Talk:Versor

message: According to the classic book Clifford Algebra to Geometric Calculus (CAGC), a versor is a product of vectors, and those vectors need not be invertible. Some texts, such as Geometric Algebra for Computer Science (GACS) require a versor to be invertible. However, the non-invertible concept is more general, and more useful, since many results hold for these 'k-products' of vectors, rather than just for invertible ones. A case in point is the formula for the contraction over a geometric product of vectors. Unit-versors are exactly the pinors, even unit-versors are exactly the spinors, and rotors are exactly the special spinors (GAGC has an erroneous claim here, which GACS propagates). I was hoping to get the non-invertible definition through as the most useful definition. What do you think?
--Kaba3 (talk) 01:41, 8 April 2013 (UTC)
thread_title:  Versors in Clifford Algebra 

message: ===This objection is a work in progress===
I think I remember when the article on versors was practically empty!  I want to complement you on the outstanding job you are doing.  I like the way you have noted hamilton's terminology!
This is just a draft objecting, I want to come back later and refine it.
===Nit picking a little on finer points of classical theory===
In classical notation as your outstanding article points out there is something called the tensor of a versor.
If the versor is written in quadranomial form it is just
 Tq  = \sqrt{w^2 + x^2 + y^2 + z^2} \,
The above operation always equals one for a versor. 
What hamilton called the common norm of a versor is could be written as
common norm = Nq = qKq
In other words the common norm is quaternion times its conjugate.  The 'common norm' and the 'tensor' of a versor were two completely different operations.  The common norm of a quaternion is equal to the square of the tensor of the same quaternion.
The common norm of a versor is also equal to one.
 
One approach I considered, but don't like very well was that since there is more than one 'norm' if you called the tensor of a quaternion the Euclidian-Hamilton norm, that would distinguish it from the 'common norm', as well as the corresponding norm operations in R^4
Along these lines terms used by both Hamilton in his classical theory and in other branches of math could be made less ambigious by affixing the word Hamilton in front of them.
For example of this nomenclature consider the section that you link to the Euclidian norm which you might consider writing  a formula identical to that of the 'hamiltonian tensor' of a quaternion.
By affixing the term hamiltonian to the word tensor I mean the an abreviation of what hamilton called a tensor.
===Your treatment of modern quaternions===
Your problem is confounded by the fact that they keep changing the names of the operations on quaternions on the main page.  Thus leading to some major ambiguity.
While the classical quaternion point of view is very well defined, and can be well documented, and its definitions of every word are very precise, the same is not true about modern thinking on the subject.
Sometimes they are calling the classical tensor, the modulus, and sometimes the absolute value.  
Eucledian norm is an operation on real vector spaces and the vector space associated with H is three complex numbers C3. So it is an operation on a different space, and therefore not identical to any operation on H  
On the other hand the word tensor has been taken over these days by a whole branch of math.  So using the word tensor, for the euclidian norm would confuse your readers, in your introductory paragraph.  
Hobojaks (talk) 03:51, 21 July 2008 (UTC)
replies: text: You are so right about tensor now being a part of multilinear algebra and the term is hardly suitable for modern quaternion study. The article mentions norm as appropriate for those developing the metric space concept. 
The whole idea of rethinking the quaternion as some type of normed vector space is a point of view that needs to be documented in terms of the versor concept. 
However some of these view are very much non-classical views. 
I can provide you with some good references to material by writers who held this view, of which one of the important early ones is Gibbs.  Gibbs book vector analysis devotes considerable time to his view of what a versor is, but it is really really different from hamiltons.  What makes it interesting is that it is written by a writer well versed in classical quaternion thinking.
In my view the most interesting candidate corresponding to the idea of a norm in a normed vector spaces when it comes to quaternions is a version of the lorentz invariant.  In other words take the product of each individual element of quaternion with itself, keeping the i j and k.  You get 
w^2 -x^2 - y^2 - z^2 
Which as Rgdboer points out is the scalar part of the square of a quaternion. 
The thing is that the whole idea of a normed vector space to my way of thinking is pretty much quaternion negitive nomenclature.  
A classical quaternion has only one product, and since it is a quaternion and not a scaler, it is excluded from the definition of a normed vector space.  
It seems to me that the motivation behind normed vector spaces is to incorperate the idea of an angle into them, but every quaternion already has a built in angle characteristic, so elevating one characteristic of a quaternion above the others and bestowing the name norm on it so that you can define an angle, or for that matter define what you mean by distance seems a bit contrived.  
In my way of thinking, and of course this is just my opinion and probably could not be included on wikipedia, a quaternion has a number of quantities that could be thought of as norms.
(1)The distance idea to my way of thinking is the tensor of the vector of the quaternion.
This is the true eucledian norm.
(2)The lorenz invariant, as I am pleased to discover, in other words the four dimensional distance in space time is the scaler of the product of a quaternion with itself.
(3)The tensor of a quaternion, looks a lot like the norm of a real four space, so I can understand why you guys coming from the non-classical point of view that advocates remaking a quaternion into a normed vector space like it so much.
Once you start to extend the classical quaternion idea, depending on how far you go, you can't really be sure any more that the common norm is still equal to the square of the tensor.  
Hamilton proved this was true for a classical quaternion, however I was thinking that if you wanted to avoid quaternion negative nomenclature and felt that the term tensor was to obscure for most of your readers, may I suggest simply in the classical section of your article."Square root of the common norm".  
In another approach I was thinking of including the sentence:  The square root of the common norm of a quaternion is a positive definite tensor of rank zero.  This characteristic of a quaternion written Tq has been found so useful that in contexts where no other kinds of tensors are being discussed it is simply called "the tensor".  The term Hamiltonian tensor is preferable when discussing this characteristic in a mixed context to distinguish it from other uses of the word tensor.  A versor rotates any vector in its plane through its angle, but preserves its length.  A positive definite zero ordered tensor preforms an act of tension either streaching or shrinking a vector.  If you pick two arbitrary vectors, you can multiply the first by a quaternion consisting of the proper choice tensor and versor and the result will be the second.
These two operations are commutative.  In symbols.
qβ = Tq.Vqβ = Vq.Tqβ
Hobojaks (talk) 21:20, 25 July 2008 (UTC)
thread_title:  What is a 'norm' of a versor 

message: Your problem is confounded by the fact that they keep changing the names of the operations on quaternions on the main page.  Thus leading to some major ambiguity.
While the classical quaternion point of view is very well defined, and can be well documented, and its definitions of every word are very precise, the same is not true about modern thinking on the subject.
Sometimes they are calling the classical tensor, the modulus, and sometimes the absolute value.  
Eucledian norm is an operation on real vector spaces and the vector space associated with H is three complex numbers C3. So it is an operation on a different space, and therefore not identical to any operation on H  
On the other hand the word tensor has been taken over these days by a whole branch of math.  So using the word tensor, for the euclidian norm would confuse your readers, in your introductory paragraph.  
Hobojaks (talk) 03:51, 21 July 2008 (UTC)
replies: text: You are so right about tensor now being a part of multilinear algebra and the term is hardly suitable for modern quaternion study. The article mentions norm as appropriate for those developing the metric space concept. 
The whole idea of rethinking the quaternion as some type of normed vector space is a point of view that needs to be documented in terms of the versor concept. 
However some of these view are very much non-classical views. 
I can provide you with some good references to material by writers who held this view, of which one of the important early ones is Gibbs.  Gibbs book vector analysis devotes considerable time to his view of what a versor is, but it is really really different from hamiltons.  What makes it interesting is that it is written by a writer well versed in classical quaternion thinking.
In my view the most interesting candidate corresponding to the idea of a norm in a normed vector spaces when it comes to quaternions is a version of the lorentz invariant.  In other words take the product of each individual element of quaternion with itself, keeping the i j and k.  You get 
w^2 -x^2 - y^2 - z^2 
Which as Rgdboer points out is the scalar part of the square of a quaternion. 
The thing is that the whole idea of a normed vector space to my way of thinking is pretty much quaternion negitive nomenclature.  
A classical quaternion has only one product, and since it is a quaternion and not a scaler, it is excluded from the definition of a normed vector space.  
It seems to me that the motivation behind normed vector spaces is to incorperate the idea of an angle into them, but every quaternion already has a built in angle characteristic, so elevating one characteristic of a quaternion above the others and bestowing the name norm on it so that you can define an angle, or for that matter define what you mean by distance seems a bit contrived.  
In my way of thinking, and of course this is just my opinion and probably could not be included on wikipedia, a quaternion has a number of quantities that could be thought of as norms.
(1)The distance idea to my way of thinking is the tensor of the vector of the quaternion.
This is the true eucledian norm.
(2)The lorenz invariant, as I am pleased to discover, in other words the four dimensional distance in space time is the scaler of the product of a quaternion with itself.
(3)The tensor of a quaternion, looks a lot like the norm of a real four space, so I can understand why you guys coming from the non-classical point of view that advocates remaking a quaternion into a normed vector space like it so much.
Once you start to extend the classical quaternion idea, depending on how far you go, you can't really be sure any more that the common norm is still equal to the square of the tensor.  
Hamilton proved this was true for a classical quaternion, however I was thinking that if you wanted to avoid quaternion negative nomenclature and felt that the term tensor was to obscure for most of your readers, may I suggest simply in the classical section of your article."Square root of the common norm".  
In another approach I was thinking of including the sentence:  The square root of the common norm of a quaternion is a positive definite tensor of rank zero.  This characteristic of a quaternion written Tq has been found so useful that in contexts where no other kinds of tensors are being discussed it is simply called "the tensor".  The term Hamiltonian tensor is preferable when discussing this characteristic in a mixed context to distinguish it from other uses of the word tensor.  A versor rotates any vector in its plane through its angle, but preserves its length.  A positive definite zero ordered tensor preforms an act of tension either streaching or shrinking a vector.  If you pick two arbitrary vectors, you can multiply the first by a quaternion consisting of the proper choice tensor and versor and the result will be the second.
These two operations are commutative.  In symbols.
qβ = Tq.Vqβ = Vq.Tqβ
Hobojaks (talk) 21:20, 25 July 2008 (UTC)
thread_title: Your treatment of modern quaternions

message: The term evil tensor for the useful quantity we are discussing would not only be consistent with quaternion negative nomenclature, but helpful for remembering things because it starts with the letter e, helping people to remember that it corresponds with the idea of what is called in quaternion negative nomenclature the euclidean norm of real space.  If the first basis vector of quaternion space is thought of as an 'imaginary' vector, then you might explain to your non technical readers that tensors are terrifying numbers.  It would help them to rememberTq.  This might also be reflective of a commonly held view of students of multi-linear algebra when first encountering the tensor of a quaternion concept.
It would unify the concept basis vectors and square roots for the non-technical readers since it is consistent with calling the square roots of minus one purely imaginary numbers, and with the naming conventions that generate expressions like real number.
It would give an introduction to the article a little less technical tone, than using the convention of naming concepts after the people who invented them, as in Hamiltonian Tensor Norm, or Euclidean norm.
I find 'the norm' very confusing because this could also mean the Lorentz invariant or scaler of a quaternion.  Perhaps affixing some term with a negative connotation like satanic scaler would be in keeping with the traditional naming conventions of multi-linear algebra?  A more neutral term starting with the letter for Sq might be special realitivistic norm.—Preceding unsigned comment added by Hobojaks (talk • contribs) 12:10, 27 July 2008 (UTC)
thread_title:  Would the term 'evil tensor' be less confusing? 

message: A great need exists to explain the basic properties of a quaternion in a way people with out a phd in math can understand.
Below is an attempt that I started working on, but I am not really sure that I like it very much or if it would be allowed on wikipedia.
The main difficulty is finding words that are at the same time non-technical and non-ambigious.
Also this text is incomplete, because I quit working on it for today at the point were it took up the subject of the angle of a versor.  I notice that no one has commented on the technique.  —Preceding unsigned comment added by Hobojaks (talk • contribs) 02:10, 28 July 2008 (UTC)
thread_title:  Some ideas I have been having on the subject. 

message: Is a versor just another word for a member of SO(3)? —Ben FrantzDale (talk) 02:42, 4 December 2008 (UTC)
replies: text: Short answer: Yes. Caveat: versor is an antiquated term from a time before continuous groups were classified. The versor concept represents a seed-idea of Lie theory: group elements can be described as evolving out of the identity (1). Thus the close association with a great circle arc. The term drags behind it the Hamilton-algebra with its controversial allusion to four dimensions. One who considers using the term versor might do well to clear with the audience the admissablility of the meta-physical 4-space.Rgdboer (talk) 00:49, 7 December 2008 (UTC)

text: No. SO(3) is not a 3-sphere. You both probably need to read the Spin group article. Incnis Mrsi (talk) 19:34, 23 January 2014 (UTC)
thread_title:  SO(3)? 

message: Why is there is a very bold unbalanced closing parenthesis in the first formula of this article? It bothers me if this is an example of someone's real notation for something.  Please tell me it was a mistake and clean it up or enlighten me about why it could possibly make sense.  — Preceding unsigned comment added by 209.66.90.5 (talk) 23:55, 10 May 2012 (UTC)
replies: text: This is a standard notation, explained in Interval (mathematics).  — Quondum☏ 07:25, 11 May 2012 (UTC)
thread_title:  Very Prominent Unbalanced Parenthesis 

message: I created a separate article for the meanings of the term versor in geometry and physics. A disambiguation page already existed, and at the very beginning of the article we say this article is only about an "algebraic rotation representation". In geometry and physics a versor does not represent a rotation, but the direction of an axis or vector. Paolo.dL (talk) 10:58, 16 August 2013 (UTC)
replies: text: I'm happy for the separation: I'm not a fan of putting multiple uses of a term into one article simply because they share a name: an article should be about a topic, not a name.  I'm not convinced that "(physics)" is the appropriate disambiguating parenthesis for the new article – "(geometry)" might have been better – but that can be debated there. — Quondum 11:59, 16 August 2013 (UTC)
thread_title:  Versor in geometry and physics 

message: A comment to 
No so simple, indeed. When you get  as a Riemannian manifold, fix the 0 point there and the orientation, then you obtain the complex projective line up to U(1) symmetry only. You can multiply points to complex numbers, and you know their complex absolute values, but you do not know their exact values as . To establish an isomorphism you have to fix the 1 point somewhere halfway between 0 and ∞.
With versors we have an analogous (but not identical) fable: when you get  as a Riemannian manifold, fix the 1 point in it and the orientation, then you have a Spin(3) structure. You instantly define the scalar part as the cosine of the distance from 1, hence have the plane (unit sphere) of unit vectors, and the arc structure described above, where the orientation specifies the direction of arcs, hence the order of multiplication of versors. But the entire picture is still SO(3)-invariant. Unlike the situation with  you can multiply  points, but not a point to a unit quaternion (specified with 4 components), nor can you get/put quaternionic values. You have to fix a correctly oriented orthonormal basis i,j,k in the space of 3-vectors for an isomorphism between a geometric picture and numeric components. Incnis Mrsi (talk) 10:13, 26 January 2014 (UTC)
replies: text: I was not intending to reference any structure other than topological and metric (i.e. the mapping to which I refer is an isometry and a diffeomorphism). We know that the multiplicative group of the unit quaternions is isomorphic to SU(2)=Spin(3) (lede of Special unitary group), but it is not clear how to me how to depict the multiplication on S3 even though we have a mapping, so your introduction of arcs in S3 does not help me understand how these relate to composing them (corresponding to quaternion multiplication or composition of Spin(3) group elements). —Quondum 03:11, 27 January 2014 (UTC)
replies: text:  Metric space and Riemannian manifold structures on a -sphere imply each other. Each of them virtually describes an embedding of the -sphere into Euclidean -space. But none of them specifies an orthonormal basis there. If you fix the 1 point on , then you choose only one its element of four. Then you have the scalar part function and its zero set, the  of unit 3-vectors. Then you can encode points of , except 1 and −1, as "directed great arcs"  (up to known equivalence) using geometric means: find a great circle on  orthogonal to the versor (i.e. lying at π/2 distance), choose the direction according to the orientation structure, and its length should be equal to the distance of the versor from 1. Now you can multiply versors although you miss 3/4 of the basis. Then you can observe that multiplication of a versor  and any right versor orthogonal to  gives another right versor orthogonal to : it just demonstrates that the great circle is multiplication-invariant. Incnis Mrsi (talk) 09:29, 27 January 2014 (UTC)
thread_title: S<sup>3</sup>, Spin(3), and the group of unit quaternions: subtle differences

message: T.Y. Lam, in his paper Hamilton's Quaternions p. 22 says: "Any quaternion  can be written in the form , where , and   is called the polar angle of .  (Hamilton, who had a penchant for inventing strange names, called  the versor of .)"  He makes no reference to the idea of versor as being used as in this article. If the term versor has come to mean a quaternion of norm 1 (which is not the same thing at all), a clearer picture of the origin of this use of the word would be appropriate.  The two linked online dictionaries define the term versor compatibly with Hamilton's use (in a sense as part of a quaternion), and not in the main sense as used in the article.  As the distinction is subtle, I'll labour the point: Hamilton's use appears to have described an operation applied to any quaternion and its result, not to an arbitrary quaternion that may be the result of such an operation (just as it sounds strange to refer to an arbitrary real number  as a polar angle), nor to a quaternion representing a rotation of a vector.  Under his usage it would be proper to say that  is the versor of some quaternion, but it would not be proper to say that  is a versor.  —Quondum 17:23, 17 February 2014 (UTC)
thread_title:  Etymology and meaning 

message: Otherwise versors aren't closed under multiplication. I'll make the change, and if anyone can find a source that disagrees, they can roll it back.  — Preceding unsigned comment added by 62.172.100.253 (talk) 14:00, 16 August 2018 (UTC)
replies: text: Hello London. Yes, a half-turn is enough given the choice of r and –r. Compare elliptic geometry with the geometry of a sphere. Also right about ambiguity of angle when r = 0. BTW, why not register as a WP:User and get a WP:watchlist as well as a user page? — Rgdboer (talk) 23:14, 16 August 2018 (UTC)
thread_title:  Pretty sure 'a' should be between 0 and 2pi in the introduction