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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?

they are covered in previous points. E.g. the product of two vectors in three dimensions the binary product in three dimensions, the cross product, so to generate additional products we only need consider dimensions higher than three. The product of three vectors in four dimensions is an instance of n-1 in n dimensions, so is covered by the second point.--JohnBlackburne^words_deeds 12:04, 23 November 2011 (UTC)[reply]

The lead claims that the seven-dimensional product is more general with 480 such products. Intuitively, I would say that in a seven-dimensional vector space with no preferred orthonormal basis, there seems likely to be an infinite number of such distinct products. Can we source/prove/disprove this? — Quondum^☏ 22:08, 31 August 2012 (UTC)[reply]

It relates to the number of octonion products: there are 480 distinct multiplication tables for them, and the seven dimensional cross product is just the octonion product with the real part omitted. Here's one source for the 480 octonion products: [1], lifted from octonion.

I can't prove there are only 480 myself, but I suspect there's also a geometric reason for it, much like there is a geometric reason for there only being two cross products. While that is invariant under SO(3) the seven dimensional product is only invariant under the group G₂, which may also be related.--JohnBlackburne^words_deeds 22:37, 31 August 2012 (UTC)[reply]

Thanks for pointing me to that reference. My confidence is boosted: use of the figure 480 cannot be extrapolated to the 7D cross product, despite the relationship with octonions. The reference refers specifically to "permutation of the indices of the pure octonions", by which is meant the imaginary basis elements. In a 7D vector space, there is no preferred basis, and arbitrary bases must be permitted (though I'll limit my argument here to orthonormal bases); consequently any sane "cross product" must yield a consistent result independent of an arbitrary choice of rotated (or reflected) basis. In particular, if the statement were true, the same 480 possible cross products would result regardless of the choice of orthonormal basis in the 7D space. Most such rotations (in particular, any rotation that does not only permute indices (i.e. basis vectors) and/or negate directions) will produce incompatible cross products. The group you mention appears to be a Lie group, which would be continuous and not discrete. Conclusion: the claim of 480 cross products is incorrect, and the correct figure is ∞. It might be more correct to say something like the family of possible cross products forms a k-dimensional real manifold, where k is probably something like 28, and the manifold has more than one disconnected part.

The geometric reason for only two cross products in 3D relates to the number of distinct unit 3-forms in 3D. From Seven-dimensional cross product#Using geometric algebra, one can see that the cross product depends on the choice of a value from a space of 35 dimensions, with some constraints. — Quondum^☏ 07:35, 1 September 2012 (UTC)[reply]

Okay, here goes. I know WP:OR is frowned upon, but when it is to replace false information in an article, perhaps it'll be received a little more kindly. I'll start with a proof of my proposition.

Proposition: The cross product operator in 7D euclidean space defined by an octonion algebra has several real degrees of freedom in its choice (and thus the number of such distinct operators is infinite).

Premise: Given a 7D euclidean space V, octonions can be used to form at least one bilinear map V × V → V, which we will call a cross product, where the octonian imaginary basis vectors are identified with an orthonormal basis of the 7D space.

Proof: Given an identification of two orthonormal basis vectors e₁ and e₂, we are free to identify (by rotation) any unit vector orthogonal to both as their octonian/cross product e₁ × e₂ = e₃. Since the space orthogonal to both e₁ and e₂ has 5 dimensions but the vector is constrained to being a unit vector, there are 4 dimensions of freedom in this choice, with every choice of e₃ being distinct, and hence making the cross product distinct. There are further degrees of freedom in the choice of the other basis vectors.

Given the directness of this proof, I'll remove the claim of 480 cross products. Note that the text does mention that "That leaves open the question of just how many vector pairs like x and y can be matched to specified directions like v before the limitations of any particular table intervene." Following my line of logic further, I arrive at 24 (=4×3×2×1) degrees of freedom and a choice of orientation. — Quondum^☏ 06:37, 2 September 2012 (UTC)[reply]

I believe the correct statement is that given an orthonormal basis for R^7 there are 480 distinct possible products that preserve that basis (in fact I think this is what the article said at some point in the past ...). I don't know how to prove that is the correct number off the top of my head, but from reading them a few years back I seem to remember that this number is supported by one or more of the references. Holmansf (talk) 21:52, 3 September 2012 (UTC)[reply]

Interpreting "... that preserve this basis" as meaning "... yields another basis vector or its negation as the product of any pair of basis vectors" perhaps. Such a fact is hardly interesting, and to state it without stating the more salient fact that there are an infinite number of products satisfying the criteria stated in the article risks obscuring this more significant fact. What is missing is something to show that in choosing a cross product, one is imposing a structure on the vector space that has a large degree of freedom of choice (24 real dimensions of it!) that it requires independent motivation and cannot be regarded as natural. In the 3-D case, this motivation is that the choice corresponds to a choice of orientation in the space. As an aside, the mention of 480 tables is also a bit esoteric, since in that context if they are isomorphic they are not distinct. I find it strange that an author should choose to make anything of it. — Quondum 08:39, 4 September 2012 (UTC)[reply]

Saying there are 480 distinct multiplication tables is the simplest way to express the concept, and I think it deserves mention somewhere if not in the lead. If you have a reference which discusses the space of 7-D cross-products as a manifold that would be interesting to include. Holmansf (talk) 16:18, 4 September 2012 (UTC)[reply]