Talk:Orientation (mathematics)

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[edit] Left- or right-handedness in this context

I replaced a short reference to handedness in the lead by this sentence:

"In the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed (or right-chiral and left-chiral), respectively. However, the choice of orientation is independent on the handedness or chirality of the bases (although right-handed bases are typically declared to be positively oriented, they may also be assigned a negative orientation)."

This sentence may not be perfect, but the previous one seemed to be much more questionable. Paolo.dL 20:42, 26 August 2007 (UTC)

[edit] Difference between chirality and handedness

From handedness (disambiguation):

Handedness can refer to different things.

I thought that chirality (mathematics) and handedness (mathematics) were synonyms. For instance, I really cannot see the difference between the chirality of a basis set and the handedness of a basis set. However, I see that in Wikipedia there are several articles somehow referring to the two concepts, and in none of them there's an explicit description of the difference between them.

Etimology. There's no difference between the two words, from an etimological point of view. The prefixes chir- (greek) and hand- (english) mean exactly the same thing.

I suppose that the meaning of the words depends both on the context and on the "object" which they refer to (spiral, screw, basis set, subatomic particle...). Namely:

  • Chirality and handedness (of a basis set on a vector space) are synonyms in the context of linear algebra. The above copied disambiguation says that also orientation is a synonym of handedness, in this context, but I doubt this is exactly true (orientation is only positive or negative...)
  • Chirality and handedness (of geometrical objects in general) are also synonims in geometry and physics (where orientation means for sure something else!)
  • Left- or right-handedness or chirality is only a special case of handedness or chirality, possible only in 3-D and only for
  1. some objects which have both a standard direction of translation along a given axis associated with a prescribed sense of rotation about the same axis (a screw, a propeller) or
  2. sets of directed and ordered objects (tern of vectors).
  3.  ? (is there something else?)
  • I mean, there are chiral or handed objects in N-D and even in 3-D that can neither be assigned a left- nor a right- chirality or handedness.

I am not familiar enough with this concept to give a final answer. However, whatever is the truth, in my opinion it should be made clear both in this article and in the article about chirality (mathematics).

The disambiguation pages Chirality and Handedness (disambiguation) should be also revised.

Do you agree? I need your opinion. With regards, Paolo.dL 21:25, 26 August 2007 (UTC)

[edit] Low-dimensional cases

I think the article would be more helpful to a neophyte if it had special discussions of the lowest-dimensional cases, where one can visualize what a choice of orientation means. Specifically, there should be discussions of the 2-dimensional and 3-dimensional cases. Ishboyfay 18:49, 16 September 2007 (UTC)

Having explored Wikipedia further, I now see that what's needed here is a citation to the "Orientation and handedness" section of the "Cartesian_coordinate_system" article. Ishboyfay 18:55, 16 September 2007 (UTC)

[edit] Should cite related article

The related article "Orientation (geometry)" cites this article. There should likewise be a citation in the opposite direction. Ishboyfay 18:49, 16 September 2007 (UTC)

[edit] Defining an orientation from scratch

Is it possible to define an orientation without using any "reference" to something that has chirality already? In other words, is it possible to describe to someone who knows no conventions what a chiral object looks like, say a right hand? Danielkwalsh 08:07, 22 September 2007 (UTC)

[edit] Λk V are not k-forms

In Alternate Viewpoints, Multilinear Algebra, it is stated that ω in Λn V is an n-form and can be evaluated on a basis of V. I think this is mixed up: We either want ω in ΛnV* or we want to know whether e1 ∧…∧en is a positive multiple of ω. 78.52.195.151 (talk) 15:06, 7 February 2008 (UTC)