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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)[reply]

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)[reply]

By the way, I think you actually mean 6 vectors multiplied together in 7D.Holmansf (talk) 23:23, 11 July 2010 (UTC)[reply]

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

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^words_deeds 14:27, 13 July 2010 (UTC)[reply]

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)[reply]

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)[reply]

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^words_deeds 17:15, 14 July 2010 (UTC)[reply]

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:

translates to a 7D cross product table as:

Would you disagree? Brews ohare (talk) 17:47, 14 July 2010 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

(edit conflict)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^words_deeds 18:56, 14 July 2010 (UTC)[reply]

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)[reply]

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^words_deeds 23:04, 14 July 2010 (UTC)[reply]

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 e_i × e_j = −e_j × e_i. 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: . 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 e_a × e_b = ±e_c”, 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)[reply]

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^words_deeds 11:32, 15 July 2010 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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^words_deeds 17:08, 15 July 2010 (UTC)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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 e_i 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

"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.

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 e_i 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_deeds 20:08, 16 July 2010 (UTC)[reply]

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)[reply]

Blackburne has raised a number of new issues unrelated to the introduction. Here they are again:

• 1. : 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.--JohnBlackburne^words_deeds 20:08, 16 July 2010 (UTC)[reply]

There is some utility in having figures showing two tables. The one in the Example section is useful in setting the stage with a concrete example early in the article, as has been explained several times already. It also happens to be the historically first table due to Cayley and Graves, and gets a lot of play in the literature. The one in the Coordinates section has several functions: (i) It provides another example, and two examples clarify that the tables can be very different. (ii) It also is a much used table in the literature. (iii) It is the table used for the coordinate discussion that happens to be the subject of this section. Of course, Lounesto's table could be deleted and Cayley's used instead throughout the article. I think that weakens the presentation and there is no harm done in returning to the topic of multiplication tables here. Brews ohare (talk) 23:45, 16 July 2010 (UTC)[reply]

• 2. :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 e_i 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. --JohnBlackburne^words_deeds 20:08, 16 July 2010 (UTC)[reply]

Well, I don't really think that this failure of comprehension is genuine. But if you can't understand it, it can be deleted.Brews ohare (talk) 23:45, 16 July 2010 (UTC)[reply]

I rewrote this section to avoid your objections. Brews ohare (talk) 16:10, 17 July 2010 (UTC)[reply]

• 3. :It's not clear how the following

"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.

etc. so any two of the above follows from the other one.--JohnBlackburne^words_deeds 20:08, 16 July 2010 (UTC)[reply]

The algebraic expression given for the multiplication table needs some elaboration to show just how the whole table can be found from it. If this particular attempt is distracting, replace it with another. Brews ohare (talk)

To illustrate the issues, the rule given e_i×e_i+1 = e_i+3 immediately provides e₁×e₂ = e₄ but how does one arrive at e₁×e₅ = e₆ using this rule? Some help is needed, for example, use of an identity. Brews ohare (talk) 16:16, 17 July 2010 (UTC)[reply]

• 4. :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 e_i 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_deeds 20:08, 16 July 2010 (UTC)[reply]

The table of other notations is helpful to the reader, as the literature uses them all. It takes no space, and can be ignored by any reader that doesn't have an interest.

The use of in-line references is a very well established practice and is not used just to settle disputed points, as any writer or reader of technical literature is well aware. This argument is Blackburne's personal preference, which is not the practice of most writers.

The quotes have been "rewritten" in no manner whatsoever. The change noted by “...one given by [Cayley, say]” is identified by brackets, and refers to Cayley's table in place of the author's numbered footnote to a reference where a table can be found. That does not change the meaning. Omissions of asides are denoted by ellipsis ‘...’ and again do not affect the meaning. Any suggestion that liberties have been taken should be presented in specific detail, and the argument for distortion clearly presented, instead of using yellow journalism and vague innuendo. Brews ohare (talk) 16:29, 17 July 2010 (UTC)[reply]

The text says

   * as a binary product in three and seven dimensions
   * as a product of n - 1 vectors in n > 3 dimensions
   * as a product of three vectors in eight dimensions

My questions are: 1) in the second item I think it is meant n >= 3 2) in the third item, what about in four dimensions?