Talk:Coquaternion

From Wikipedia, the free encyclopedia

	Physics Portal This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
???	This article has not yet received a rating on the assessment scale. [FAQ]
???	This article has not yet received an importance rating within physics.
Help with this template Please rate this article, and then leave comments here to explain the ratings and/or to identify the strengths and weaknesses of the article.

I think split-quaternion would be a better name for this page; mostly for naming consistency with split-complex number. We could then do split-octonion as well. As you've pointed out all these structures have numerous names; we should, however, strive for some consistency. -- Fropuff 16:52, 2005 Feb 6 (UTC)

[edit] "Hyperbolic Quaternion" ambiguity

(19 June 2006)

I see that there is a potential ambiguity about the term "Hyperbolic quaternion". The reference I've added to the split-quaternion site refers to hyperbolic quaternions from the hypernumbers program only, and not to the hyperbolic quaternions as they were defined in the 19th century. I hope that my wording is clear and the reference is therefore not ambiguous. In addition, I've explicitely added hyperbolic quaternions to the referenced hypernumbers page (as an example), and am also referencing split-quaternions / coquaternions from there.

While I am open to any wording or placement adjustment, I see this as a correct and valid statement, and see the reference therefore as applicable, useful, and reasonable. I would be glad to discuss any concern. Hypernumbers in general are actively discussed in the "hypernumbers" Yahoo(R) group. Thanks, Jens.

I've done some layout clean-up, as a step in the right direction, hopefully. Comments are invited. Thanks, Jens Koeplinger 02:08, 19 August 2006 (UTC)

[edit] re: Clifford algebras

Since Clifford algebras are a more advanced concept than a particular 4-dimensional real algeba, the introduction of the extra concept in the introductory paragraph amounts to obscurantism. Clear expository principles should prevail over the urge to link in an already well-funded research area. The better we describe the lower-dimensional algebras, the firmer will be the foundation for higher study. So, I have removed the Clifford algebra link from the first paragraph and replaced it with a more elementary and concrete reference.Rgdboer 01:59, 30 March 2007 (UTC)

I've put it back again. The point is that identifying the Clifford algebra(s) immediately tags what this algebra is about, what it is likely to be useful for, and how it compares to other hypercomplex numbers or quaternion variants. It means that somebody who can read a Clifford algebra classification knows exactly what to expect to find in the article.

So seeing Cℓ_2,0(R), one can read off immediately that this is going to be the natural algebra for describing vectors and directed areas in 2d, and their behaviour under reflections and rotations; and co-ordinate transformations on hyperbolic surfaces in 3d through the isomorphism with Cℓ⁰_1,2(R). Seeing Cℓ_1,1(R), one can read off immediately that this is going to be a natural algebra for describing vectors and areas on surfaces with one real and one hyperbolic coordinate; and through the isomorphism with Cℓ⁰_2,1(R), it's going to relate co-ordinate transformations on hyperbolic surfaces in 3d with the alternative metric convention (+,+,-), rather than (-,-,+).

As for Clifford algebras being obscurantist, I agree that IMO the current version of the Clifford algebra article starts at much too abstract, too all-encompassing, too mathemematically sophisticated a level. But that is a problem with the current article, not with the topic. If you look at hypercomplex number, you can start to see how useful the Clifford algebra classifications can be for making sense of all these algebras, and linking them all together with a systematic contextualised approach. I'm also going to put in some work on the geometric algebra article, which has potential to become the consolidated entry-level way in to study algebras like this one, and the geometric uses to which they can be put. See for example the first couple of chapters of the Cambridge "Physical applications of geometric algebra" course, linked at Geometric_algebra#Further_reading for how it ties everything together.

If you read that link, I hope you'll agree that it ought to be much more easy for students to study all the examples of geometric algebras together, understanding their geometric uses in a systematic way, rather than trying to pick them off in an ad-hoc unconnected way one by one. Jheald 07:48, 30 March 2007 (UTC)