Talk:Eight-dimensional space

Recent Revisions

I made some revisions to the article which were reverted by User:Arthur Rubin. I'd like to discuss here. The reason for the revisions are as follows:

• Clarifying the difference between an 8 real dimensional vector space and an 8 real dimensional Euclidean space. The earlier definition of the main topic muddled this point.
• Removing undue focus on hyperbolic and elliptic spaces. Such a classification makes sense in the context of 2-dimensional compact orientable surfaces, because of the Uniformization theorem; however 8 dimensional spaces are generally neither elliptic nor hyperbolic nor Euclidean. Hyperbolic geometry in specifically eight dimensions is not an area where significant research has been done (hyperbolic geometry is studied mainly in 2, 3, and n dimensions.) Likewise I am totally unaware of 8-dimensional elliptic geometry attracting any interest whatsoever.
• Removed bizarre reference to spacetime being possibly eight dimensional. Although I can't rule out the possibility that this is asserted in some fringe theory, basically nobody believes this. Significant dimensionalities of spacetime are 4 in the standard model, 5 in the Kaluza-Klein theory (which is not considered plausible but is historically significant), 10 in fermionic string theories, 11 in M-theory, and 26 in bosonic string theory (again, not plausible but important.) Theories in virtually every integer dimension are studied but that doesn't mean anyone believes the actual universe has such a dimensionality.
• I included a correct definition of a normed division algebra. The previous one made no mention of the division part, which is crucial. Normed algebras of arbitrary dimension exist.
• Included a paragraph on the biquaternions, which deserve mention just as much as the octonions.
• Removed the paragraph on superspace, because it was misleading and wrong. However the topic of Minkowski superspace may deserve mention here and I should replace it with a new paragraph.
• The "notes" section was pure OR, and for that matter nonsense. --Sammy1339 (talk) 19:42, 9 November 2014 (UTC)

You do not quite have the relationship between Euclidean space and vector spaces correct. Euclidean space is affine space with a consistent metric satisfying the parallelogram law, not exactly a normed vector space. (Whether a vector space is an affine space with a 0, or whether an affine space is a vector space acting on a space of points, is subject to debate. Regardless, it is misleading, if not inaccurate to say that a Euclidean space is a vector space with certain properties.) There are theorems about regular (and uniform) hyperbolic polytopes and tessellations, but I don't know what they are; in dimensions where we know them, elliptic and hyperbolic space should be in the lead.

What we actually have in the article are 8-dimensional Euclidean space and 8-dimensional algebras. In 7 dimensions, we only use Euclidean space, cross product (as an example of 7-dimensional algebra), and exotic spheres (7-dimensional differentiable manifolds).

• The biquaternions do not deserve as much as you've given. They may deserve a sentence under octonians, but I think a {{see also}} should be adequate.
• It's not your fault, but kissing number should be moved up under geometry.
• For 7, we should possibly retain one of the fringe physical theories in 7-space, and probably mention the 7 "rolled-up" dimensions in (at least some versions of) M-theory.

Others may disagree with other edits you've made to the articles. I'll make some of the body changes I've indicated as necessary, but the lead is not good, in either form, and as an expert, I don't feel qualified to determine what is understandable. — Arthur Rubin (talk) 21:35, 9 November 2014 (UTC)

1. n-dimensional Euclidean space is $\mathbb {R} ^{n}$ (or an n-dimensional real vector space) with the Euclidean metric aka the dot product, or equivalently with the Euclidean distance function. I think that's the simplest definition - vector spaces are more accessible than affine spaces. Of course you could do it your way as well; you could also say it's a Riemannian n-manifold that is flat, homologically trivial, and simply connected at infinity, but that doesn't mean the other definitions are incorrect. I'm not sure I understand your concern: are you worried that stating it's a vector space may imply there is a canonical choice of zero? Note that I did not say it was a normed vector space. Maybe that's where the confusion is coming from.

1a. The point about tilings of hyperbolic space is interesting. This is totally outside of my area and I'm unaware of any results in dimensions as high as 7 or 8 - I'd be interested (in a happy way) to see that, since already in four dimensions the problem seems intractable. If you know of nontrivial results of this kind they probably could be included. I still disagree with giving top billing to hyperbolic 8-space as there are many more significant 8-dimensional objects.

2. Both articles could do with some expansion. I think it's reasonable to talk about algebras, and manifolds, and Euclidean geometry in each article. Right now there's not a lot there. Until we have enough material to fork, we might as well include all of that.

3. I disagree with this; the biquaternions see far more frequent use (under various names) than the octonions. Anyway by what crazy logic does super Minkowski space belong here but not the biquaternion algebra?

4. This I just don't get at all. M-theory is 11-dimensional. Your argument for mentioning it in the seven dimensional space article is that 11-4=7? As for the fringe theories, I actually did a literature search seven- and eight-dimensional models and found very, very little. I'm not trying to get rid of stuff but there's really just not a lot there. I also somewhat disagree with including insignificant theories in a very general article like this.

5. Can you clarify more specifically what's wrong with the lead?

--Sammy1339 (talk) 22:27, 9 November 2014 (UTC)

@Arthur Rubin: I reverted an edit of yours in which you wrote:

The octonions are a normed algebra with a quadratic norm over the real numbers, the largest such algebra. Mathematically they can be specified by 8-tuplets of real numbers, so form an 8-dimensional vector space over the reals, with addition of vectors being the addition in the algebra. A normed algebra is one with a product that satisfies

$\|xy\|=\|x\|\|y\|$

for all x and y in the algebra. Hurwitz's theorem prohibits such a structure from existing in finite dimensions other than 1, 2, 4, or 8.

Your edit summary seemed to indicate that this was not a simple oversight. I'm not sure how to interpret this. --Sammy1339 (talk) 22:44, 9 November 2014 (UTC)

Euclidean space v. vector space v. ???

In regard the lead, it is misleading to assert that a vector space is an abstraction of Euclidean space or that Euclidean space is an example of a normed vector space. I still assert that the most common uses are Euclidean space and algebra over the reals, where neither is an abstraction of the other. — Arthur Rubin (talk) 23:14, 9 November 2014 (UTC)

Once again, I did not assert that Euclidean space is a normed vector space. I did not assert that a vector space is an abstraction of Euclidean space either. I asserted that Euclidean space is an 8 dimensional real vector space with the Euclidean distance function. Obviously such vector spaces can also have an algebra structure, without having a distance function. --Sammy1339 (talk) 23:34, 9 November 2014 (UTC)

I think Euclidean space and vector space should be parallel constructions, without explaining the difference. The way you've described the difference is not accurate. If an accurate simple way of describing the difference could be added, I would be in favor. — Arthur Rubin (talk) 23:44, 9 November 2014 (UTC)

I am not in favor of not explaining the difference. Neither can I see a simple way of defining Euclidean space without reference to vector spaces - your proposal was to define it as an affine space, which is a concept that relies on vector spaces. A definition as a manifold also requires vector spaces. Euclid managed without vector spaces, but with modern standards of rigor his formulation requires thirty-something different axioms. Feel free to make a proposal, though. Also please clarify what you believe is inaccurate in the current version. --Sammy1339 (talk) 00:20, 10 November 2014 (UTC)

Kissing number

Clearly the kissing number is a geometry question, and should be placed under geometry. — Arthur Rubin (talk) 23:14, 9 November 2014 (UTC)

I agree. --Sammy1339 (talk) 23:28, 9 November 2014 (UTC)

Octonions

Hurwitz's theorem states that a finite-dimensional quadratic-normed algebra over the reals has dimension 1, 2, 4, or 8. There is a different theorem, mentioned in division algebra, which states that any finite-dimensional (not necessarily associative) division algebra over the reals has dimension 1, 2, 4, or 8. We don't mention the requirement of a norm, there. The section has an incorrect formulation of Hurwitz's theorem. — Arthur Rubin (talk) 23:15, 9 November 2014 (UTC)

Oh I see. You simply confused normed algebras with composition algebras, which is an understandable mistake. As for the theorem you're referring to, that's a difficult result that uses K-theory, but note that Hurwitz's theorem implies that 8 is the largest possible dimension for a normed division algebra, which is what I wrote. --Sammy1339 (talk) 23:27, 9 November 2014 (UTC)

Hurwitz's theorem covers composition algebras, which include quadratic normed (aka inner-product spaces) division algebras. It does not, as described here, apply to any normed division algebra. — Arthur Rubin (talk) 23:39, 9 November 2014 (UTC)

(edit conflict) It does. John Baez gives a good introduction to the subject:[1] and also provides references. However with due humility, I was the one who introduced the confusion. My definition of normed algebra was slightly wrong: I wrote "equals" instead of "less than or equal," when in fact the equality is derived from the fact that it's a division algebra. That is, a normed division algebra must be a composition algebra. (Baez glosses over this too with a careful choice of words. It is almost trivial - consider the norm of $xx^{-1}$.) --Sammy1339 (talk) 23:57, 9 November 2014 (UTC)

It appears that we have defined normed division algebra to mean that there is norm N such that.

1. $N(xy)=N(x)N(y)$

   (where a norm usually only has ≤)

2. N is a quadratic form in the coordinates.

   If the field is not of characteristic 2, then we can define an inner product

   $(x,y)={\tfrac {1}{2}}(N(x+y)-N(x)-N(y))$

That's not the same as a normed algebra which is a division algebra, which is what I would have expected. — Arthur Rubin (talk) 23:53, 9 November 2014 (UTC)

Actually it is the same as a normed algebra which is a division algebra. However my apologies for the mistake, which has already been corrected. --Sammy1339 (talk) 00:00, 10 November 2014 (UTC)

Biquaternions

To be perfectly honest, I'd never come across the biquaternions except as a generalization of the bicomplex numbers, which are, in turn, not of actual use in mathematics any more. The tessarines were considered of mathematical interest at one time. If they match GL₂(C), that might be of interest, but I have my doubts. — Arthur Rubin (talk) 23:14, 9 November 2014 (UTC)

Withdraw. It does appear to be M(2,C). Although of little mathematical interest, it is an 8-dimensional associative algebra of some interest. — Arthur Rubin (talk) 23:23, 9 November 2014 (UTC)

Rated article

Since this article is slightly long, I'm rating it Start-class; however, since it's relatively minor (similarly to Four-dimensional space and Five-dimensional space), I'm rating it Low-importance. Duckmather (talk) 21:15, 21 May 2021 (UTC)