a multiplicity of cross products for any selected two vectors in a plane.

Perhaps these sentences are meant to convey the different fact that a particular direction in ℝ7 may be realized by a variety of cross products, that is, a variety of vector pairs? For example, using the table in the Intro, e1 = e2 × e3 = e4 × e5 = e7 × e6, which demonstrates three different cross products have the same result.

Whatever meaning these two sentences intend, it must be rephrased to make sense. Brews ohare (talk) 18:21, 9 July 2010 (UTC)

It may be that I am misled by intuition based upon 3D, and multiple x × y do exist by virtue of the failure of x × y to be invariant under SO(7). If that is the case, a more elaborate explanation is needed than the fact that any two-space has five normals. Brews ohare (talk) 17:59, 9 July 2010 (UTC)

A multiplicity of x × y has the interesting implication that various multiplication tables are not simple changes in coordinate systems as they are in 3D, where they all can be mapped into each other by SO(3). Brews ohare (talk) 18:21, 9 July 2010 (UTC) For example, coordinate changes in 7-D due to rotations are related by SO(7). Suppose a multiplication table in one coordinate system leads to a particular x × y. Then in a rotated coordinate system, despite x and y being the same vectors, x × y is not the same vector cross-product found in the first coordinate system, unless the rotation belongs to G2. Brews ohare (talk) 19:07, 9 July 2010 (UTC)
 * your reasoning is wrong: the 3D cross product is unique up to a sign, i.e. you also need a right-handed rule" to specify the orientation. Geometrically this comes down to choosing one of the vectors normal to the plane x and y lie in. Algebraically it's the dual of the bivector x ∧ b. So it makes sense looked at in all those ways. In 7D none of that is true and the 7D cross product is not unique. That's what it means: there's more than two opposite choices for the product, so more than just a single product, or two differing only by handedness or orientation. It may not be clear to you but it's a perfectly clear statement.-- JohnBlackburne wordsdeeds 19:14, 9 July 2010 (UTC)
 * John: Simple assertion of non-uniqueness doesn't aid understanding, for me or for readers. It seems probable that the issue is as I have described it: Suppose a multiplication table in one coordinate system leads to a particular x × y. Then in a rotated coordinate system, despite x and y being the same vectors, x × y possibly is not the same vector cross-product found in the first coordinate system. Do you disagree with that statement? Brews ohare (talk) 19:57, 9 July 2010 (UTC)
 * I'm sorry if you don't understand it, but the benefit of this being WP is you can look at other articles, look at the references, and ask here if you don't. As for the behaviour under rotation this is already covered in the section titled Rotations is a good and well sourced way. Your interpretation would add nothing to it.-- JohnBlackburne wordsdeeds 20:07, 9 July 2010 (UTC)
 * A yes or a no would be an answer to my question. I gather that you have nothing to say on this matter. It is odd that an obvious point such as there being multiple vectors x × y for every pair of vectors x and y receives no mention in the literature. That makes such an assertion OR, and suggests that your statement in the article should be removed. I have placed a request for a citation in this regard, and of course, if none can be found, the statement will be removed according to WP:OR. Brews ohare (talk) 20:38, 9 July 2010 (UTC)
 * This is beginning to look like a clear case of WP:POINT. DVdm (talk) 20:40, 9 July 2010 (UTC)
 * DVdm: What the above requests is clarification and documentation. So far the assertion that x × y is not unique is not clarified (for example, does it actually mean there are multiple vectors x × y for every pair of vectors x and y, or not), and also is not sourced. There is no need for you to escalate matters beyond that. Brews ohare (talk) 21:48, 9 July 2010 (UTC)
 * It is my opinion that the statement does indeed mean that there are multiple vectors x × y for every pair of vectors x and y based on the view that there are multiple normals to the plane of x and y and that besides orthogonality to the plane, all that the definition requires is a particular length, which it would seem could be achieved by a vector in any of the infinity of directions available in the 5-D space of vectors orthogonal to the plane. However, for reasons unknown, Blackburne will neither confirm nor deny such an extended explanation, and no source that says such a thing can be located. Brews ohare (talk) 22:07, 9 July 2010 (UTC)
 * Now each and every multiplication table provides its own uniquely directed cross product. An intriguing question then is the relation between these various multiplication tables that quite possibly lead to different choices for the direction of the cross product even for exactly the same vectors x and y. Are there an infinity of such tables corresponding to the infinity of possible normals to a plane? I'd guess there are limitations, because the Fano plane will restrict choices. However, Blackburne remains adamant that he will not exert himself upon this question. Alas! Brews ohare (talk) 22:22, 9 July 2010 (UTC)
 * Don't be so dramatic. If you want answers to questions about the topic they are in the sources, don't expect anyone else to do the work of reading them for you if you can't or won't take time to understand them yourself. Insisting other editors carry out your requests is very poor etiquette.-- JohnBlackburne wordsdeeds 23:03, 9 July 2010 (UTC)
 * I do not feel that trying to elicit your help exhibits lack of etiquette; however, I gather you are not able to help without doing some reading and understanding that is not to your taste. I understand that. You might simply have told me so. I had thought you were more conversant. Brews ohare (talk) 12:15, 10 July 2010 (UTC)

The Introduction
Brews, that was very interesting about the number 480. I had actually begun to wonder just how many conventions we could have in total. But much as this material in the introduction is very informative, I was thinking that perhaps some of it really ought to be blended into the section entitled 'coordinate expressions'. I can se a bit of duplication arising, and we do need to keep the introduction to key facts. See what you can do and I'll help out. If you get the details down to the lower section, I'll attempt to write a short summary of the same information in the introduction. David Tombe (talk) 16:50, 11 July 2010 (UTC)
 * I'm in accord with streamlining the introduction as you have done. The article now covers the basics, but the deeper connections are hardly touched upon.
 * I moved the Lounesto quote to what seemed to me a better spot. Brews ohare (talk) 17:17, 11 July 2010 (UTC)
 * I don't know what else you want to move out of the intro: most of it simply explains the properties of the table. As this is where the table is located, it can't be put elsewhere. An alternative is to describe the later Lounesto table instead, and leave this material out of the intro. Is that your suggestion? Brews ohare (talk) 18:09, 11 July 2010 (UTC)

Brews, That's better now. It's necessary to have some of that stuff in the introduction because the reader needs to have a rough picture in their mind of the end product before starting at the first principles. Hence that material has to be split between the introduction and the special section further down. But the most of the details should be further down. It's just a question of striking the correct balance. David Tombe (talk) 22:47, 11 July 2010 (UTC)

Nonuniqueness
The edit here is intended to show a multiplicity of cross products x × y exists for each pair x, y. It states that a particular table produces e1 × e2 = e3. To make the point, it must be noted that the Lounesto table produces e1 × e2 = e4, which obviously is different when e3 ≠ e4. This approach avoids making any unsupported statements, however logical, but requires some changes from the reverted edit, and of course, is much less basic. Brews ohare (talk) 21:30, 11 July 2010 (UTC)


 * Brews, I think the current wording is very bad. Really this non-uniqueness is a minor point since all the cross products are the same up to isomorphism, and I don't want to waste my time on it much longer. I will try to explain to you one more time my thoughts on this, and then you can just do whatever you want.


 * A cross product is a map (or a function if you like) that assigns to any vectors x and y a third vector x×y and satisfies some properties. So when I say (or you say in the article) that the cross product is unique, I mean that there is only one map with these properties. If I say the cross product is not unique, then I mean that there is more than one map with these properties.


 * Now let me carefully parse some of the current wording and attempt to explain in detail why I think it's bad. The current wording says, "In contrast with three dimensions where the cross product is unique (apart from sign), the cross product of two vectors that exists in seven dimensions is not unique." The first part of the sentence says the cross product in three dimensions is unique up to sign. This means that there are exactly two maps that satisfy the properties required to be a cross product in three dimensions, and they differ by a factor of -1. That is clear. Now the second part says, "the cross product of two vectors that exists in seven dimensions is not unique." Recall a cross product is a map, so when you say the cross product of two vectors you are (presumably, although I think it gets unclear here) referring to a particular cross product applied to a pair of vectors. This is the same as how we might call a function f, and then talk about that function acting on an element of its domain as f of x. Just like in the case of a function f, it makes no sense to then say f of x is not unique. Similarly, it makes no sense to say, "the cross product of two vectors that exists in seven dimensions is not unique." Really what you're trying to say is that the cross product (the map) is not unique, but your doing it in a rather confusing way.


 * Next sentence, "Instead, there are multiple vectors x × y for every pair of vectors x and y because there are multiple normals to the plane of x and y and, apart from the requirement for orthogonality to the plane of x and y, all the definition requires of the cross product is a particular length." To write x × y you are implicitly (in my reading) saying you have chosen a particular cross product, and so the beginning of this sentence suffers from the same problems as the previous sentence. Really it's trying to say that for any x,y, and v orthonormal, it's possible to define a cross product so x × y = v. This is the language I've tried to put in.

Next, let me discuss "There is also another aspect of non-uniqueness with the seven-dimensional cross product which does not arise in three dimensions." Only in a very obscure manner is this an "aspect of non-uniqueness." The quote is somehow a statement about non-uniqueness of the inverse of the cross product considered on some sort quotient space (ie. the set of oriented planes). I don't think this should be referred to as non-uniqueness.

Finally, let me point out that the multiplication table currently in the intro says e_1 × e_2 = e_3. Holmansf (talk) 22:33, 11 July 2010 (UTC)
 * Holmansf: First, I appreciate your effort to explain to me your position. Second, I hope your patience might extend further, as I don't yet fully understand. Let me take things slowly, and hope for eventual clarity.
 * Map vs. cross product. You raise an interesting contrast in view. As an engineer I am very used to thinking of x × y as a real thing, independent of coordinates. Hence my notion that e1 × e2 is a "thing" and the result that e1 × e2 = e3 (Cayley table) and e1 × e2 = e4 (Lounesto table) struck me as the meaning of Blackburne's phrase that cross-products were not unique. However, I get your point that what is involved here is multiple mappings, so a better statement is to avoid the "uniqueness" idea, and talk about 480 possible multiplication tables with who knows how many different results for e1 × e2; certainly more than one and probably more than the two exhibited. One can say that for any chosen map, e1 × e2 is unique, but that there are many maps, and they don't all do that. Would that be clear?
 * "Instead, there are multiple vectors x × y for every pair of vectors x and y because there are multiple normals to the plane of x and y and, apart from the requirement for orthogonality to the plane of x and y, all the definition requires of the cross product is a particular length." This sentence would translate into a comment about why different multiplication tables (mappings) are allowed to have different entries for e1 × e2. How's that?
 * I'd say that Lounesto's statement is illustrated by the observation that e1  occurs six times in every multiplication table (as does every unit vector), as opposed to the three-D case where it can occur only once. I said exactly that in earlier attempts, and Blackburne and yourself called it OR. However, it is an easier statement to follow and verify.
 * Are we on the same page?? Brews ohare (talk) 23:22, 11 July 2010 (UTC)


 * "One can say that for any chosen map, e1 × e2 is unique, but that there are many maps, and they don't all do that." This statement is close to being correct, although the second part should be changed to " ... and not all of them take the pair e1 and e2 to the same vector." As you wrote it it contradicted itself (it seemed by "do that" you meant "map e1 × e2 to a unique vector").


 * I'd also like to stress that a cross product is a map by the definition in the article. This is not some new idea I'm bringing up, it's what the article is talking about. Further, the non-uniqueness of these maps is exactly what both I and John Blackburne have been talking about for some time (we should not avoid it).


 * "Instead, there are multiple vectors x × y for every pair of vectors x and y because there are multiple normals to the plane of x and y and, apart from the requirement for orthogonality to the plane of x and y, all the definition requires of the cross product is a particular length." This sentence would translate into a comment about why different multiplication tables (mappings) are allowed to have different entries for e1 × e2." I do agree with this, although I think part of the point with putting the definition in terms of a map with certain properties is that we don't need the crutch of the multiplication table to talk about the existence and uniqueness of a cross product.


 * On the last point, if you look back through the comments you will see that I never claimed this was OR. I just don't think it should be referred to as "non-uniqueness" and I thought the way you worded it was confusing and obscured certain facts as I mentioned earlier.Holmansf (talk) 23:59, 11 July 2010 (UTC)

OK, I've rewritten these sections in an attempt to meet your objections. I'm inclined to feel that it shows a love affair with the words "map" and "mapping" that was previously implied by the term cross product, and really didn't need special emphasis. However, now we have it. Please comment. Brews ohare (talk) 00:54, 12 July 2010 (UTC)


 * Your recent revision states that for any v perpendicular to a plane, an x × y = v can be found with x and y in that plane. Translating the statement in to your more careful language, this would read: ‘for any v perpendicular to a plane, a mapping can be found such that x × y = v with x and y in that plane’ I believe this wording is preferable, in that it makes clear that the statement does not apply to a particular multiplication table but to the ensemble of such tables.. Brews ohare (talk) 01:55, 12 July 2010 (UTC)


 * I think it's better how I have it now. In particular, I think "an x × y = v can be found with x and y in that plane," is rather confusing. I am not going to edit this page anymore. Holmansf (talk) 02:16, 12 July 2010 (UTC)

I changed this statement to refer to a general choice of x, y and v, not restricted to orthonormal triads. Brews ohare (talk) 14:09, 12 July 2010 (UTC)

Multiplication tables that preserve the basis
The phrase “that preserve the basis” has been inserted as a modifier of multiplication tables. Frankly, I cannot imagine a multiplication table that changes the basis, assuming the "basis" refers to the index column and index row that frame the table, and are by no stretch of imagination considered to be altered by doing the multiplication.

I removed this phrase as just padding. Brews ohare (talk) 21:36, 11 July 2010 (UTC)


 * The reference given there states that "... there are 480 distinct multiplication tables for which e_a e_b = e_c ... " The phrase you removed is expressing the requirement e_a e_b = e_c in a different way. Probably it or something like it should be added.Holmansf (talk) 21:49, 11 July 2010 (UTC)

I'd suggest putting it in as done in the source. Brews ohare (talk) 23:25, 11 July 2010 (UTC)


 * Fine, do it that way. Holmansf (talk) 00:02, 12 July 2010 (UTC)


 * By the way, this is a real requirement. You can have multiplication tables that do not meet this requirement. Holmansf (talk) 00:38, 12 July 2010 (UTC)

Implemented as in the source. Brews ohare (talk) 14:07, 12 July 2010 (UTC)

Non-bilinear cross products
Holmansf, There is another kind of 7 dimensional cross product which involves seven vectors being multiplied together at once, and which is established from the determinant of a 7x7 matrix. This is mentioned in the main article cross product in the section entitled 'extension to higher dimensions' and there are plenty of sources about it in the literature. That is why I made that minor amendment to the wording. The way that you have worded it wrongly implies that a cross product has to be bilinear. It doesn't. But the particular cross product of this article is bilinear. David Tombe (talk) 22:43, 11 July 2010 (UTC)


 * Okay, well then change it back with a reference, and a comment in the definition section about alternate uses. I would suggest something like this, "For the purpose of this article we define, following Massey(reference), a cross product to be ... " Then after the current definition give a brief mention of and perhaps a reference to the "generalizations" section.Holmansf (talk) 23:18, 11 July 2010 (UTC)
 * By the way, I think you actually mean 6 vectors multiplied together in 7D.Holmansf (talk) 23:23, 11 July 2010 (UTC)

These recommendations have been implemented. Brews ohare (talk) 14:06, 12 July 2010 (UTC)

OR & synthesis
A lot has been added to the article in the last few days that does not seem to be properly sourced: in particular a lot of material seems not to be based on sources on the 7D cross product but rather on other areas, such as the octonions. This causes multiple problems. First without proper sources it's impossible to confirm the accuracy of the content. Second use of inconsistent and confusing notation, from different sources and some entirely original, makes the article much more confusing and less clear. Third it misrepresents the importance of the topic: the reason why no source covers this topic at length is it's a really obscure topic that merits a few paragraphs even where it's relevant. If that is the case, i.e. it really is that obscure, then the article should reflect that. It should not be extended using sources on other topics, with results from those topics manipulated to make them look like published results on the 7D cross product.

So I propose going through the article and removing all such content, at the same time removing the irrelevant sources it seems to be drawn from. I am happy to do this, as I already have a fair idea what can be written from the sources that cover the 7D cross product.-- JohnBlackburne wordsdeeds 14:27, 13 July 2010 (UTC)


 * As I anticipate little agreement over your proposed actions, I'd like to see them presented here first, and moved to from the Talk page only after adequate discussion.
 * To begin, there is little doubt that there is a strong connection between octonions and the 7D cross-product, some of which is already mentioned in the article. I'd anticipate any confusion over what is appropriate to octonions and not readily extended to the 7D cross product can be readily assessed on this Talk page and any confusions straightened out. Brews ohare (talk) 05:32, 14 July 2010 (UTC)


 * I find the section on the octonions most appropriate. This is actually how I am most familiar with the 7D cross product, so I think the concern over WP:OR is perhaps unwarranted, at least as far as this section goes.   Sławomir Biały  (talk) 13:51, 14 July 2010 (UTC)


 * I've no problem with the sections on octonions either. Clearly the 7D cross product and octonions are related, and one way to generate the product is using octonions, while the cross product is the pure imaginary part of the octonion product as in 3D and quaternions. But they are different things, usually use different notation, and have different properties. But a lot of octonionic source seems to have been used in other sections, synthesised into material on the 7D cross product. E.g. I've not seen multiplication tables or Fano planes for the 7D cross product in any source, only in Octonion sources (and even there only one of each, and without the shading and observations added here).


 * I should add that there's another reason to do this, which is fix the lead which has grown to big and far technical. I've mentioned this already, and the manual of style is clear: the should provide an accessible overview. A lot has been added which apart from being badly sourced makes the lead overly detailed and technical, with content that should be presented later if at all. E.g. a coordinate expansion is in some ways the least interesting thing about the product, as it's not unique so can't be used to establish properties. But it's presented here as if most important. So even if some of this material should be kept it should be moved to the coordinate section and merged.-- JohnBlackburne wordsdeeds 17:15, 14 July 2010 (UTC)


 * Multiplication tables: Blackburne, are you suggesting that these tables are incorrect? Or, are you suggesting that their connection to the octonion tables should be made clearer? I think it is very obvious that an octonion multiplication such as:
 * $$e_i e_j = - \delta_{ij}e_0 + \varepsilon _{ijk} e_k \, $$
 * translates to a 7D cross product table as:
 * $$\mathbf{e_i \times e_j} =  \varepsilon _{ijk}\ \mathbf{ e_k }\ . $$
 * Would you disagree? Brews ohare (talk) 17:47, 14 July 2010 (UTC)


 * Shading of multiplication tables: Are you suggesting that tinting of the squares along the diagonals of the table for emphasis is objectionable? Or that tinting of the squares corresponding to a few particular vectors to show where they occur in the table and to emphasize structure is objectionable? Brews ohare (talk) 17:47, 14 July 2010 (UTC)


 * Lead is too technical: Can you say specifically what is too technical? The fact that a × b is a vector? That it is related to the octonions? Is an example multiplication table too technical, the historically first one due to Cayley? I'd say the opposite: it is very readily understood. Brews ohare (talk) 17:47, 14 July 2010 (UTC)


 * Removal of table to the coordinate section: As you know, it was felt by some on this Talk page that the naive reader would find a concrete example of the cross product an easy entry point to the general presentation beginning with the section on Definition. The reader is cautioned that there are many tables, and that great reliance upon only one is hazardous. It is not used to establish properties, and the reader is cautioned about that too. So really your objections that the table is given too much importance aren't valid: it is there simply as a concrete entry point, and as such will be helpful to readers that prefer proceeding from the specific to the general, rather than vice versa, a group found in educational circles to be predominant in the population. Brews ohare (talk) 18:15, 14 July 2010 (UTC)


 * It's not whether they are correct, but whether they appear in sources on the 7D cross product. The same applies to the shading, and the conclusions drawn from it. Without sources it's impossible to be sure it's correct. Or even if correct it's impossible to tell whether the conclusions drawn are the most interesting and so notable ones. There may be other, more important symmetries which are being obscured by the table and shading - to me the only pattern that's obvious from it is that the product is anti-commutative. At worst content is added to the article so it grows far beyond the notability of it's subject, misleading and confusing readers who on trying to check the material in the sources find it's not there as presented.


 * As for the lead being too technical, please read MOS:LEAD which discusses it at length. Or look at e.g. Special relativity which has far more mathematical content than this article but only one forumla, E = mc2, in the introduction. And putting things in an illogical order is not fixed by cautions to the reader. Better to organise the article so the order makes sense, so readers are not confused by it and non-encyclopeadic editorial comments can be avoided.-- JohnBlackburne wordsdeeds 18:56, 14 July 2010 (UTC)

Blackburne: As you know, various multiplication tables are in use, and making tables to show two of them is hardly a controversial act: it's just a mode of display. And you're worried whether the coloring of the squares is "correct"!! What does that mean?

Just what "conclusions are drawn" from the table? On one hand, you worry that the "most notable" conclusions haven't been stated, and on the other hand wonder just what these "notable" conclusions are. Or, do any exist? This is not critique, but mulling about.

I don't know what formulas in the lead worry you. Maybe the algebraic summary of the multiplication table? Be specific.

The organization of the article has a rationale provided just above, and you have made no comment upon it.

John, please make some substantial comments or let this matter drop. Brews ohare (talk) 21:06, 14 July 2010 (UTC)


 * The problem with the tables are they are very poor way to show the product, as already noted. First the product is not commutative, i.e. order matters, which is clear when the product is written a × b = c but not in a table. Second while there are a number or ways to write out the product term by term, to show e.g. the symmetry of it, there's only one way to show a table. That's why the only symmetry that's obvious from either table is that it's anti-symmetric, even with the shading. If the tables appeared in a reliable source then it would be clearer how they should be used, but they appear in no source - no source thinks they are a good representation of the 7D cross product. The same applies to the Fano planes, the alternative forms of notation, and the detailed discussion of these. There's nothing like them in any source so it's OR and synthesis, whether or not it's correct maths.


 * It's not particular formulae in the lead that are of concern, it's all of them - there should if possible be none. Compare it to e.g. cross product, which despite having far more algebra in the body of the article has none in the introduction, and hardly any in the first section after it. The lede is meant to be an accessible introduction to the subject, so should avoid detailed technical exposition. And again, the majority of the properties of the product follow from it's definition, not from any particular representation. A page of discussion on one particular representation will confuse readers into thinking it's far more important than it is, something that's not fixed by a last sentence which in effect says "now is the maths you can actually use to solve problems withe the product".-- JohnBlackburne wordsdeeds 23:04, 14 July 2010 (UTC)


 * Blackburne:
 * Figures: You say figures are a poor way to show the products, because they don't show that the products are not commutative. Of course, they do show that, because they show ei × ej = −ej × ei. In any event, an example makes the point. You say that the figure doesn't show the symmetry well, aside from anticommutativity, and in some ways that is true. The symmetries of the multiplication table are best demonstrated by listing the seven groups that define the table, as in 123, 145, 176, 246, 257, 347, 365 for the Cayley table. These numbers are provided. the members of the first group are tinted in the figure. None of this is “detailed technical exposition”, but rather simple description, not logic, not analysis. And as already said over and over, the reader is cautioned that the figure is only one of many such tables and the basis independent formulation of the Definitions section should be used to derive general properties. I think the bottom line here is that you don't like the figures, and you don't like the figures, and you don't like the figures.
 * Formulas: John, the principal formula in the lead is: $$\mathbf{e}_i \mathbf{\times} \mathbf{e}_j = \varepsilon _{ijk} \mathbf{e}_k \  $$. That hardly seems excessive to me, and describes exactly the symmetry you want to have shown. There also is the description of the multiplication table as “such that ea × eb = ±ec”, which was put in for editor Holmansf, and which in my opinion is superfluous. And lastly there is a description of how to translate the figure into the components of the vector x × y, again, only descriptive, no logic, no analysis. Reading components off a figure as described here is far easier than using an algebraic result, and one good reason for having a table.


 * In short, John, you are simply repeating yourself without taking into account anything said to counter these statements of yours, and without responding to any of the arguments supporting the present lead section. Brews ohare (talk) 01:21, 15 July 2010 (UTC)
 * Perhaps I seem to be repeating myself as you are repeatedly ignoring what I'm writing. The tables are unclear as the order of the product is unclear from them. E.g. the top-right corner of the first table is e6. But does that mean e1 × e7 = e6 or e7 × e1 = e6 ? It would not matter if the product is commutative but as it's not the tables are ambiguous. That is why tables are a very poor way to show the product, and probably why none of the sources use them.
 * And again please read MOS:LEAD in paticular the section on providing an accessible overview. Or look at the articles already mentioned, Special relativity and Cross product. Or one you are familiar with, Pythagorean theorem. Or a similarly advanced article to this one (and a featured article), Laplace–Runge–Lenz vector. There should be little or no maths in the introduction: perhaps a statement of the product, but nothing else. Certainly not the page of detailed mathematical working and argument that there is now.-- JohnBlackburne wordsdeeds 11:32, 15 July 2010 (UTC)
 * You do not appear to be repeating, you are repeating. And nothing you have said has been ignored; all has been responded to but not heard by you. In terms of an accessible introduction, feel free to post an RfC to see how others react. IMO it contains only descriptive material, and not "technical development" requiring attention or knowledge of the reader. And as for your insistence that the table is ambiguous regarding the order of factors, it has been pointed out to you over and over that this matter is described for the reader and an example given. In fact, if the reader were to use the table backwards, it still would be a valid table as all that would happen is all products would flip sign. It would be a different table but a valid one. And finally, the intro is devised to respond to readers that prefer to proceed from specific example to general theory instead of the reverse, a large group of individuals to judge from the educational literature. Brews ohare (talk) 14:05, 15 July 2010 (UTC)

Since John asked for comments- I certainly agree that the introduction is way too long. I think the introduction should 1) give an informal definition of 7D cross products in line with what is in the definitions section, 2) explain how this is a generalization of the well known 3D cross product, 3) summarize some of the differences and similarities between the 3D and 7D cases, and 4) state that nontrivial binary cross products only exist in 3 and 7 dimensions. IMO it should not have details about how to calculate a 7D cross product using a multiplication table, the number of possible tables, or observations about any particular multiplication table. In fact, I would say almost everything after the first two paragraphs should be incorporated in the "Coordinate expansions" section.

A big issue above, it seems, is whether a multiplication table should appear in the introduction. If you're taking a vote, I say no. Holmansf (talk) 15:50, 15 July 2010 (UTC)


 * Holmansf: Perhaps you would agree that your points 1-4 are covered in the first two paragraphs of the introduction? That restricts disagreement to the figure presenting one of the possible multiplication tables. The basis for putting such a table in the introduction is to provide the reader with a concrete example, which happens to be the historically first example. Perhaps you would care to discuss the merits of proceeding from the concrete to the general, rather than the reverse? That is the underlying issue. It is a pretty common observation that many prefer that type of organization, as they find having a concrete example in mind helps them to follow the more general approach, as provided in the Defintions section. Brews ohare (talk) 16:43, 15 July 2010 (UTC)
 * An alternative to the present organization is to follow the example of the Octonions article and place the figure representing the table in the Definitions section, in combination with the abstract postulates? Brews ohare (talk) 16:57, 15 July 2010 (UTC)


 * IMO the multiplication tables should go, if anywhere, in the Coordinate expansions section. I sympathize with what you are saying about having a concrete example in mind, but in this case I think the concrete example is the 3D cross product. That is, when you start reading about the 7D cross product you have the 3D one in mind, and you want to know in what ways the 7D one is different or similar to the 3D one, which is the concrete example that you can visualize. All the multiplication table does is show that the product is anticommutative, and perhaps convince some people that it actually exists. I personally doubt that anyone gains any real insight into the 7D cross product from considering that table. It's only useful for calculating, and should not be given a top spot in the article. It is not one of the most important aspects of the subject IMO. Holmansf (talk) 17:55, 15 July 2010 (UTC)


 * I would mention again, as no-one seems to have addressed it despite my highlighting it in my last post, that the problem with the multiplication tables is that they are ambiguous as it's not clear what the order of multiplication is for row and column headings. That is, for example, does the top right corner for the first table mean
 * e1 × e7 = e6,
 * or does it mean
 * e7 × e1 = e6 ?
 * This is why the tables are such a poor choice for the product, and I would think why they are not used in the sources. They also do little to help understanding as they obscure the symmetry of the product, obvious when it is written out as in the coordinate section. It's difficult even to pick out the triplets that make up the product (e1 × e2 = e4, e2 × e4 = e1, e4 × e1 = e2 etc.). All that the table shows is it's antisymmetric, but that's part of the definition so is hardly useful.-- JohnBlackburne wordsdeeds 17:08, 15 July 2010 (UTC)


 * IMO John is right that tables are a not the best choice. However it seems there is a strong feeling that they should be included, and I personally think that's okay (ie. to include them). A single sentence can explain the anticommutation and how to interpret the table. As I said above though, I do think they should only appear in the Coordinate expansions section. Now you know my opinions, and I do not want to get in an extended argument about this as I see y'all like to do. I will not respond on this topic any longer. Holmansf (talk) 17:55, 15 July 2010 (UTC)

New section: Example
As a compromise, I've moved the table out of the intro into an introductory Example section. Although Blackburne seems to be unable to read such remarks (this being the third or fourth mention), “the problem with the multiplication tables is that they are ambiguous as it's not clear what the order of multiplication is for row and column headings” is a problem only for Blackburne. The use of the table is described carefully, and even if the reader ignores instructions and runs the table backward, a valid multiplication table results. Brews ohare (talk) 18:01, 15 July 2010 (UTC)


 * I'll go along with that compromise. David Tombe (talk) 21:47, 15 July 2010 (UTC)


 * You could also use a notational trick by defining new symbols for the same quantities. E.g. you can define e'_r = e_r  and then list the primed quantities on e.g. the top of the table and write in the caption that the table gives the quantities e_r times e'_s. Count Iblis (talk) 01:44, 16 July 2010 (UTC)


 * On the ambiguity of the table one of them now has a note that tries to clarify it but that was not there when the table was first inserted, and is hardly clear. I still contend that the tables, even with such notes, are worse than just writing out the algebra, which do not require such explanation and are much better able to show the symmetries of the product.


 * The problem now is there two sections that do the same thing: show one way the product can be expressed in coordinates, both of which could be called "coordinate expression" or some such. I agree with Holmansf that there should be only one such section, which can then cover the various ways that the product can be expressed. This means other parts could be merged: the line starting (x × y)1 = is just the first line of an expression like that for the whole product given later on, for example.


 * It's also not clear what some things mean. What does this
 * "It also may be noticed that orthogonality of the cross product to its constituents x and y is ensured by the table's property that all interior columns are orthogonal to each other and to the left-most (index) column and all interior rows are orthogonal to each other and to the uppermost (index) row"


 * mean ? The "columns" of the table are sets of seven vectors, not vectors. So a "column" is not something that can be orthogonal to anything else. If it means the individual cells are orthogonal, as each ei appears only once in each row and column, that's true but it does not imply the product as a whole is orthogonal. It doesn't even imply it in 3D.
 * It's not clear how the following
 * $$\mathbf{e}_i \times \left( \mathbf{e}_i \times \mathbf{e}_{i+1}\right) =-\mathbf{e}_{i+1} = \mathbf{e}_i \times \mathbf{e}_{i+3} \ ,$$
 * "produces diagonals further out".
 * The same expression appears later with indices 6,5 and 1. There it seems to be used to permute the indices but that's unnecessary: the product occurs in triplets, e.g.
 * $$\mathbf{e}_1 \times \mathbf{e}_2 = \mathbf{e}_4, \quad \mathbf{e}_2 \times \mathbf{e}_4 = \mathbf{e}_1, \quad \mathbf{e}_4 \times \mathbf{e}_1 = \mathbf{e}_2,$$
 * etc. so any two of the above follows from the other one.
 * Other things that are unnecessary include the quotes and names of sources and the table of different notations. On the latter it's usual to pick a notation and stick with it, and as the sources all use ei or something like it there's no need to give anything else. More generally there's no need mention sources in the text, unless there's a particular reason such as a disputed point ("X says this while Y says that"). Adding source names to different parts of the article suggests there is such a difference, but there's not. There's just a single product, mostly derived in different ways in different sources but they are all describing the same thing. The quotes are unneeded, and seem to have been partly rewritten which is strictly against the rules: see e.g. WP:QUOTE. There's no reason to include the first quote and the second isn't even on the topic.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 20:08, 16 July 2010 (UTC)


 * John, I don't really see what you are complaining about here. Everybody now seems to agree about the underlying theory. The table which you are objecting to gives a very good initial mental image of the what the topic is about. You are correct when you say that it doesn't make clear whether the row term or the column term comes first in the operation, but that is only a very minor detail. Anybody looking at the table will quickly get the main point and it will all be explained in full as we move down the article. David Tombe (talk) 20:41, 16 July 2010 (UTC)