Talk:Orientation (vector space)

message: 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)
thread_title:  Left- or right-handedness in this context 

message: From handedness (disambiguation):
:Handedness can refer to different things.
:*handedness of human beings
:*in mathematics, handedness is a common synonym for orientation or chirality  —Preceding unsigned comment added by Paolo.dL (talk • contribs) 21:42, August 25, 2007 (UTC) 
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 
:#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
:#sets of directed and ordered objects (tern of vectors). 
:# ? (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)
thread_title:  Difference between chirality and handedness 

message: 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)
thread_title:  Low-dimensional cases 

message: 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)
thread_title:  Should cite related article 

message: 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)
thread_title:  Defining an orientation from scratch 

message: 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)
thread_title:  &Lambda;<sup>k</sup> V are not k-forms 

message: Clockwise (CW) and counterclockwise (CCW) is a concept which can be defined in 2-D, without the need of a defined "point of view" (or "normal vector") in 3-D. 
Do you know Flatland, the 2-D universe? People in Flatland cannot see the third dimension, but they can see the difference between {1, i} and {i, 1}, provided the two frames are drawn in their flat universe! Indeed, they can see that no (2-D) rigid rotation can "match" the two frames (i.e. put the two frames in contact with each other along both axes). They completely ignore the existance of the third dimension, and cannot think of a rotation in 3-D (e.g. a "rotation about the real axis"). But they can tell you the two frames have a different orientation, as well as we can state, in our 3-D universe, that our two hands have a different orientation, without reference to a fourth dimension. This is also correctly stated in the very first sentence of the introduction of our article: "orientation is a geometrical notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise". And this means that it is not a matter of apparence. 
Stated more simply: You don't need the third dimension to see the difference between three o'clock and nine o'clock (provided the hour hand is shorter than the minute hand, of course). So, people in Flatland can use a clock.
Paolo.dL (talk) 10:50, 22 January 2011 (UTC)
replies: text: How would you define clockwise in R2? Ozob (talk) 15:55, 22 January 2011 (UTC)
thread_title:  Example: CW and CCW orientations on Flatland 

message: My main point in this whole discussion is that the orientation of a basis cannot be totally defined by "its sign" (or the sign assigned to the consistently oriented equivalence class, if you prefer). Adam can tell Eve that the basis lives in a positively or negatively oriented equivalence class, but this is not enough for Eve to deduce whether the basis is CW or CCW. This is the concept explained in the removed text about "non-numerical information" (see above), which in my opinion was insightful and useful, although improvable.
And the idea that the existance of a "natural orientation" can solve this problem in complex space appears illogical to me, since Ozob repeated that there are two possible "natural orientations". For instance, even if a given orientation on the complex plane is natural, I cannot see how you can deduce from its sign whether a consistently oriented basis (e.g. the standard basis) is CW or CCW.
In this discussion, there might have been a terminological misunderstanding. Perhaps, you are used to call differently (possibly, "chirality of a basis") what I called, and the article calls, in the introduction, the "orientation of a basis" (right-handed or left handed, CW or CCW). 
— Paolo.dL (talk) 20:30, 4 February 2011 (UTC)
thread_title:  Orientation of a basis not totally defined by its sign 

message: I think it is correct to consider a manifold like a special vector space. From wikipedia:
:" manifold is a mathematical space that on a small enough scale resembles the Euclidean space"
Therefore a manifold can be considered as a space in the mathematical sense. About the vectorial character, from a mathematical point of view the elements of a manifold satisfy the vector space definition. Therefore manifolds can be considered as an special case of vector space with a specific metric tensor.
:--Juansempere (talk) 18:10, 14 February 2011 (UTC)
replies: text: An orientation on a vector space is not the same as an orientation on a manifold. An orientation on a manifold is a continuous choice of orientations on the tangent spaces; each of those orientations is an orientation of a vector space. Some manifolds cannot be oriented; this is because of their topology (it has nothing to do with a Riemannian metric).  All that means is that there is no way to continuously choose orientations on their tangent spaces. E.g., this happens for the Möbius strip. But it cannot happen for a vector space because the vector space has no interesting topology. Ozob (talk) 00:49, 15 February 2011 (UTC)
thread_title: Manifolds as vector spaces

message: What, if any, is the difference between an "orientation form" and a choice of determinant function? As far as I know, a determinant function on V is nothing more than an element of Λn(V*), in other words, a linear form on Λn(V) (due to the canonical isomorphism between Λn(V*) and Λn(V)*.) What is the point of introducing another term "orientation form" that sounds strictly weaker, when all you are really talking about is a determinant function on V? Lit-sky (talk) 21:33, 23 August 2012 (UTC)
replies: text: There are a number of answers to this, the essence of which being that they inherently distinct concepts. Even if in some contexts the two might be directly related, this relationship is by no means natural in a general context. For example, a determinant on a linear transformation V→V may always be defined for V of finite dimension (i.e. even without a metric tensor), whereas its correspondence with a volume form requires a metric tensor (making your definition incorrect in the broader context). An orientation form is also an n-form (a type (0,n) tensor), whereas a determinant is a non-linear function V⊗V∗→K (K being the underlying field for V), which can in general at best be identified with a type (n,n) tensor. — Quondum☏ 07:00, 24 August 2012 (UTC)
replies: text: I'm not familiar with the usage of the term "tensor" for elements of exterior powers -- exterior powers are quotients of tensor powers, not subspaces, so how does this terminology make sense? I don't know what a metric tensor is. This article is about linear algebra and I am talking about vector spaces, not manifolds. I am aware that in the finite-dimensional case, the determinant of a linear operator V→V can be defined and is basis-independent. My definition of a determinant-function on V is an alternating, n-linear form on V. That is, it takes in n vectors from V (not an endomorphism of V) and produces a scalar. My point is that the article defines an orientation form as an element of Λn(V)* which is certainly the same thing as an alternating n-linear form on V, i.e. a determinant function on V. Lit-sky (talk) 18:55, 24 August 2012 (UTC)
replies: text: One can accommodate/define an exterior algebra within as a subspace of a tensor algebra without using quotients; however, it does not make sense to refer to tensors for illustration/argument if that is not familiar ground. My objection does not arise with n-forms, but with the description you use of a determinant: "a determinant function on V is nothing more than an element of Λn(V∗)". Perhaps you could point me to where this (IMO incorrect) claim is made? (Why on V? It is on V⊗V∗.) BTW, an orientation form is not strictly weaker than a determinant: the determinant does not define an orientation on Λn(V∗), whereas an orientation form does. — Quondum☏ 21:13, 24 August 2012 (UTC)
thread_title:  Orientation forms and determinants