Talk:Orientation (vector space)

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)

Difference between chirality and handedness

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

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)

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)

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)

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)

Λ^k 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 ∧…∧e_n is a positive multiple of ω. 78.52.195.151 (talk) 15:06, 7 February 2008 (UTC)

Non-numerical information

I have moved the following here for discussion. I'd rather not throw around phrases like "original research", but this is clearly something that would benefit from some references. Apart from seeming to be speculative (the phrase "it is impossible..." appears three times with no real justification), I don't think it is entirely accurate, or at least not complete. For instance, any complex vector space comes with an orientation for free. I think the orientation here is described "only with numbers" (those numbers happen to be complex numbers). Similarly, any vector space over the quaternions also carries a natural orientation. The paragraph seems to be written from a purely phenomenological perspective, in describing the naive geometry of human experience. To a certain extent, perhaps it is fair to extrapolate this idea to orientation more generally. But I don't quite see precisely how to do this, in light of the observation that there are vector spaces that do carry a natural orientation. I think to be on the safe side, we should insist on a reference before it is restored. Sławomir Biały (talk) 19:19, 16 January 2011 (UTC)

Non-numerical information

It is impossible to define the orientation on a vector space without referring to the orientation of a basis, and hence to the orientation of a coordinate system. In turn, it is impossible to unequivocally describe the orientation of a coordinate system only with numbers. For instance, the definition of a right-handed Cartesian coordinate system must be based on non-numerical information, such as the shape of the right hand of a human being, or the combined concepts of down-up, South-North, and West-East, or down-up, left-right, and backward-forward. Similarly, it is impossible to explain which is the right hand or side of a human being without either showing it, or referring to other non-numerical knowledge, such as the definition of a Cartesian coordinate system. This is also true for the definition of the clockwise and counterclockwise senses of rotation, which cannot be only based on numbers, as it is related to the definition of left-handed and right-handed coordinate systems (in a right-handed coordinate system xyz, counterclockwise is the sense of the shortest rotation from axis x to y, as seen from a point in the positive z semiaxis).

I should also add to the above critique that, even if one raises some objection about the complex numbers (e.g., that the orientation is not invariant under the Galois group), then there are still natural orientations that don't require any appeal to oriented bases. For instance, any phase space from classical mechanics carries a natural orientation as well. It's not clear how someone defending the above text would address this, but it does seem to be rather misleading as it stands. Sławomir Biały (talk) 20:25, 16 January 2011 (UTC)

I don't think the above paragraph is intended to be a mathematical statement in any ordinary sense. I think it's an interesting point, but perhaps not very clearly expressed. For example, consider the natural orientation on the complex numbers dx∧dy. We always think of this as counterclockwise orientation because we always draw the complex numbers with the positive real axis pointing right and the positive imaginary axis pointing up. But there's no reason why right and up are positive. We could just as well declare the positive real axis to point left or the positive imaginary axis to point down, and if we do either one (but not both) of these, then the natural orientation appears to reverse itself; it's now clockwise. Of course, the natural orientation hasn't changed (it's still dx∧dy) but our perception of it has. I wonder if orientation—in the philosophical, not mathematical sense—might be one of those primitive, irreducible concepts that can't really be explained. If that's the point the paragraph is trying to make, it's interesting. It would be nice to have a source for that. Ozob (talk) 02:39, 17 January 2011 (UTC)

Agreed. Sławomir Biały (talk) 18:58, 17 January 2011 (UTC)

I think the concept is extremely interesting. How would you explain to an alien the orientation of a vector space without showing or referring to the shape of some 3-D object? Is there another way? Just asking this question is thought provoking and useful to better grasp the concept of orientation. It is also interesting to understand that the concept of clockwise-counteclockwise is related, and equally difficult to decompose into more elementary concepts. Also, how can you define the third axis of a 3-D Cartesian coordinate system (or second axis of a 2-D system), without showing, selecting somehow, with some non-numerical method (e.g. the right hand rule), the relevant semispace (or semiplane)?

The section requires a gentle introduction. For most readers, an explanation limited to the 3-D Euclidean space would suffice. A new subsection could be added to explain the concept of natural orientation for other kinds of vector spaces. Paolo.dL (talk) 23:19, 17 January 2011 (UTC)

How would you explain to the layman the concept of natural orientation? Is there a simple way? I know nothing about this concept, and can't help you. Paolo.dL (talk) 21:31, 18 January 2011 (UTC)

An orientation is an arbitrary choice of handedness. It specifies which side of a mirror you're on: If you're on the same side as me, then we have the same orientation, but if you're on the opposite side, then we have the opposite orientation. We can distinguish the two by taking ordered bases and computing the sign of the determinant of the change of basis matrix. The choice of basis, however, is arbitrary, and usually no basis is better or worse than any other. But occasionally that's not true. For example, consider the complex numbers (or more generally almost complex manifolds). The real reason (if you'll pardon the pun) for the natural orientation is that the real and imaginary axes give us a non-arbitrary basis. Because our vector space comes with a non-arbitrary basis, we can use that basis to define an orientation! Usually we orient things so that the real axis comes first when you try to apply the right-hand rule and the imaginary axis comes second, but this is arbitrary and not really relevant to why the orientation is called natural. Note, for example, that we get the other orientation on C by flipping this choice. (It's more complicated in higher dimensions; if you flip all the choices and the manifold has dimension divisible by four, then nothing happens; if the dimension is 2 mod 4, then the orientation flips.) Ozob (talk) 02:54, 19 January 2011 (UTC)

Thank you for your explanation. However, I can't see the logic of this sentence of yours: "the real and imaginary axes give us a non-arbitrary basis". You need an ordered basis, and to build it you need:

1. An order
2. A decision about the direction of both axes

How does the mere fact that an axis is real, and the other imaginary, entail as a logical consequence the choice of 1 and 2? As you wrote, it may suggest the real-imaginary order because of a widely used convention, but a convention is just an arbitrary choice accepted by group of people. It is not a logical consequence. And as you wrote above it may suggest a conventional direction (first axis to the right, second axis upward), but this is also arbitrary. In short, how can "the real and imaginary axes give us a non-arbitrary basis"? Of course, to define an orientation, we don't really need the absolute direction of both axes, but the direction of the second with respect to the first. But how can can the real and imaginary axes give us a non-arbitrary orientation?

You also wrote: "Note, for example, that we get the other orientation on C by flipping this choice." This seems to confirm that the "natural" orientation is arbitrary. Am I missing something???

What is the meaning of the word "natural"? If you define "natural" as "most common", then even a "right-handed" frame in R³ is naturally oriented. But I am sure this is not what you and Sławomir Biały meant, so, I am puzzled. To obtain an useful definition of "natural", we need to prove whether or not it is based on some arbitrary initial assumptions or axioms.

— Paolo.dL (talk) 21:06, 20 January 2011 (UTC)

Ah ha, you caught me!

As you've noticed, it really depends on what we mean by "natural". There's a technical definition, natural transformation. We can shoehorn the current discussion into that framework (I'll say how in a moment), but in a way that doesn't really capture the spirit of things. We're using "natural" in a somewhat vague way. It means something like, "There are no arbitrary choices that affect to the situation at hand." So as I noted above, we need an order, and the choice of order is arbitrary. But we make this choice once for the complex numbers, and then we don't have to choose anything more for complex vector spaces or complex manifolds or almost complex structures or anything else; and the particular choice that we make doesn't affect anything else we do.

Of course, you've probably just noticed that the last claim isn't precise, either; I've just shifted my ambiguity about the meaning of "natural" onto the ambiguity of "anything else we do". Getting around this requires using natural transformations. I don't know how familiar you are with them, so the following may not make much sense. Also I'm coming up with this off the top of my head, so no guarantees that it's correct. Let I be the identity functor on the category of n-manifolds and local diffeomorphisms, and let O be the orientation double cover functor. This means that for a manifold M, OM is the set of all pairs (x, o) where x is a point of M and o is an orientation at x; and for a map f between two manifolds, Of maps OM to ON by sending (x, o) to the pair consisting of f(x) and the pushforward of o (this is why we're working with n-manifolds and local diffeomorphisms). I and O is a functor from our category to itself. There is no natural transformation I → O, meaning that the choice of orientation is not natural. But! If we instead work in the category of almost complex manifolds, then there is such a natural transformation. That natural transformation is the natural orientation. In fact there are two such natural transformations, one for each of the natural orientations.

Does that help? Ozob (talk) 22:33, 20 January 2011 (UTC)

Example: natural orientation for complex numbers

Sorry, it does not help. Especially your last sentence! We need to discuss the simplest possible example. Would you mind to start from the complex plane? I am so confused that I cannot even answer this simple question: do we have a natural orientation for this frame? This is what you seemed to state above. You wrote: "For example, consider the complex numbers...". Let me state it in a simpler way: If I impose that the real axis points to the right, is there a "natural direction" for the imaginary axis? If a complex plane does not have a natural orientation, as I believe, what's the simplest example in which a natural orientation exists? Paolo.dL (talk) 23:51, 20 January 2011 (UTC)

No, there is no natural direction for the imaginary axis. There are two natural directions: i and −i. This is why I said above that there are two natural orientations and two natural transformations. Put another way, both the possible orientations on the complex plane are natural, but neither of the possible orientations on R² is.

You can get an orientation as follows: Choose your first basis vector to be 1. Then pick any complex number z which is not collinear with the real axis. In other words, z is in C − R, which is a union of two open half-planes. If z's imaginary part is positive, then you get the same orientation as if you picked z to be i, and if it is negative, then you get the same orientation as if you picked z to be −i.

Remember that there are many ways of defining the complex numbers. For example, they are the field extension R[z]/(z² + 1); they are also a certain set of 2×2 matrices; they are also the algebraic closure of R; etc. All of these ways are isomorphic as fields. All these isomorphisms preserve orientation: The real axis is sent to the real axis and i is sent to i (and −i to −i and z to z). Because these fields are also one-dimensional complex vector spaces, there are also vector space isomorphisms between them. These are all equivalent to multiplying by a non-zero scalar and applying a field isomorphism. Multiplying by a non-zero scalar leaves the real axis and the imaginary axis in the same relative position and direction as when they started: By writing the non-zero scalar in polar form, we see that it is a scaling (which doesn't change the real and imaginary axes at all) and a rotation (which transforms them both in the same way).

The complex numbers are also a two dimensional real vector space. There are many isomorphisms of the complex numbers as a two dimensional real vector space which are not isomorphisms as a complex vector space. For example, complex conjugation is one of these. This leaves the real and imaginary axes in the same location, but it switches the direction of the imaginary axis. So the two axes do not end up in the same relative direction: If, before conjugating, you turned one direction to go from 1 to i, you now after conjugating turn the other direction. Therefore complex conjugation changes orientation.

The natural orientation is natural because when we are working with complex vector spaces, all of the linear transformations preserve the relative positions of 1's and i's: If you choose a copy of C embedded in your source vector space, and if your linear transformation maps onto another copy of C (and not zero), then the relative position of 1 and i in the source copy of C is the same as in the target copy of C. This means that orientation is preserved (no matter what orientation you started with). If you tried to do this with a copy of R instead, then you would not have an i available, so you would not have a second basis vector to work with. You are working with less information, and in fact it is not enough information to guarantee that orientation is preserved (as the example of complex conjugation shows).

How's that? Ozob (talk) 00:40, 21 January 2011 (UTC)

Great job, thank you. I still have huge doubts, though. For instance, in your first contribution, you wrote: "We could just as well declare the positive real axis to point left or the positive imaginary axis to point down, and if we do either one (but not both) of these, then the natural orientation appears to reverse itself; it's now clockwise. Of course, the natural orientation hasn't changed (it's still dx∧dy) but our perception of it has." How can you call this just a "perception"? It is a perception when you observe the rotation from the other side (e.g. when you observe a clock from behind), but in your example you have actually, not just apparently, defined a different orientation. In other words, the opposite rotation sense, not the opposite point of view.

By the way, since I know little about Category theory, Manifolds, etc., I dare to insist only because I assume most readers are as ignorant as I am, and your goal is to find an elementary way to explain the concept of natural orientation in the article. To everybody, not just to me. Paolo.dL (talk) 20:19, 21 January 2011 (UTC)

Well, when I wrote that statement, I was thinking of how we usually draw complex numbers. Most often, we draw a long array pointing right for the real axis and a long arrow pointing up for the imaginary axis. Furthermore, "the" natural orientation is usually taken to be the one given by the basis {1, i}. There isn't a good reason for that, except maybe human nature liking real numbers better than imaginaries (so 1 before i) and positive-looking numbers better than negative-looking numbers (so i and not −i). When we take that orientation and draw it in the usual way, then we get the counterclockwise orientation. But there are other ways of drawing the same mathematical data: If I draw a long arrow pointing right for the real axis and a long arrow pointing down for the imaginary axis, then now the natural orientation appears to be clockwise. The orientation is the same object; the switch from counterclockwise to clockwise is mostly in our minds. It happens because we have an outside frame of reference, namely reality. Clockwise and counterclockwise mean something in reality. We relate the orientation of a plane to an orientation of reality by taking a normal to the plane and looking at the orientation we get using the orientation of the plane and the normal. Which normal we take determines which orientation we get on space, even though we started with the same orientation on the plane. Put another way, if we rotate the plane 180^ˆ around the real axis but use the same normal vector, then we get the same plane but it determines the opposite orientation of space.

Regarding category theory: It's mostly just a convenient language. I could (with effort) reduce the existence of a natural orientation down to more elementary language. The article would be better by a more elementary description than the one I gave. The manifold stuff is more appropriate at orientability. But I didn't know your background; your user page is even emptier than mine! Ozob (talk) 02:42, 22 January 2011 (UTC)

Thank you again. However, basically you just repeated what you wrote above, and better explained part of what I wrote above (about the "opposite point of view" in 3-D). Thus, your point still does not make sense to me. Paolo.dL (talk) 10:50, 22 January 2011 (UTC)

Example: CW and CCW orientations on Flatland

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)

How would you define clockwise in R²? Ozob (talk) 15:55, 22 January 2011 (UTC)

Interesting question, which brings us right at the start of our discussion. Clockwise (CW) in 2-D or right-handed in 3-D can be defined only "non-numerically"! See the text removed by Sławomir Biały. There's no way to give a definition without "showing an example" somehow, or referring to other similar concepts that in turn cannot be defined numerically, but only by showing an example (e.g. cardinal directions). These are "pre-numeric" concepts. Something that simply cannot be defined by combining other notions, as they are the logical foundation for any other definition we are using in this context. How would you define CW or right-handed to an alien? Think about it, and you'll conclude exactly what is explained in the removed text. But I would also like to know if you agree that, contrary to what you wrote above, there are actually, and not only apparently, two possible orientations in the complex plane.

Let me partly correct what I wrote above: the first sentence in our article is not 100% correct. It is true that in 2-D (e.g. Flatland) you can define two orientations and call one of them CW. And it is true that, after defining CW, you will always be able to tell CW from CCW, without reference to the third dimension. But actually CW is conventionally defined only in 3-D. I mean, a clock is a 3-D object. When you project it on a plane (Flatland), you are free to observe it from the other side. There's no way to tell the front and back side of a plane (Flatland), as even if the plane is xy, you are free to point the z axis one way or the other. A man on one side of Flatland would say "this is CW from my point of view", another one on the other side would say "this is CCW from my point of view". Thus, I conclude that there's no conventional method to define CW for an abstract plane (of course, CW is conventionally defined in some specific cases, e.g. if the plane is defined by the surface of a non-transparent wall, as only a crazy man would hang a clock face toward the wall, ...). However, you can arbitrarily define CW on a given plane (e.g. Flatland), with or without reference to the third dimension. After you define CW for that plane, CCW is actually, not apparently, defined as the opposite sense.

In 3-D, CW is also arbitrary, but at least there's a world-wide convention which defines it. This is because in 3-D you can tell the front from the back of a clock.

— Paolo.dL (talk) 02:28, 23 January 2011 (UTC)

Adam and Eve standing on the opposite sides of Flatland

This brings us back to natural orientations. Suppose that we are looking at C. Let's imagine that C is Flatland for a moment. You are standing on one side of C and I am standing on the other. You look down through the plane and see me. You say, "Hello there! Which direction do you think clocks go?" And I say, "Clockwise!" You ask, "But what does that mean to you?" I gesture and say, "Do you see those points marked zero, one, and i?" You nod. "Of course." I continue, "To me, clockwise means, 'The direction we travel if we go in a circle around zero, taking the shortest path that visits i first and then one.' But what does clockwise mean to you?" "The same! But what would we do if we didn't have zero, one, and i here to help?" "We'd have to agree on something in advance. If we were standing on R² instead, we'd have zero, but we wouldn't have any other landmarks. So the best we could do would be to call one direction clockwise and hope it caught on."

I don't think I'm saying anything really different from what I've said before. Besides the question of natural orientations, I think we are saying mostly the same thing but in different words. Ozob (talk) 03:10, 23 January 2011 (UTC)

I am sorry, unfortunately your conclusion does not make sense to me. But thank you for exploring the example of Flatland. I am curious to understand your point, and discussing this example will be very useful, as it is simple enough for us to find out the reason why we disagree.

Since CW is just an arbitrary label, and defining CW is just a terminological decision, let's call Adam and Eve the two persons standing on Flatland (in the Book of Genesis they are described as those who gave names to everything in the world). Here's my interpretation of your example:

1. I am sure you agree that CW is arbitrary, so two persons standing on the opposite sides of Flatland, and even two persons on the same side may disagree about the definition of CW.
2. Let's say Adam decides he will use Eve's arbitrary definition of CW. CW on Flatland becomes a convention between Adam and Eve.
3. Eve can show Adam his own definition of CW with a practical example. She doesn't need axes for +i and +1! She can just draw an L, or a curved arrow.
4. But if she decides to use +i and +1, she first must draw on Flatland two axes to represent them. Most importantly, she needs to arbitrarily select a direction for +i, relative to +1! Again, this is a "pre-numerical" decision. This is because there are two arbitrary directions of +i, relative to +1, on Flatland.
5. It does not matter whether Adam and Eve are standing on opposite sides or on the same side. Even Flat people can tell the difference! In other words, if Eve didn't show Adam an example of her preferred orientation of the frame, Adam might decide to imagine the opposite orientation (i.e., the opposite direction of +i relative to +1)!
6. If Flatland were R², rather than C, Adam and Eve would have to follow a similar method to reach an agreement about CW. For instance, Adam might say to Eve: "Can you see these two axes x and y, that I drew? Tell me whether you want to arbitrarily define CW as the rotation from x to y or vice versa, and I'll accept your arbitrary decision".
7. I conclude that, both on R² and C, there are in my opinion two possible arbitrary orientations of the frame, and contrary to what you wrote, they are actually (not apparently) different from each other.

It is crucial for me to know if you agree about the last point, and also about the fact that this is the opposite of what you wrote above. Otherwise, I'll never be able to understand your point.

— Paolo.dL (talk) 13:07, 23 January 2011 (UTC)

I think that point (4) is where we start to disagree. You say, "But if she decides to use +i and +1, she first must draw on Flatland two axes to represent them." No! In C, i and 1 are already there. That is what is natural about this orientation. That is also why there is no natural orientation on R²: Nothing like i and 1 are already there.

Regarding your last point: There are two possible orientations of R² and of C, and they are actually different. I wrote that above in my very first reply to you; nowhere have I said otherwise. My later point was about the visual perception of these orientations. Neither of these orientations is a priori clockwise or counterclockwise. In order to identify either of them with the everyday meanings of clockwise and counterclockwise, we need to make an arbitrary choice of visual presentation. Because that choice is arbitrary, there is no way to identify either of the possible orientations as being "real everyday clockwise" or "real everyday counterclockwise". Ozob (talk) 14:42, 23 January 2011 (UTC)

Two possible orientations... but only one is possible

Good! We finally found the root of our disagreement! How can you say that "in C, i and 1 are already there", and at the same time maintain that there are two possible orientations on C? Let's be precise, please: are you saying, as it seems, that their relative orientation is "already there", and hence there's no need for an arbitrary decision by Adam or Eve to select an orientation? Do you realize how counterintuitive is this statement?

You wrote that, when you flip either i or 1, "the natural orientation hasn't changed... but our perception of it has". Although it did not make sense to me, this seemed to be consistent with your point, as it seemed to imply that there is only one possible orientation (i.e that the other one is just an illusion, somehow). But since you confirm that there are two possible orientations your statements remain (at least from my point of view) inconsistent! Either you are wrong (and I hope you are not), or there must exist a cleaner and more intuitive way or example to explain this concept.

Note: I don't really care right now that the two possible orientations are natural, as long as they are both possible! I will probably ask questions to search for an intuitively appealing definition of "natural" later. I did not forget your outstanding contributions above. I still have doubts, but I have read them several times with attention.

Paolo.dL (talk) 16:02, 23 January 2011 (UTC)

In terms of ordered bases, the two orientations are (1, i) and (i, 1). In terms of differential forms, the two orientations are dx∧dy and -dx∧dy, where x is the real coordinate and y is the imaginary coordinate (as functions, these coordinates send a point to its real part and imaginary part, respectively). In terms of matrices relative to the ordered basis (1, i), the two orientations are the class of $({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}})$ and $({\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}})$ modulo GL⁺(R²); in terms of matrices relative to the ordered basis (i, 1), the two orientations are $({\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}})$ and $({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}})$ modulo GL⁺(R²). In terms of the two-dimensional relative homology group H₂(R², R²−{0}), note that because R² is contractible, the long exact sequence in relative homology gives an isomorphism of the relative homology group with H₁(R²−{0}), and the two orientations are the classes of the paths e^2πit and e^−2πit. Ozob (talk) 20:25, 23 January 2011 (UTC)

Thank you. This is interesting, but how is this supposed to solve the above mentioned inconsistency? Let me explain: If you suggest to Adam and Eve to select (1, i), rather than (i, 1), without drawing the axes on Flatland, how are they supposed to understand the relative orientation of the two axes? And without that, how are they going to understand your definition of CW? I'll list a few statements. Please try to understand them all, and tell me if they make sense to you:

1. The purpose of our discussion is to non-ambiguously define CW for both Adam and Eve.
2. The order of the elements of the ordered basis (1, i) is not enough to define the orientation (CW or CCW) of the basis. You also need relative direction.
3. The orientation of a basis is not the same as the orientation on a vector space (the first is "pre-numerical", the second is based on the first).
4. We are interested here in the orientation of a basis (see title of this section).
5. We arbitrarily decided to call this orientation CW. We agree that the definition of CW is arbitrary, but we don't care. We want a conventional, non-ambiguous definition.
6. We need "non-numerical" information, i.e. we need to show an example. There's no other way.
7. Similarly, in R², if you say you select (x, y), rather than (y, x), this is not enough to non-ambiguously define CW.

Here are my questions, based on what I wrote above:

Q1 How can the axes (1, i) be "already there"?

Q2 Do you understand how counter-intuitive is (at least for me) your statement that they are "already there"?

Q3 Is your statement wrong? If not, then what invisible (and for me inimaginable) force of nature or logic deduction could fully determine their relative orientation on Flatland?

Q4 How can you give a non-ambiguos definition of CW?

Q5 What's the difference with (x, y), as far as the non-ambiguous definition of CW is concerned?

— Paolo.dL (talk) 23:47, 23 January 2011 (UTC)

I'll reply first to your statements.

1. True.
2. False. The order is the relative direction.
3. False. There is no such thing as an orientation of a basis; it must have an order. Even then, orientations are only defined up to orientation-preserving (which means by definition positive determinant) change of bases. Orientations are objects on vector spaces, but, similarly to how congruence classes have representatives, orientations have representatives which are ordered bases.
4. I guess so; but such a thing does not exist.
5. If you are referring to the ordered basis (i, 1), then yes.
6. I do not know what you mean by "non-numerical". One of the definitions of the natural orientation on C that I gave above involved no numbers, namely the one using homology.
7. I do not know what you mean when you write (x, y) and (y, x). Could you use one of the definitions on the article page?

Regarding your questions:

1. C is the complex numbers, so it has by definition 1 and i.
2. Clearly not.
3. No. 1 and i exist in C and are linearly independent over R; therefore they determine an orientation.
4. For the case of C, I gave many such definitions above. For R², there is no such definition because there is no natural orientation.
5. I do not know what you mean by (x, y).

I think you might be well served by attempting to put your thoughts into the language used by the article. It may look technical, but it is really quite easy once you get the hang of it. Ozob (talk) 01:52, 24 January 2011 (UTC)

At least it is clear now to which (huge) extent we do not understand each other. Have you read the introduction of the article? I used the language of the article when I wrote there is a difference between the orientation of a basis (e.g. CW or right-handed) and the orientation on a vector space (positive, associated with the arbitrarily selected, and hence declared and known, orientation of a standard basis). You agreed that we are discussing about the definition of CW, and even asked me about it. And in the "language used by the article" CW is the orientation of a basis, not the orientation on a space!

"Non-numerical" refers to the concepts explained in the article section titled "Non-numerical information", which is the text we are discussing. Is this so difficult to understand? It means you cannot use vectors to define the orientation of a basis. E.g. the versors (1,0) and (0,1) of a 2-D standard basis do not change when you change the orientation of the basis, so you can't use them to define the orientation of the basis. This should be obvious to you. The definition of the orientation of a basis might be called "non-vectorial".

I used (x, y) to indicate a sequence (ordered set) of two Cartesian axes x and y which provide the orientation of a standard basis for R². Sorry if my notation was improper. Can you understand now what I meant?

— Paolo.dL (talk) 09:59, 24 January 2011 (UTC)

I just read the article carefully again. Nowhere does it talk about unordered bases, except perhaps implicitly when it says that the ordering of a basis is crucial. It never defines the "orientation of a basis", and I stand by my claim that there is no such thing. I believe I am being trolled, so I will not continue this conversation any further. Ozob (talk) 11:51, 24 January 2011 (UTC)

Why are you mentioning "ordered bases" now? We have always agreed that the basis must be ordered. There's no reason to discuss something about which we have always agreed. You are not being trolled, you are just assuming bad faith because you did not read the article with attention. The article says, right in the introduction: "In linear algebra, [...omissis...] the orientation of an ordered basis is a kind of asymmetry...". If CW is not the orientation of a basis, then what is it, in your opinion? According to the article, the orientation on a vector space is either positive or negative (which, in R² does not imply CW or CCW, and in R³ does not imply right-handed or left-handed).

I really can't understand the reason why this idea is for you inacceptable, as being aware of the orientation of the standard basis (and hence of the corresponding Cartesian coordinate system) is extremely important in 2-D and 3-D Euclidean space, e.g. in 3-D computer graphics.

— Paolo.dL (talk) 15:02, 24 January 2011 (UTC)

Orientation of a basis not totally defined by its sign

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)

Manifolds as vector spaces

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)

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)

Orientation forms and determinants

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)

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)

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)

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)

Why is orientation a boolean?

While the idea of orientation generally makes sense, it is not intuitively obvious to me when describing high dimensional objects that there should be at most two choices of orientation. Why is this so? Is this ultimately a consequence of the fact that the only finite non-trivial subgroup of the real numbers under multiplication is $\{+1,-1\}$ ? Woscafrench (talk) 13:21, 21 December 2015 (UTC)

Yes. Recall the setup of the article: If V is a finite-dimensional vector space and b₁ and b₂ are ordered bases, then there is a unique linear transformation A of V that sends b₁ to b₂. Therefore A is an element of GL(V). The article treats the case where V is a real vector space, so GL(V) is isomorphic to GL(n, R) for some n. So you ask yourself: What kind of information can I extract from GL(n, R)? Well, it's a Lie group. All Lie groups are extensions of discrete groups by connected groups (Proof: If G is a Lie group with connected component G^o, then G is an extension of G/G^o by G^o, and one can check that G/G^o is discrete). And in the case of GL(n, R), the connected components are the positive determinant matrices and the negative determinant matrices. It's easy to see that there are at least these two connected components because the preimages of (0, ∞) and (-∞, 0) under det must be disjoint. And it can be shown that these two sets are connected because you can draw a path from any positive determinant matrix to any other. (Proof: Consider the row operations of multiplication by a scalar and adding a scalar multiple of one row to another. Using these operations, any matrix can be reduced to a permutation matrix. These operations can be turned into a path from the initial matrix to the permutation matrix by shrinking the parameters of the row operation to 1 (for a scalar multiplication of a row) or 0 (for an addition of a scalar multiple of one row to another). The permutation matrix has positive determinant if and only if the permutation is even. Such a permutation can be written as a product of pairs of transpositions. One can write down an explicit path from a matrix representing a pair of transpositions of basis vectors to the identity matrix. So the permutation matrix of interest can be reduced to the identity by concatenating some explicit paths. This proves that the positive determinant matrices are connected; hence the negative determinant matrices are also connected because they all arise from positive matrices by flipping the sign of the first row.) Therefore GL(n, R) / GL(n, R)^o = Z/2Z is the group of orientations. And this group is really the same as sign because the quotient factors as (Im det) / (Im det restricted to positive determinant matrices), and that's R^× / R⁺.

This can all be generalized if you have a topological field k. Then a generalized orientation of a k-vector space is an element of GL(n, k) / GL(n, k)^o. Since GL(n, C)^o = GL(n, C), there is no such thing as a complex orientation. Since GL(n, Q_p)^o = 1, there is such a thing as a p-adic orientation, but it's too fine an invariant, because matrices are in one-to-one correspondence with their p-adic orientations. So over most common fields that are not R, orientation is not a useful concept. Ozob (talk) 17:39, 21 December 2015 (UTC)