Talk:Seven-dimensional cross product

New complaints

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)

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)

• 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)

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)

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

• 3. :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.--JohnBlackburne^words_deeds 20:08, 16 July 2010 (UTC)

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)

• 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)

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)

Mistakes?

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)

480 such products?

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)

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)

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)

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)

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)

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)

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)

It is already mentioned in the Examples section, where I have left it, in a form that reasonably crisp and correct. I only removed it from the lead, where I feel it does not belong in any form and was stated incorrectly. I am not aware of a reference of the manifold of cross products in 7-D; that was OR. It is already clear from the article that there is at least 5 dimensions of freedom in the choice, though. — Quondum 20:08, 4 September 2012 (UTC)

Generalizations

I took issue with both parts of this change so reverted it.

• First n ≥ 3 is better, as the n = 2 case is trivial: n − 1 is in this case 1, so there's only one vector. The 'exterior product' of it as a set of vectors is just itself, trivially so, while the dual is also trivial. On other words it's not something most people would recognise as a product.

• The other change is more subtle. It's correct that a product of three vectors exists in four and eight dimensions, but the four dimension case has already been mentioned, in the line mentioned above. The product of three vectors is an exceptional case and so is listed separately to complete the list.

This is based on Lounesto pg 98, which explicitly identifies the 2D case (and describes how it generalises to all even dimensions) as trivial. Looking again at that it's clearer with the order given there, with 3 and 7 dimensions first then the generalisation and exceptional case, so I've swapped those lines.--JohnBlackburne^words_deeds 08:19, 17 February 2014 (UTC)

There are a number of choices, depending on how one wishes to present it.

• First, one chooses between each bullet standing on its own, giving the solutions to a constraint (e.g. number of factors), or listing cases until all are covered, exceptional cases coming afterwards. You evidently prefer the latter.
• I regard the main case as being the regular family of n − 1 factors. There is good reason to prefer this as the main case over the binary case under the heading of generalizations: they are natural (being unique up to a sign), and include the familiar case. In contrast, the seven-dimensional case is unnatural, since it breaks the symmetry of the space massively (i.e. it requires the arbitrary choice of several real numbers to specify).
• One can then (i) present further cases not yet covered, (ii) have an exposition in which every bullet is the answer to a question: all cases that satisfy the given constraints (number of k-ary operation in n dimensions), or (iii) have a "natural family" followed by "unnatural cases". It would be nice if the selected pattern was clear from the tresentation.
• The "trivial" (unary) cases have been excluded in the article without even defining what is meant by the term, which should be corrected. They are not technically trivial, given that aside from the member (n = 2) that is also in the "main family", they are unnatural in the same sense that the n = 7 and n = 8 exceptions are. For this reason they should be of no less interest than those exceptions. Lounesto seems to realize that for completeness of an answer to the question of generalization, they must be given, and has found them interesting enough to give an example construction (they are so non-trivial that I could not readily find examples other than for n = 2 before reading Lounesto). And you presume about "most people": perhaps the idea of an empty product is reasonably accepted in the context of a k-ary product? IMO, the nullary operation is really the only case truly deserving the appellation "trivial" in this context. Whatever terms one chooses (Lounesto even uses the term "degenerate" synonymously, which jars for me), this does not imply that they should not be mentioned.

I am aware however that my preferences for regularity, completeness and simplicity of structure are stronger than that of many mathematicians. In particular, I do not have an aversion to including trivial cases to enhance regularity. However, mine is but one opinion. —Quondum 14:52, 17 February 2014 (UTC)

I've added a paragraph on the trivial products. It is fairly subjective what's trivial, or degenerate (a term I don't like in this context either), but I've included some information to clarify why the even dimension products are trivial. I don't think of in terms of natural and unnatural: it's on a fairly abstract mathematical topic which as far as I know has no real world applications, but we still study it. And we certainly have trivial examples/results even in the most abstract areas of mathematics.--JohnBlackburne^words_deeds 18:52, 17 February 2014 (UTC)

The word "trivial" is being overworked here as it has already been given a specific meaning in this context (that the magnitude that must be matched is inherently zero – perhaps "degenerate" fits here though, freeing up "trivial"). It is not necessary to repeat that case. How would this replacement

With the lifting of the restriction to a binary operation, a further simple case arises in even dimensions that satisfies the requirements. This takes a single vector and produces an orthogonal vector of the same magnitude. In two dimensions this is a rotation through a right angle.

for your paragraph sound (if you like "simple"→"trivial" if "degenerate" is used earlier)? I've not included the bit about the bivector because that gives little insight without more information on what type of product (a contraction). It might make more sense to mention a right-angled isoclinic rotation in four dimensions instead. —Quondum 20:16, 17 February 2014 (UTC)