Talk:Quaternion/Archive 1

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More accessible intro

I wrote a short introduction so that non-mathematicians stumbling across this article, or curious folks wondering what quaternions are, would have something to grasp before being plunged into inaccessible terminology. I think this is important for math articles if they are to serve a purpose other than preaching to the converted (so to speak).

However, that doesn't mean that my introduction is particularly good - I swiped part of it from the History subsection. Any good historians/sociologists of math out there who can improve it? - DavidWBrooks 20:17, 1 Jun 2004 (UTC)

There could be more pretty pictures and geometrical intuition about e.g. quaternion as a rotation. --sigs

Correction to quaternion argument arg(p) and logarithm ln(p)

The quaternion argument arg(p) was listed as:

\arg(p) = \arccos\left(\frac{\operatorname{Scalar}(p)}{|p|^2}\right)

The norm should not be squared however, so it should be:

\arg(p) = \arccos\left(\frac{\operatorname{Scalar}(p)}{|p|}\right)

The quaternion logarithm ln(p) was listed as:

\ln(p) = \ln(|p|) + \sgn(p)\arg(p)

The sign should be taken from the vector part only not the whole quaterion

\ln(p) = \ln(|p|) + \sgn(\vec{u})\arg(p)

This follow from the exponential notation of quaternions:

p = r\exp([0, \vec{u}]\phi) = r\cos(\phi) + r\sin(\phi)\vec{u} \land |\vec{u}| = 1

Hkuiper 17:34, 30 Oct 2004 (UTC)

Formatting problem

There's something messed up with this page, starting after the equation in the history subsection. Using Firefox, everything after that point is rendered in a much smaller font.--Starwed 20:26, 2 May 2005 (UTC)

ijk = -1 does not imply associativity

The article says the equation ijk = -1 implies the quaternions are associative. Actually it assumes it, but does not imply it.

John Baez

The statement is now removed. -- Fropuff 01:57, 2005 May 2 (UTC)

Quaternion multiplication typo?

I think there's a typo in quaternion multiplication. I haven't figured out how to get the images working here, but I hope I can make myself clear with my own ascii notation.

The article defines two quaternions

q = a + u_

p = t + v_ [the underscores denote the vector part]

The article gives the proper form for

pq = at - u_.v_ + av_ + tu_ + v_ x u_

but not for qp,

qp = at - u_.v_ + au_ + tv_ - v_ x u_

which should be

qp = at - u_.v_ + av_ + tu_ - v_ x u_

A look at the source for the section says that both the img and alt versions are faulty.

Quaternion multiplication
pq :

The usual non-commutative multiplication between two quaternions is termed the Grassmann product. This product has been described briefly above. The complete form is described below:

<img class='tex' src="/math/c8439d2102f59aa407da93e2da708782.png" alt="pq = at - \vec{u}\cdot\vec{v} + a\vec{v} + t\vec{u} + \vec{v}\times\vec{u}" />
pq = (atbxcydz) + (bt + ax + czdy)i + (ct + ay + dxbz)j + (dt + az + bycx)k

Due to the non-commutative nature of the quaternion multiplication, pq is not equivalent to qp. The Grassmann product is useful to describe many other algebraic functions. The vector portion of the multiplication of qp follows:

<img class='tex' src="/math/b3cff5a7d4f6c36c9cb09d393887c3c9.png" alt="qp = at - \vec{u}\cdot\vec{v} + a\vec{u} + t\vec{v} - \vec{v}\times\vec{u}" />

I think you're right, there's a sign error when compared to this page --Anniepoo (talk) 23:33, 28 May 2008 (UTC)

The products given are p*q and q*p which seem correct. I corrected case p = q claim. Generally, this business of alternative quaternion products is too arcane, of little interest, since the basic product is implied by the basis products. However, I'd rather not be the one to trim down the content, though it needs doing.Rgdboer (talk) 23:25, 29 May 2008 (UTC)

The Grassman product is used to do rotations in computer graphics. Googling it will find a large number of explanations of how to multiply quats. Certainly it's encyclopedic for that reason. If that's what you mean. The inner product stuff is probably less important, and certainly needs reorganized Anniepoo (talk) 05:54, 30 May 2008 (UTC)

Quaternions and quantum theory

Small particles do not obey the laws of classical logic. Instead these particles obey the axioms of a weaker logic, the quantum logic. Classical logic has the structure of an orthocomplementary modular lattice. Quantum logic has the structure of an orthocomplementary weakly modular lattice. Classical logic is isomorphic with the Venn diagrams. Venn diagrams are often represented by a series of overlying circles. The structure of quantum logic is far more complex. There exists a mathematical representation of this structure in the form of the subspaces of a Hilbert space. A Hilbert space is a collection of analytical functions. A countable but still infinite number of mutually independent functions can span this space. The space can be defined over the real numbers, over the complex numbers and maximally over the quaternions. Linear operators that work on these functions have eigen-values that belong to eigen-functions of these operators. The eigen-functions of such operators span the Hilbert space. The eigen-values are the things that appear to us as our physical world. Group theory applied to these operators reveals the basic formula’s of quantum theory. When quaternions are used as the applied number system, then the quaternionic quantum theory results.

With quaternions, concepts like spin and parity become a trivial interpretation. When the multiplication rule for quaternions is written as:

pq = at ± (u,v) + av + tu ± uxv

then the sign of the second and the last term can be taken at will, while still a proper multiplication rule holds. (u,v) stands for the inner product of the vectorial parts. u x v stands for the outer product of the vectorial parts. In physical sense the first sign choice directly relates to the physical feature: parity. The second sign choice relates to spin.

The real part of the physical quaterion can often be interpreted as time, while the vector part represents the three dimensional space we live in. However if you take a quaternionic Fourier transform of that world, you end up in another view of that world where time is replaced by energy and the location vector space is replaced by the impulse space. (Impulse has a close relation with force).

(source: Jauch (1968) Quaternion Quantum Mechanics)

The Fourier transform based relation between the two views causes the Heisenberg’s uncertainty relation. You cannot precisely know time and energy of a particle and you cannot know precisely the location and the impulse of elementary particles. An individual in time-location space has a huge extension in energy-impulse space and vice versa. A rather vague object in time-space will be rather compact in energy-impulse space. This may give a physics based support to the fact that sensitive people can use the associative capabilities of their brains to collect information from ‘the other view’ and interpret it so that it becomes a meaning in a wider time-location frame.

J.A.J. van Leunen


I am a novice when it comes to quaternions and am struggling to read Altmann's book on rotations quaternions and double groups. Interestingly this book is quoted in the article but there is no mention of Rodrigues who -if I must believe Altmann- has come up with a more rigorous version of quaternions than Hamilton. Can anyone comment to that?


About a reference

I'd like to note that (currently fourth) external reference ("Doing Physics with Quaternions") points to a website of the same title, which is unfortunately nothing but one wild mess of gibberish and nonsense. [The author does not appear to ever have heard of vector bundles and gauge theories when he states thats "quaternion multiplication is reminiscent of spin", for instance.] There is nothing encyclopedic about it, and (not mentioning the potential harm that website could have on the innocent laymen, or on the laughing muscles of visitors with some mathematical knowledge) the reference should be deleted.

Division over reals

The article states that only the real, complex, and quaternion numbers form a finite-dimensional associative division algebras over the field of real numbers. There should be *four* division algebras - we need to add octonion numbers.

error in quaternion inverse

I was writing some code using this page as a guide. My code didn't work. It took a long time to track down the problem.

The definition of the inverse is wrong. It shows a grassman product in the denominator. Uses same notation as earlier for grassman product.

But this should be instead the dot or inner product.

Observe that for any suitable product Prod, you can compute

q_{prod} \, ^{-1} = \frac{\overline{q}}{Prod(\overline{q}, q)}

When you multiply this fraction with q using the same kind of product, you get the same quantity in the numerator as in the denominator. Of course, this only works for products that actually produce a non-zero scalar  Prod(\overline q , q ) when  q \not = 0, and have suitable associativity and commutativity properties for the multiplication with scalars.

If q = t + \vec v, the inner product with the conjugate is \overline{q} \cdot q = t^2 - \vec{v} \, ^2, which can be zero for q \not = 0.

This contrasts with the Grassman product, which has positive scalar  \overline q \cdot q for q \not = 0.

Of course you could compute  q_{inner}^{-1} = \frac {q}{q \cdot q} and then compute  q_{inner}^{-1} \cdot q . In this way you again get the same positive number in the numerator and the denominator.

PerezTerron 12:44, 29 December 2006 (UTC)


I have noticed that a lot of the information in this article is repeated in more than one place, and the logical coherence from start to finish isn't tight. This is not uncommon when many editors contribute. I want to make known that I started to reorganize it to be more concise and better structured. Usually, when I do this, I rarely add/delete/change much. Usually it is just a matter of changing the outline (section titles, etc.) and moving content appropriately.

On a side note, my apologies for inadvertently marking my first reorganizational edit as a minor edit; it was clearly not. Baccyak4H 20:35, 14 November 2006 (UTC)

Halfway through the "Outer-product" section, the nicely-defined A, B, Q notation is dropped, and undefined p, q, u, v are used. Evidently author wars. The change to A, B, Q should be completed, although this is a major undertaking.

Bruce Jerrick, Nov 02 2007 —Preceding unsigned comment added by (talk) 21:33, 2 November 2007 (UTC)

The base quaternion products are Grassman products

You can arrive at the Grassman product by formally multiplying polynomials in "unknowns" i, j, k assuming multiplication with scalars commute, and using the base products of the quaternion group,  i^2 = j^2 = k^2 = i j k = -1 .

In this sense, the Grassman product is the "native" product that has the equations i^2 = ... = -1 as special cases. Perhaps this should be pointed out?

Actually this could be made part of the definition section. This section declares that

Quaternions are a generalization of complex numbers, obtained by adding the elements i, j, and k to the real numbers, where i, j, and k satisfy

 i^2 = j^2 = k^2 = i j k = -1

and where multiplication is assumed to be associative.

Here the multiplication seems to be defined, but the definition is in a way incomplete. The section goes on to say that quaternions are real linear combinations of the four base elements. In light of the fact that the multiplication is not commutative in general, it should be included in the definition of this multiplication that the products of these base elements with real numbers do indeed commute.

PerezTerron 14:11, 29 December 2006 (UTC)

I would like to extend Terron's argument one step farther, and suggest that the "grassman product" which appears to be a formulation of quaternion multiplication in terms to Cross product and dot product operations should be moved to the "critisism of quaternions" section.

The operation of "Grassman multiplication" is apparently the operation of starting with two quaternions interchanging the imaginary numbers i + j + k of Hamilton for the i=[0,0,1] j=[0,1,0] and k=[0,0,1] of Gibbs and Wilson. Then performing the Wilsonian dot and cross product operations on them.

The final step in "Grassman Multiplication" is unclear. Do we then substitute the Hamiltonian imaginary numbers i,j,k back into the resulting four-tuple? Or like Wilson have we found a method of dispensing with quaternions once and for all and replacing them with a scaler, in an ordered pair with a Wilsonian Modern Vector?

I agree that the "Grassman Product" is an elegant proof that we can formulate physics with out resorting to quaternions. It shows that Wilson's elegant defense of three dimensional Cartesian space, has now become so deeply embedded in our state school's math system, that an argument against quaternions, is now used as the definition of their fundamental cardinal operation of multiplication.


January 25 2008

With the addition of the new section under historical section about classical 19th century quaternion notation, I don't feel as bad about the Grassman product. Maybe in the 19th century and early 20th century representing everything in terms of dot and cross products was part of a crusade to eleminate the quaternion. Now that the historical section lists Multiplication as one of the two Cardinal operations of classical quaternions I feel my objection has been satisfactory resolved.

Possible material for inclusion in classical multiplication

Hamilton first introduces his quaternion, in his 1853 lecture series as the quotient of two "Hamiltonian Vectors". He writes


Hamilton also quickly introduces the idea that the product of two Hamiltonian vectors is a quaternion. Using a single cardinal multiplication operation defined in his famous i, j, k multiplication table, as was pointed out by the original founding author of this thread.

(ai + bj + ck) x (ei + fj + gk)

Where I use the Juxtaposition of two of Hamilton's Quaternions to with out an intervening dot or cross to mean Hamilton's original cardinal operation of multiplication.

Since Hamilton postulated that his cardnal multiplication operation was distributive, all we need to do to arrive at the "Grassman Product" with Hamiltons three imaginary i, j, k substituted back in for the three Cartesian "Unit Vectors" of Wilson is to just multiply everything out.

We get nine terms.

aeii + afij + agik + beji + bfjj + bgjk + ceki + cfkj + cgkk

The most interesting result of this multiplication is that the three terms which contain ii, jj and kk result in a scaler component!

Hence the product of two Hamiltonian Vectors is a quantity with four terms, one of them being a scaler component or time component.

Hamilton multiplied Quaternions and then de-constructed the results.

I was reading an article where the author claims that quaternion with zero for its scaler value is "proper quaternion", what I am calling a "hamiltonian vector"

—Preceding unsigned comment added by Hobojaks (talkcontribs) 02:35, 21 January 2008 (UTC)

Hello - thanks for sharing your thoughts here. In the early 20th century, there has been considerable effort with quaternions and other hypercomplex numbers, due to their apparent relation with space-time geometry. I thank User:Rgdboer for having maintained and updated a historical record here in Wikipedia. You may want to look at Hypercomplex number, Coquaternion, or Hyperbolic quaternion, just to mention a few concepts that were looked at. Personally (please note: POV), I don't think that quaternions have been "argued against" in any way, it's just that descriptions of physics using quaternions turned out to either not work, or to not yield "new" physics (i.e. one might just as well continue to not use quaternions). The conclusion is that if quaternions are not required, they are a "nice-to-have", a mathematical curiosity - at least from the viewpoint of physics. Therefore, I don't agree that this has been forgotten. The historical development went to Clifford algebra for multi-dimensional analysis, tensor algebra for description of gravity, and Lie algebra for describing internal (non-spacetime) symmetries. All three approaches (Cliffor, Lie, tensors) include quaternions, so in that respect they've become quite "mainstream", so to speak. ... Maybe we need a new article, history of the hypercomplexes? Or expand into a new "history" section on hypercomplex number? Thanks, Jens Koeplinger (talk) 22:06, 21 January 2008 (UTC)

Note Some comments deleted here have now been added to the historical section

Thus making this thread shorter, and thanks for all the help Koeplinger

Hobojaks (talk) 18:43, 23 January 2008 (UTC)

Trying to keep short (sorry): You seem to be interested in what's called a Normed division algebra (to "divide two vectors"). The largest one of that kind are the octonions. Koeplinger (talk) 20:30, 23 January 2008 (UTC)


Is "Distributative" a British spelling or a misspelling? It's not in my dictionary.--Jwwalker 22:14, 2 January 2007 (UTC)

It's certainly not a peculiarly British word; I would always use the term "distributive". I suspect that it's a common (incorrect?) variant, perhaps based on falsely extending the pattern "commute"->"commutative" therefore "distribute"->"distributative" by analogy. If you Google the term, it pops up quite a lot of examples, but sometimes in contexts where it is incorrect; for example, one website refers to the British trade union USDAW as the "Union of Shop, Distributative & Allied Workers" whereas they are definitely "distributive" not "distributative".

I would recommend changing it to "distributive", which I believe to be completely correct, but I'm no professional mathematician so I hesitate to do it myself! -- SamTheCentipede 20:39, 1 February 2007 (UTC)


is it: Quat-er-ni-on where Quat = "Kw-at" or Qu-a-ter-ni-on

The latter in general speech... the same with quaternary, quattro, etc. --Lionelbrits 03:29, 27 September 2007 (UTC)

Using Euler Notation to represent Quaternions

Has anyone even considered putting this in there? They have it for regular complex numbers so it should be here also. Here is how you get it: (i have no idea how to use the math markups so somebody can edit those in here; i just use alt characters)

z = A + Bi + Cj + Dk

r x eiθ+jφ=r x (cosθ+ isinθ)(cosφ+ jsinφ)= r x (cosθcosφ + isinφcosθ + jcosφsinθ + ksinφsinθ)

thus r = √A²+B²+C²+D²

and rcosθcosφ = A

rsinφcosθ = B

rsinφcosθ = C

rsinφsinθ = D

you can finish assigning the angles. it's relatively simple. just use the arccos of sums and differences of the halves of the coeffecients


arccos(A+D)= θ - φ

arcsin(B+C) = θ + φ


r x e(arccos(A+D)+arcsin(B+C))/2 x i + (arccos(A+D)+arcsin(B+C))/2 x j = A + Bi + Cj + Dk

SOmebody with a better knowledge of markups should post this in there. —The preceding unsigned comment was added by The Roc 1217 (talkcontribs) 03:15, 3 April 2007 (UTC).

ij = k and ji = k ... recent example changes

Hi - there's been some recent improvement to the example calculations given, but apparently also some confusion. Since there's been quite a bit of back-and-forth within the last week, please make sure to check consistency with above definitions. I've just had to revert a calculation that I cannot verify, by the simple definition of ij = k, which is also the one I'm familiar with. If there's a different definition, it would be good to know, and maybe we can drop a note at the top of the page, like "Please note that the computer program so-and-so uses a different definition with this-and-that.". Thanks, Koeplinger 12:48, 2 May 2007 (UTC)

Maxwell and Quaternions

Hi - a recent "citation needed" placement next to Maxwell's equations alerted me to an inaccuracy in the text: Already in the very treatise in which Maxwell published his famous equations, he also suggested a quaternion representation of the same. I've updated the text and referenced the treatise (though it looks like I did a mistake there; I'll fix it). So - whoever placed the tag: Good catch! Thanks, Jens Koeplinger 01:43, 14 May 2007 (UTC)

As far as the citation ref goes: I give up. It shows the alleged footnote reference, but then it doesn't work when I click on it. I'm using Firefox on Linux. Has anyone seen this behavior before? If so, please fix the reference link! (PS: I'll try to dig-up the section / page number where the quaternion rep is demonstrated; also, if someone has a 1st edition reference, please replace). Thanks, Jens Koeplinger 01:57, 14 May 2007 (UTC)
The citation isn't working because the article is presently using the wiki footnote system (there is no <references/> tag anywhere). You might do better to use a Harvard style reference for now, i.e. (Maxwell 1904) and then manually add the reference at the bottom. --Fropuff 02:21, 14 May 2007 (UTC)
Ok, thanks. I also got the 1st edition reference but don't have the publication at hand. Other sources claim that is already uses quaternions so I'm ... er ... putting it there as a starting point? The publication year of the 1st edition, 1873, worries me, though. Well, one more item on my to-do list. Thanks again, Jens Koeplinger 03:05, 14 May 2007 (UTC)

Quaternion division

A proof recently added to the page simply shows that right division, i.e. the division operator that makes (P/Q)*Q=P true, is equivalent to P*Q^-1. But that is obvious. Left division, i.e. the division operator that makes Q*(P/Q)=P, is equally obviously equivalent to Q^-1*P. The proof does this in its first step, where it multiplies out. If this multiplication were on the left, you'd get the other division. Anyway, your professor's telling you he liked your proof doesn't constitute WP:Verifiability. Bring examples in the published literature where P/Q is defined to mean P*Q^-1 -- that would be much more compelling. --Macrakis 22:56, 30 May 2007 (UTC)

The definition of Q^-1 is defined such that Q^-1*Q = 1 (the unit quaternion).
(P/Q)*Q = (P*Q^-1)*Q = P*(Q^-1*Q) = P*1 = P
I believe you are saying that with left division Q*(P/Q) = Q*(Q^-1*P) = (Q*Q^-1)*P. Now what? Q*Q^-1 does not necessarily equal Q^-1*Q
Please show how you get Q*(P/Q) to equal P. --MarkMYoung 05:09, 31 May 2007 (UTC)

For all associative operations with identity (monoids), left and right reciprocals are the same. Proof: Call aR the right reciprocal and aL the left reciprocal. Then aR = 1 aR = (aL a) aR = aL (a aR) = aL 1 = aL. --Macrakis 13:42, 31 May 2007 (UTC)

Thank you. So, if aR = aL, then what is wrong with defining P/Q as being either and both P*Q^-1 and Q^-1*P? This would also allow the definition of a quaternion inverse to be Q^-1*Q = 1 = Q*Q^-1. To me, it's like saying a rigid solution to quadratic equations doesn't exist because there are two possible solutions (but I don't have a doctorate in mathematics). BTW, the professor was my professor about 8 years ago. --MarkMYoung 15:52, 1 June 2007 (UTC)

Though Q^-1 is unique, P*Q^-1 is in general not the same as Q^-1*P, since quaternion multiplication is not commutative. I think you need to review your elementary abstract algebra book before you edit more on algebra topics.... --Macrakis 16:39, 1 June 2007 (UTC)

That was my point exactly (reference to quadratic equations having 1 or more solutions). Instead of showing how Q*(P/Q) equals P, you posted that aR = aL, where a*aR = Q*Q^-1 and aL*a = Q^-1*Q. Now you've come full circle saying they are not generally the same (reference my original reply where I said, “Q*Q^-1 does not necessarily equal Q^-1*Q”). How can the inverse be rigidly defined as Q^-1*Q = 1, but not define Q/Q = 1 (similarly Q^-1*P = R and P/Q = R)? Shouldn't it read \mathbf{Q_L}^{-1}\mathbf{Q} = 1 or \mathbf{Q}\mathbf{Q_R}^{-1} = 1, where \mathbf{Q_L}^{-1} does not necessarily equal \mathbf{Q_R}^{-1}? --MarkMYoung 18:13, 1 June 2007 (UTC)
Mark, please read more carefully. The left and right inverses are the same. Left and right division are not. I'm sorry, but Talk pages really aren't the right place to study abstract algebra. I've made my point. --Macrakis 18:29, 1 June 2007 (UTC)

Bearden quaternion quote

Hi - just saw that the quaternion quote from Bearden was removed [1]. I actually liked the quote collection before, because it ranged from "hard-headed physical scientist" to pseudoscience advocated by Bearden. I love the wide spectrum of attention that quaternions and certain primitive algebra types are receiving and I find it notable. I suggest to keep the Bearden quote, but on his article Tom Bearden possibly add a classification Category:Pseudoscience. Agreeable compromise? Too controversial? For an encyclopedic entry, I find it notable enough to keep; we just have to put it into the correct context.

I'll go ahead and revert for now.

Thanks, Jens Koeplinger 01:04, 8 June 2007 (UTC)

The problem is, it's out of context, and doesn't indicate that it's not from a generally respected scientist.--Prosfilaes 13:18, 10 June 2007 (UTC)
Disagreed and agreed.
I disagree that the quote is out of context; indeed, I really love the context: Taken the last two quotes together, of today's article, it seems as if Bearden's quote is the example of "air of nineteenth century decay" that the predecessor quote is eluding to.
I agree that Bearden is a generally not accepted scientist. Just follow the Tom Bearden link and you'll see this stated quite bluntly. Not sure this needs change here, but if you have a suggestion on how to improve the context, you're quite welcome to propose: There's already been discussion about D. Sweetser's "Doing physics with quaternions" link, to content that is similarly questionable (you'll find that link further down).
One thing I'd suggest to keep in mind is that a classification as pseudoscience doesn't necessarily means it's wrong, neither does it endorse a certain way of "out-of-the-box" thinking. It simply means that it doesn't follow the scientific method. As you know, if you don't follow the scientific method, you can propose pretty much everything and nothing, which is an obvious concern. Bearden's inability to produce a working sample of an otherwise revolutionary and grandious concept is another major concern, not mentioning the difficulty in explaining why a narrow physical effect would violate a profound physical law (to allow Perpetual motion) for which there is otherwise no substantial evidence.
Full of concerns, there is yet a fair amount of attention to "free energy" ideas and advocates, which should support notability and therefore be grounds enough to keep the reference in Wikipedia. What is the best way to do so? That's debatable ... Good luck. Thanks, Jens Koeplinger 02:35, 11 June 2007 (UTC)
It is out of context; all the other quotes are about quaternions, but this quote is really about the unified gravitation/EM part of Maxwell's theory. The fine points of the philosophy of science are irrelevant; Bearden's theories are believed by almost no one, and have no evidence to back them up. Sure, there's a fair amount of attention to "free energy" ideas and advocates; that means there should be articles on them, or possibly even that this article should discuss that quaternions are used heavily in these ideas and advocates, if they in fact are, but that doesn't mean we should have the quote.--Prosfilaes 13:02, 11 June 2007 (UTC)
Your point is valid, I agree. Too bad, I loved the contrast :) Thanks for your patience. Jens Koeplinger 01:16, 12 June 2007 (UTC)

Basis for multiplication

The article says:

ijk = -1

But it also says:

ki = j and ik = -j

Shouldn't it be:

ijk(j) = -1(j)
-ik = -j
ik = j and thus ki = -j


Hi - the article is correct. In your example, you've not accounted for non-commutativity between k and j. Specifically: ijk(j) = -1(j) yields +ik = -j, because kj = -jk. Note that this also uses associativity of the multiplication, because you would otherwise have to consider the placement of the brackets. Jens Koeplinger 01:28, 21 August 2007 (UTC)
Thanks a lot! I'd have never figured this detail out by myself. You should consider adding a note to the article on that, it'd be really helpful. —The preceding unsigned comment was added by (talk) 23:07, August 22, 2007 (UTC)
I agree with this suggestion -- I see nothing in : i^2 = j^2 = k^2 = i j k = -1 , \,\! which requires the use of non-associative multiplication. Documenting the motivation for making quaternion multiplication non-associative would have more then mnemonic value. —Preceding unsigned comment added by (talk) 19:36, 27 September 2007 (UTC)
Hmm - multiplication is associative; it's just that when you have products of the form abcd you must obey the order in which the factors are. I.e. for (associative) quaternions, the following are all the same ((ab)c)d = a(bc)d = a(b(cd)) = (ab)(cd). Once you mess with the order, you're effectively commuting elements - because in general abcd \ne acdb; which is what led to the initial confusion, I believe. ... But how to word that? It's more an explanation of commutativity, so it should be really short on the quaternion page. Koeplinger 12:36, 28 September 2007 (UTC)

Cayley-Dickson construction?

I believe the first formula should be q=u+vj (instead of current q=u+jv). In this way it would expand into q=a+bi+cj+d(ij), which together with notation for dyad (ij)=k would fit the standard definition at the top of the page. In current form there seems to be some sort of confusion: either k=ji or we have unpleasant minus sign before d in the expansion. —Preceding unsigned comment added by (talk) 23:18, 14 October 2007 (UTC)

I'll expand it a little to illustrate -

q=u+jv - Our quaternion
u=a+ib - first complex term
v=c+id - second complex term
q=(a+ib)+j(c+id) - substitution of u, v
j(c+id)=jc+jid - distributive law a(b+c)=ab+ac
q=a+ib+jc+jid - substitution of j(c+id) for jc+jid
ji=-ij - but we want ij
q=a+ib+jc+ij(-d) - so we negate d, as ij(-d)=jid

Hope that helped. Nazlfrag —Preceding comment was added at 11:36, 19 October 2007 (UTC)

Hi - your demonstration is good, and since I don't see a reference that would prefer the one over the other approach, and since both are entirely equivalent, they can be replaced. The section also needs a bit of formatting (the i and j symbols show as bold i and j instead of italic). By the way, your concern in mentioned very briefly at the very end of that section, where it currently says:
Note that if u = a + i b, v = c + i d, and p = a + i b + j c + k d then p′s construction from u and v is rather
 p = u + v j = u + j v^* \,.
Which is exactly to your point, I believe. Thanks, Jens Koeplinger 12:27, 19 October 2007 (UTC)

possible misplaced text after outer-product paragraph

In the paragraph titled "outer-product":

The outer-product can be rewritten using the Grassmann product: (equation) and the absolute value of z is the non-negative real number defined by (equation)

I don't think there is meant to be any connection between the outer product and the absolute value. This stuff on absolute value appears to be from an older version of this article. The equation given is the same as that listed in the "quaternion modulus" section further down. Here though there is a note about interchanging conjugation with multiplication.

Another thing I noticed is that at this same point in the article the notation switches to using p=(a,u), q=(t,v) instead of A=(At,Ax,Ay,Az) and star is used for conjugation instead of a bar over the variable. —Preceding unsigned comment added by (talk) 03:58, 26 November 2007 (UTC)

Problematic use of the word "element"

Some quotes from early in the article, as at 3-12-07:

"every non-zero element has a unique multiplicative inverse"

"Only quaternions with real elements will be discussed here"

"Quarternions ........ contain the complex numbers"

"The t element represents the scalar quantity"

There seems surely a problem with this terminology, if it is intended to define quartenions for a beginner. The ambiguity could throw a beginner's attempts to follow various comments surrounding the early definitions. At the least, a warning about possible misinterpretation needs to be inserted. "Element" seems to be used: (a) for any individual member of the quarternion algebra (e.g. 1+2i-2j+5k is an element); and (b) for each one of the four coefficients (1,2,-2,5) that are used to express such a quarternion as a 4-vector in terms of the basis "elements" (1,i,j,k). Would it be clearer to scrap one of these usages of "element" and maybe replace the second one by "coefficient", or "coordinate" or something? Or else to pad out each occurrence with extra explanatory words to make the text less ambiguous? Or just to insert a note to warn that in the article that follows "elements" is used with different sets in mind: there are "elements", and there are "elements"? —Preceding unsigned comment added by (talk) 21:56, 3 December 2007 (UTC)

I very much agree that the two uses of "element" are confusing, and that using either "coefficient" or "coordinate" to mean the "pieces" of a quaternion is a good idea, reserving "element" to mean a member of a quaternion algebra. JackSchmidt 22:39, 3 December 2007 (UTC)

determinant of 2x2 matrix vs. quaternion norm

In the section "Matrix representations" it is stated that the norm of a quaternion is the determinant of its 2x2 matrix. It seems to me that the determinant is equal to the *square* of the norm of the quaternion. —Preceding unsigned comment added by (talk) 00:56, 14 December 2007 (UTC)

A lot of typing gone?

Unfortunately a section on historical notation of quaternions I was attempting to create was deleted. Along with my attempt to divide the controversy about quaternions into two sections one about the historical notation controversy and the other about the more metaphysical controversy.

Oh well, lots of work for nothing. Not sure if did something wrong, or if someone just took the liberty to delete it. So much for my attempt to contribute to man kinds understanding of quaternions. Hobojaks (talk) 02:24, 25 January 2008 (UTC)

Hello. I reverted a bad edit that introduced incorrect formatting to the article - no problem, I didn't need to let you know. I then reverted my own edit so as to lose _only_ the last one or two or your edits - the one containing the bad edit. Please re-check. I think you will find that the adjustment was very much less than you think. Thanks, Ian Cairns (talk) 02:29, 25 January 2008 (UTC)
Hi Hobojaks - as Ian says, your edits are all there, this is a link that shows your changes so far: - Thank you for your gift of knowledge! You may want to explore the "history" tab on articles, there's a lot of great things you can do with it. Things are very rarely ever deleted, and even if this would happen there's a whole process with voting etc around that. No worries, your contributions are safe and well taken. Thanks, Jens Koeplinger (talk) 02:59, 25 January 2008 (UTC)

Thanks for the help guys. I agree that in my ignorance I have messed up the heading structure.

But there are other problems. This first edition about Hamilton's notation is based on the understanding of someone with very limited knowlege on the subject. I need help with it.

Second, I have vastly extended the section on the controversy about quaternions.

I have attempted to divide it into two parts. One being the historical notation controversy, about issues such as if the cross product of two vectors is should be written as taking the vector of the quaternion product.

Some of my assertions in this section, about what the issues debated at the time were about notation issues are largely educated seculation. For example the part about with two kinds of multiplication instead of one introducing square roots of one and zero. I don't have any citations to prove that this was actually debated. It just seemed so obvious an example of a potential issue over notation to debate that I used it with out proof. I feel confident that within the volumes of debate on the subject this must have come up.

The quadrential versor argument is the only one I have managed to document.

Another problem I am having, which may be due to my own ignorance is that my conception of complex numbers, which I owe to a high school math teacher, is that the a complex number is either a sum of a real number and a purely imaginary number, or else it was an ordered pair consisting of a real number and an imaginary number. Wikapedia currently states last time I checked that an complex number is an ordered pair of real numbers? Sure there are two real numbers in a complex number, one being a pure real, and the other being the real number coefficient to an imaginary unit equal to a square root of negative one. But my math teacher thought the pair was one real and one imaginary.

I boldly took out the section in the history of quaternions which claimed that Hamilton said that a quaternion was a fourtupple of real numbers. I have not read all of Hamilton's writings but it does not really sound like something he would say, given my limited understanding of his writings. So I changed it to one scaler, and three imaginary numbers.

So anyway a lot of typing, but what I have typed still needs a lot of work. —Preceding unsigned comment added by Hobojaks (talkcontribs) 03:26, 25 January 2008 (UTC)

That's perfect: You added a lot of content, changed a bit, and (now) described on the talk page what you did and why. In not too long, people will pick-up on your contributions and do things to it, mostly editorial. Thanks for pointing to the Hamilton statement, this needs to be sourced. Just a few tricks you may find helpful: When you feel something is obvious but you also feel that a better reference should be provided, you may want to consider putting the following string instead of a reference: {{fact}}. This will produce Wikipedia's ever-so-famous "citation needed" tag. Also, when you're doing a long edit that could risk a edit conflict (if someone else attempts an edit at the same time), you can put the following at the top or at the section where you're editing: {{inuse}} (and save the page right-away). This will put a visible notice to readers that the article is undergoing a longer edit. For more info, see e.g. Template:Inuse. And - again - no worries, as far as things you can do, Wikipedia is a bottomless pit. Thanks again for contributing! Jens Koeplinger (talk) 03:36, 25 January 2008 (UTC)

Extensions to Quaternions.

extensions into higher dimensions like octonions and Clifford algebras. are an important subject that needs a section of its own.

Extensions of quaternions is not a relic of the past to be recorded historically, but rather an ongoing living breathing thing.

I feel that extensions to quaternions deserves to be taken out of the 19 century history section, and rather deserves its own section.

In the 19th century people didn't like the idea of four space, and wanted to go back to three space. Extensibility was not an issue for Gibbs and Wilson who say noting about higher dimentions in their book vector analysis.

20th century extentions

In the early 20th century, there has been considerable effort with quaternions and other hypercomplex numbers, due to their apparent relation with space-time geometry. Hypercomplex number, Coquaternion, or Hyperbolic quaternion, just to mention a few concepts that were looked at.

Descriptions of physics using quaternions turned out to either not work, or to not yield "new" physics (i.e. one might just as well continue to not use quaternions).

The conclusion is that if quaternions are not required, they are a "nice-to-have", a mathematical curiosity - at least from the viewpoint of physics.

The historical development went to Clifford algebra for multi-dimensional analysis, tensor algebra for description of gravity, and Lie algebra for describing internal (non-spacetime) symmetries. All three approaches (Cliffor, Lie, tensors) include quaternions, so in that respect they've become quite "mainstream", so to speak. ... Maybe we need a new article, history of the hypercomplexes? Or expand into a new "history" section on

Extenstions to Quaternions Argument

Below are some important yet offand ideas in my latest edit.

The extensibility argument is sometines used against quaternions but this has not stopped continuing interest in not only the quaternions but also what are viewed as its many extentions.

While the superiority of modern vector, matrix and tensor notation over classical Hamiltonian quaternion notation is debatable in three and four dimensions, Hamilton's classical quaternions being defined as four dimensional cannot be directly applied in higher dimensions.

Technically models of actual space based on the tri-vector and 3 x 3 tri-matrix are by definition trapped in three space as well. Likewise Hamilton's classical quaternion notation is trapped in four space by definition.

Advocates of other modern notation argue that they have better extensibility to higher dimensions.

Hobojaks (talk) 01:23, 26 January 2008 (UTC)

History section is overlong - split into new article

It might be better to create an article, perhaps History of quaternions, to deal with the details of the history, and to include a 2 or 3 paragraph summary in this article, Quaternion, along with {{main|History of quaternions}} to indicate the main article on the history.

I tried to sort the headings in the current history section, but there are just too many. In particular, the Quaternion#Quotes section is probably not part of history, but I am not sure.

If this sounds like a good idea, then I think Hobojaks should create the new article, copy paste his text into it (preserving the history insofar as he is clearly the primary author of that section). Once this is done, someone needs to create a 2 or 3 paragraph summary of the *entire* history of quaternions, and adjust the headings again. Assuming the information is already in the article, I can shrink the history and do the technical heading stuff. JackSchmidt (talk) 17:33, 26 January 2008 (UTC)

Perhaps this is already done in Classical Hamiltonian Quaternions. JackSchmidt (talk) 17:36, 26 January 2008 (UTC)

History section is looking good!

I think that the link to Classical Hamiltonian Quaternions fits right where it should in a brief history of quaternions, in this article about modern state of the art quaternions.

Don't forget that I think there needs to be an extensions to quaternions section in the modern article, including the long list things like Octonians and the clifford algebras and the ongoing development of new ideas rooted in the quaternion idea.

I think that there is a distinction to be drawn between a classical 19th century system of notation and vocabulary, and the history of it.

What if someone wanted to actually attempt to read Lectures on Quaternions? Its confusing because the meanings of words have changed, and notation has changed so much that it looks at first like Hyrogliphics.

Then there is the history of it, what happened to it, who was in favor of it and why, who was against it and why, and so on.

Hamilton_quaternions also links to Quaternions not Classical_Hamiltonian_Quaternions given the poor state of development of a day old page, maybe that should stay the same for a while. However maybe some day, if the section on Classical Hamiltonian Quaternions gets cleaned up enough, this might be worthy of a change. —Preceding unsigned comment added by Hobojaks (talkcontribs) 19:26, 26 January 2008 (UTC)

Quaternion Euclidian product

In the part of the quaternion euclidian product, the article states:

"When p = q, the result is the conjugate"

When p = q, the result is just a scalar. Shouldn't this be "When p and q are interchanged, the result is the conjugate" ?

Furthermore, in the sentence "The norm of a quaternion (the square root of a product with its conjugate, as with complex numbers) is the determinant of the corresponding matrix", the norm of the quaternion should be the square root of the determinant.

Akarai (talk) 17:51, 16 February 2008 (UTC)

Trifonov's edits

I removed a section added by User:Trifonov citing papers by Vladimir Trifonov. The writing was very dense and hard to follow, but my impression is that there are two claims being made:

  1. You can define an inner product on the quaternions by \langle x,y \rangle = \phi(xy) where φ is a linear functional on the quaternions (treated as a real vector space) and xy is quaternion multiplication. Symmetry of the inner product demands that φ(v) = k Re(v) where k is any (nonzero) scalar, which gives a scalar multiple of the Minkowski inner product. This is perfectly true and it might be worth mentioning, but I'd like to see some evidence that someone other than Mr. Trifonov has used this inner product in a published paper.
  2. The group of nonzero quaternions has a natural FLRW metric. I daresay this is true too, but the supposed connection to cosmology doesn't exist. The FLRW metric is simply the most general homogeneous isotropic metric with Lorentzian signature. Having the right signature and sufficient symmetry is not enough to make a viable cosmological model. It might be worth mentioning this as a purely mathematical fact without the physical claims, but again I'd like to see some evidence of interest in this metric outside Mr. Trifonov's work.

-- BenRG (talk) 16:07, 21 March 2008 (UTC)

I undid the changes by BenRG for the following reasons.

1. This section, like many others, requires some preliminary knowledge (elementary linear algebra, diff geometry, first or second semester) and of course may be hard to follow otherwise. The reference [1] cited in the section is chronologically the first one on the subject, and therefore has a certain right to be cited. Also see, for example, I. Raptis, IJTP 46 (3) (2007) 688-739. In addition, the “linear functional” \phi above is not a linear functional at all, for several reasons, for example, because it is undefined, since a definition would require an assignment of a real number to each quaternion in a linear fashion, while it assigns to the product of an ordered pair of quaternions (x,y) something denoted \langle x,y \rangle.

2. No claim with regards to a construction of a "viable cosmological model" is made in the section. The statement is that the metric in question is closed FLRW which is an important solution of the Einstein equations, and this statement is true. The reference [2] cited in the section is chronologically the first one on the subject, and therefore also has a certain right to be cited. Although it was published only a year ago, a quick search on the Web resulted in at least one published paper citing [2]. I do not list it because I have not read it yet.

3. It should also be mentioned that the results are somewhat unexpected – the metric in question was considered by many incompatible with algebraic structure of quaternions (see, e. g. D. Widdows, PhD Thesis, St Anne’s College, Oxford, UK). Since the results are "purely mathematical facts" and "might be worth mentioning", I mention them in a section of a mathematical wikipage. A complete removal seems unjustified, although sensible revisions and additions are welcome.

Trifonov (talk) 00:43, 22 March 2008 (UTC)

I'm not going to search through a 120-page PhD thesis for something that backs up what you said; I need a page number. Nor will I conduct a literature search on the basis of your "e.g." or "for example". I couldn't find the Raptis paper online. I could try to track down a physical copy at the library, but that would take time. You can't get something into Wikipedia by making it such a hassle to verify that no one will bother. It's your responsibility to make verification easy. For the reference of anyone else reading this thread, here's Trifonov's paper and Widdows's thesis.
I don't understand your objection to my use of φ. What do you mean that it's undefined? It's supposed to stand for a generic linear functional, not a specific one. It does assign to every quaternion a real number in a linear fashion, and it assigns to the product xy the value \langle x,y \rangle by definition of the latter. Clearly not every inner product can be defined this way—in particular the usual inner product on the quaternions can't be defined this way—but what you call the principal inner products can be defined this way. My φ(xy) is the same as your S(φ,x,y).
It is not true that "closed FLRW is an important solution of the Einstein equations". There is a class of important solutions to the Einstein equations which happen to be closed FLRW manifolds, namely those FLRW manifolds for which a(t) satisfies the Friedmann equations with k > 0. The generic closed FLRW manifold with arbitrary a(t) is not a solution of general relativity. Your claim is very much like saying that the class of all spherically symmetric manifolds is an important solution of the Einstein equations because the Schwarzschild and Reissner-Nordström metrics are spherically symmetric. You couldn't get away with that claim because everyone understands that there's no inherent connection between spherical symmetry and solutions of GR; most spherically symmetric manifolds are not GR spacetimes and most GR spacetimes are not spherically symmetric. That's just as true of FLRW manifolds, but by a linguistic accident the qualifier "FLRW" is associated with general relativity while "spherically symmetric" isn't. As a result your claim will slip under some people's radar; that doesn't make it any less wrong.
Again, the space of FLRW manifolds is simply the space of all homogeneous isotropic Lorentzian manifolds. You have homogeneity arising from the group action, and isotropy arising from the SO(3) symmetry of i, j and k, and a Lorentzian signature from your choice of inner product, so you get an FLRW manifold. It's a nice mathematical result, but that's all. Jumping from that to "a solution of the Einstein equations" is a complete non sequitur. Even with that clause deleted the mere mention of FLRW metrics will tend to mislead people into associating this result with GR, so the only way this result can reasonably be included in the article is with an explicit mention that the phrase "closed FLRW" is here used to refer to a class of manifolds having a certain symmetry with no relation to GR. -- BenRG (talk) 15:52, 22 March 2008 (UTC)

Here is the link to the Raptis paper.

Widdows - the page number is 10.

Einstein equations constitute a mathematical entity, an if something that looks exactly like one of the solutions is discovered in any area of mathematics, it is a solution. Some of them may belong to a certain class (say, closed FLRW). Even if they are not realizable in nature (whatever that might mean), they still should be called closed FLRW. To say that one class of solutions is more important than another, is a matter of taste, I suppose. Although I do not claim anywhere in the section or in the paper [2] that there is an explicit connection between [2] and GR, I certainly have no right to claim that there is none - absence of evidence is not evidence of absence, and some may be discovered in the future. In any case, as a new “nice mathematical result” on quaternions it probably deserves a mention on the Quaternion (mathematics) Wikipage. Trifonov (talk) 20:45, 22 March 2008 (UTC)

I'm a bit apprehensive about wading into a discussion about a subject I don't know much about. I wanted to record some citation statistics for the work in question. From Google Scholar: A Linear Solution of the Four-Dimensionality Problem: 8 non-self citations; GR-Friendly Description of Quantum Systems: 0 refs, Natural Geometry of Nonzero Quaternions: 0 non-self citations. The latter two articles might well prove fundamental to the subject in the future, but it's not necessary to cite such recent work, and it is hard to evaluate their relevance so soon after publication. Even the first article, the "linear solution", doesn't appear to be particularly notable as yet, but it wouldn't hurt to summarize it in a sentence or two in a "Recent Work" section. In any case, it must be clarified that this work is very recent and not mainstream, compared to the other material in the article. (I am also concerned about Trifonov's self-promotion and the policies WP:CONFLICT and WP:NOT#CBALL.) Sam Staton (talk) 10:02, 28 March 2008 (UTC)

“Natural Geometry of Nonzero Quaternions” came out in the February 2007 issue of IJTP, and considering that the average time of a publication process is about one year, it is probably a bit too early to expect citations. Yet there is one “GR-Friendly Description of Quantum Systems” came out in the February 2008 issue, and although it is completely unrealistic to expect any citations, nevertheless there is one It should be stressed that the section under consideration deals with mathematical truths about quaternions, and the fact I happened to get them first is irrelevant under the circumstances – sooner or later somebody else would have discovered them. It is an author’s obligation to promote the results (s)he considers important: there are quite a few cases in the history of science where delay in these matters did more harm than good. There is absolutely nothing non-mainstream about the discovery of Minkowski metric on quaternions (“A Linear solution…”) or the discovery of closed FLRW metric on nonzero quaternions (“Natural geometry…”). The results may prove useful in several areas, and as I noted above, are already being used by others. In any case it is difficult to understand why would anybody want to prevent them from appearing on Wikipages which is an excellent medium for this purpose, or equip with unnecessary and unjustified comments, and what would that achieve. Trifonov (talk) 18:27, 28 March 2008 (UTC)

Hello, It is important that someone coming to this article, to find out about quaternions for the first time, should find what is generally understood about the subject. For this reason I've moved the section in question to further down the article. It takes time for work to be fully accepted by the research community, and this needs to be indicated in the article -- I think it is now clear. (It is possible that the paragraph gives undue weight to this work; if so it should be abbreviated.)
You are correct that you have a duty to promote results you think are important. If you were writing an article by yourself, you could give the spin that you wanted. But since wikipedia is built collectively, by consensus, it has (necessarily) an extreme kind of neutrality. So promoting your own little-known results appears to be a 'conflict of interest', rather than a duty. In future, if you want to promote some of your work, it is best to propose it in the talk page first, see WP:COIC. Of course, there are other forums, conferences etc., that are more appropriate for publicizing your work. Sam Staton (talk) 12:07, 29 March 2008 (UTC)

The section seems to be quite neutral – it contains statements of the results with minimum waste, and the name of the author does not appear once in it. Trifonov (talk) 12:50, 29 March 2008 (UTC)

I have no opinion on the substance of this discussion. However, there are some important procedural issues.

  • Authors should not usually contribute material based primarily on their own research. That would be a little like authors refereeing their own material. The result may be correct and true and all that, but an independent editor should be evaluating the pertinence to the subject and appropriate emphasis to give. Wikipedia is not an appropriate place for authors "to promote the results (s)he considers important". The question is whether others, namely the appropriate community, consider them important.
  • It is not Wikipedia policy that references must be available on-line, despite the claim above:
I couldn't find the Raptis paper online. I could try to track down a physical copy at the library, but that would take time. You can't get something into Wikipedia by making it such a hassle to verify that no one will bother. It's your responsibility to make verification easy.
That would be absurd, excluding the vast majority of the written literature of the past as well as many modern scholarly publications which (unfortunately) are not available on-line or at least not for free.

Could we please try again on this, but follow Wikipedia rules this time? --Macrakis (talk) 21:42, 29 March 2008 (UTC)

Since many sections are written by experts in a particular field, such a demand would put them in a difficult position - they would be allowed to cite everybody but themselves, producing (possibly irreparable) gaps in the presentation. Trifonov (talk) 01:40, 30 March 2008 (UTC)

This is not usually a problem. There is a difference between citing a well-recognized standard text, and citing three articles that you've recently written. I think your articles look interesting, from the category theory, but I don't know enough physics to judge their importance, so I've put the importance tag on this subsection. Someone more familiar with the field can either remove the tag or trim/cut the section. Sam Staton (talk) 15:56, 2 April 2008 (UTC)

I think the tag is a reasonable precaution. Trifonov (talk) 23:00, 2 April 2008 (UTC)

Promotional editing is against policy

I'm concerned that User:Trifonov's activity on Wikipedia so far is entirely to promote his own work. Note this extract from Wikipedia:Conflict of interest:

Accounts that appear, based on their edit history, to exist for the sole or primary purpose of promoting a person, company, product, service, or organization in apparent violation of this guideline should be warned and made aware of this guideline. If the same pattern of editing continues after the warning, the account may be blocked.

Since I don't perceive a consensus here that Trifonov's 1995 paper belongs in the bibliography, I suggest it be removed. I'd also support removal of the section 'Quaternions and Minkowski metric.' EdJohnston (talk) 18:00, 2 April 2008 (UTC)

The list of references is too long, in any case. It is hard to argue which links should be here. (I'm not sure Trifonov is entirely to blame; I think others have been self-promoting too, e.g. User:Jemebius. That's not to say I propose those links be deleted; it's hard to say.) Sam Staton (talk) 18:37, 2 April 2008 (UTC)

These are two pieces of information about quaternions discovered relatively recently: (a) existence of Minkowski metric on quaternions, generated by their algebraic structure. (b) existence of closed FLRW metric on nonzero quaternions generated by their algebraic and differential-geometric structures. Interestingly enough, although it can be argued that the importance of the results is reflected in the role these metrics play in other areas of science, in particular, in physics (for this reason they are published in physics journals), and in the fact that a priori there are no obvious reasons why quaternions should possess them, this is probably irrelevant in this particular situation: the results are free of the burden of having to be “accepted by scientific community”; they are pure mathematical facts and cannot be “undiscovered”, even if the whole community wished for that.

Now, the question is - should Wikipedia present them? If not, the “Quaternion (mathematics)” article would be incomplete and not quite up-to-date, not a good thing for a dynamical medium such as Wikipedia. Trifonov (talk) 20:35, 2 April 2008 (UTC)

Your work needs to be commented on by others in reliable sources before it deserves coverage here. Adding links to your own work is blockable otherwise. EdJohnston (talk) 20:45, 2 April 2008 (UTC)

It has been (see the discussion). The results won't change no matter what the comments are, and no other links are available to the their origin. Trifonov (talk) 22:37, 2 April 2008 (UTC)

This article is too technical for a general encyclopedia. (talk) 11:01, 20 April 2008 (UTC)

A mistake in the absolute value definition??

Hi, it appears that there is an error in the section that defines how the absolute value function acts on quaternions. It says that p^*=a^2-b^2-c^2-d^2, but I think this is not the converse of p, since isn't p=a+bi+cj+dk?? Thus making p^*=a-bi-cj-dk the proper choice for p*? A math-wiki (talk) 08:12, 2 May 2008 (UTC)

Yes, that was strange. Maybe a leftover from previous edits. I've replaced it with the definition of the conjugate, which is exactly as you stated. Thanks! Koeplinger (talk) 21:10, 2 May 2008 (UTC)

H as a union of complex planes

In the section it mentions a unit sphere called "2-sphere" in 3-dimensional space with the coefficients of i, j, and k as the axes. It does not seem to mention anywhere that the centre of the sphere is the origin in the 3-D space, which confused me till I reread the whole thing. As a layman, I don't know how I should edit this article, so I think someone else should do it. —Preceding unsigned comment added by Anuragsahay (talkcontribs) 11:08, 8 May 2008 (UTC)

Quaternion Article looking better all the time.

Hi everyone, I have been working so hard with school for a long time now and have had little time for typing on wikipedia for a while.

I have to say that the article on Quaternions is looking great!

One section that I found very interesting is about the relationship between quaternions and Minkowski space.

In my investigation into classical works on quaternions, I was a little disappointed to see that early researchers into the subject did not seem to have paid to much attention to a very obvious potential product.

If the product of each element of a quaternion is multiplied by itself we get

T^2 - X^2 - Y^2 - Z^2

Which I believe is the Lorenz invariant, for a Minkowski space with a trace of minus two.

That is why I found the new section on the relationship between Minkowski space, and quaternions fascinating.

What I found a little bit unfortunate was that that section was just a little to technical for me. I think that very technical explanations are important, however simple explanation more accessible to us unwashed masses I think would also be helpful. —Preceding unsigned comment added by Hobojaks (talkcontribs) 04:58, 29 June 2008 (UTC)

As for the product you've found, this would be the Q_t component as mentioned in the article in section Quaternion#Quaternion_products. Indeed, if a quaternion is to be viewed at as a four-vector, then the Q_t component (or "zero component", or "t component" as it might be called also, depending on notation) is Lorentz invariant, therefore reflecting Minkowski geometry. By the way, when multiplying two quaternions, then this component is also identical to the so-called "inner product of four-vectors" (Four_vector#Mathematics_of_four-vectors); again, reflecting this geometry.

Historically, this similarity was very well known, and you may want to follow User:Rgdboer's edits and references here in Wikipedia. Just to mention one, the Quaternion Society (1899 - 1913) was actively working in the field. The historical development, however, went towards Clifford algebras and Lie algebras, because quaternions did not bring relevant benefits to the description of physical law. A simple rewrite of known laws using quaternions is not enough, as quaternions would not be required. That's about where we are today. Thanks, Jens Koeplinger (talk) 13:25, 29 June 2008 (UTC)

Quaternion444 image

Computer Rendered Quaternion

I moved Image:Quaternion444.jpg here from the main article (see picture on the right). It is not clear what it represents and what it has to do with quaternions. It looks like Koeplinger has the same issue, as he added a {{fact}} tag with the edit summary: "nice picture added; but we need reference info: What is being depicted? How was the image generated? What tools were used? ... something along these lines". -- Jitse Niesen (talk) 13:50, 29 June 2008 (UTC)

Minor random edits

Please use caution with minor edits. I undid revision 224630598 by for the following reasons. One of the first (and trivial) facts about quaternions is that they constitute a real vector space. In other words, each quaternions is a vector, and since the dimensionality of the vector space is four, it is a 4-vector. So the information added in the edit is completely extraneous. Moreover, (a, b, c, d) is not a vector, this ordered quadruple constitutes the components of a vector in a particular basis (in this case (1,i,j,k)). Please be absolutely sure that you know what you are talking about, especially now, that most experts are on vacation. —Preceding unsigned comment added by Trifonov (talkcontribs) 00:05, 10 July 2008 (UTC)

Possible Mistake in section "Quaternions and the geometry of R3"

The sentence "This is equal to the product of p and q as quaternions" should be "This is equal to the VECTOR PART of the product of p and q as quaternions". Am I right? —Preceding unsigned comment added by (talk) 19:43, 28 July 2008 (UTC)

You're right. Fixed. Ozob (talk) 20:15, 28 July 2008 (UTC)

This is an outstanding new section

Congratulations on this outstanding new section, and also deleting that long list of grassman products which I always found somewhat annoying.

Since vectors were an integral part of hamilton's original quaternion calculus, saying "vectors replaced quaternions" really grates on me, as being a misrepresentation of what replaced what. Thanks for shedding some much needed light on the subject.

Older texts before 1901, on the subject of quaternions, are really on the subject of 'quaternion calculus' which was a complete mathmatical system. These texts spend a great deal of time on the subject of vectors, which were an integral part of quaternion calculus. For example Hamilton devotes more than 100 pages of elements of quaternions to the concepts of the addition and subtraction of vectors, before he attempts to address the subject of multiplication and division of vectors in book two. The notion of a quaternion first arises when the subject of multiplication and division of vectors first arises in book two, because in classical quaternion calculus the product or quotient of two vectors is a quaternion. Your outstanding article addresses this issue.

Another way to look at the difference between the vectors of hamilton's calculus and vector analysis was that Gibbs-heavyside-wilson notation popular in the late 20th century was a dumbed down version of Hamilton's quaternion calculus. A dirty trick used by Gibbs-Wilson's book vector analysis, was to expropriate a great deal of hamilton's vocabulary. Very notable in this expropriation is the rip off of the word vector. It makes it hard to communicate because the word vector means two different things in the context of Hamilton's quaternion Calculus and Gibbs-Wilson vector analysis, and the difficulty is compounded by the fact that they both represent the same geometric idea.

Certainly Hamilton's classical vectors add like Gibbs-Wilson vectors. You make a great point there.

3-Vectors are not closed under multiplication and division, but to cover this fact up Gibbs-Wilson dumbed down Hamilton's version of quaternion calculus, by making two different 'products'. That way one product, was a vector and the other was a scaler, but these were really just two haves of Hamilton's original product. Again as your outstanding article points out.

The big loss was that the concept of division was left out of the dumbed down Gibbs-Wilson vector. I have been reading more of goldstein, and he devotes several sections to what he calls a Q matrix, which is the matrix representation of a quaternion, which after reading just a little bit of hamilton's work makes a lot more sense to me.

Looks like to me, that trouble arose in that around the turn of the century people actually discovering new things, were coming from a culture and way of thinking based on Euclidean Geometry and linear algebra, and not based on Hamilton's quaternion calculus, and that advocates of Hamilton's calculus were mostly translating works written in the context of linear algebra into their own notation, rather than formulating special relativity in terms quatenion notation to begin with.

It looks to me like from what I can understand of goldsteins treatment of using 'Q matrixes' to define lorentz transformations that Hamilton's 1842 notation

QRQ^-1 could get pretty far by itself.

Here is what I like about classical quaternion calculus, I find it easy to understand. All my life on and off I have picked up books on relativity, but the weird Einstein notation with all the strange subscripts and super scripts and trying to imagine time as an imaginary fourth dimension always tended to hurt my brain.

Hamilton's point of view appeals to me more. In quaternion calculus time is a real number! I like that, and it makes sense to me.

The elements of Hamilton's vectors were what he and Tait called 'geometrically real' numbers, which were numbers that when squared produced a negative scaler.

Hobojaks (talkcontribs) 03:39, 22 August 2008 (UTC)

A small objection

I do have a small objection, to some of the notation in the article, in that I think there might be a few small tricks that could be used to add greater clarity to the outstanding intent, but I am reluctant to offer any quick suggestions, because I want to make my suggestions more well thought out.

One small change that I did make was to change 'imaginary quaternion' to right quaternion. I believe that was the word you were looking for? A right quaternion is the proper term for a quaternion that has a zero scalar part and hence consists of only a vector. I added two foot notes citing Hamilton and Hardy with page links to the original classic texts on the subject.


"Because the vector part of a quaternion is a vector in R3"

I suppose another vague objection, is to the use of the word 'is' where a better term could be chosen to indicate that the two notions of a vector while different both work just like each other. Greater clarity in distinguishing the two uses of the word vector in your article might be a little helpful? But I want to think about this a little more before suggesting any tangible changes.

Don't get me wrong, I think you have created an outstanding article here, and my constructive critisism here is an indication of an outstanding article which with a little work can achieve total perfection.

Hobojaks (talk) 18:37, 22 August 2008 (UTC)

The modern concept of vectors is very different from Hamilton's concept. Saying that they "replaced" Hamilton's quaternions (and in particular his notion of vectors) is entirely appropriate. There is simply no natural way to multiply or divide arbitrary (modern) vectors; it requires additional structure, and, as the article states in the "Algebraic properties" section, that structure only rarely exists. Even if it does, it requires arbitrary choices: The cross product depends on a choice of an orientation for R3, and without that choice it produces only a pseudovector. This is because the appropriate abstraction of the cross product is a combination of the wedge product and the interior product with the volume form, and every space (such as R3) that admits a volume form also admits an oppositely oriented volume form; therefore the cross product of two vectors depends not only on the vectors but also on a choice of orientation.
Regarding terminology, my understanding is that "right quaternion" is entirely a historical term at this point.
I agree that there is a slight ambiguity in the statement that "the vector part of a quaternion is a vector in R3". But the article uses "imaginary quaternion" when it intends to refer to a vector in Hamilton's sense and "vector" when it intends to refer to a vector in the modern sense. With this understanding I believe that the statement is correct: The vector part of a quaternion is its projection onto the vector subspace of H spanned by i, j, and k. Ozob (talk) 19:16, 22 August 2008 (UTC)

Requested move

The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the proposal was move back. JPG-GR (talk) 02:01, 11 August 2008 (UTC)

Quaternion (division algebra)Quaternion — Proposal to revert the recent undiscussed move of "Quaternion" to Quaternion (division algebra). This is the primary topic for the title "Quaternion". — Sam (talk) 13:52, 6 August 2008 (UTC)


Feel free to state your position on the renaming proposal by beginning a new line in this section with *'''Support''' or *'''Oppose''', then sign your comment with ~~~~. Since polling is not a substitute for discussion, please explain your reasons, taking into account Wikipedia's naming conventions.
  • Revert - please go ahead and revert the move. This is a longstanding article, and the presumption should be in favor of the status quo. The burden of persuasion is on the person who wants to rename it; since that burden was not met, I think it can be reverted without discussion. —Steven G. Johnson (talk) 14:21, 6 August 2008 (UTC)
  • Revert The dab page seems to have been created to give space to an obsolete meaning of quire. A dab header would be more than enough. Septentrionalis PMAnderson 16:08, 6 August 2008 (UTC)
  • Revert, add dab instead. Koeplinger (talk) 16:26, 6 August 2008 (UTC)
  • Revert. The present article is by far the most common usage of quaternion. Ozob (talk) 19:47, 6 August 2008 (UTC)
  • Revert. Nothing a dab page can't handle. I didn't know there were any other meanings... Baccyak4H (Yak!) 19:58, 6 August 2008 (UTC)
  • Revert. False impression of a content that is too orthodox algebraic. Trifonov (talk) 20:10, 6 August 2008 (UTC)


Any additional comments:

Whatever happens, the links have unfortunately been messed up by the recent move, and need to be repaired. Sam (talk) 13:57, 6 August 2008 (UTC)

The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Customers for quaternion-explaination

Many non-mathematicians need to find the definition of a quaternion as a rotation. Only few mathematicians are interested in the "division algebra"

Stamcose (talk) 21:25, 6 August 2008 (UTC)

What's wrong with quaternions and spatial rotations? -- Fropuff (talk) 21:35, 6 August 2008 (UTC)
Rotation is a subtle topic; the division algebra contributes to the culture of rotation operators in three dimensions, especially with its concept of versor. Regardless of "non-mathematicians' need to find the definition as a rotation", the moves have not been justified. A speedy reversion is called for.Rgdboer (talk) 22:19, 6 August 2008 (UTC)

To be more explicit:

The new standard for star sensors as flown on spacecraft is to output the determined attitude as a quaternion. Spacecraft engineers serching "quaternions" are looking for the definition, most likely to make a transformation to a 3x3 rotation matrix (direction cosinus!). This new use has created a completely new "market" for quaternions that must be considered to have been a very specialized topic in mathematics before!

Stamcose (talk) 07:06, 7 August 2008 (UTC)

The formula for the rotation matrix the spacecraft engineers would be looking for (and should easily find!) is:

q_1 q_1 & q_1 q_2 & q_1 q_3 \\
q_2 q_1 & q_2 q_2 & q_2 q_3 \\
q_3 q_1 & q_3 q_2 & q_3 q_3 
q_4 & -q_3 &  q_2 \\
 q_3 &  q_4 & -q_1 \\
-q_2 &  q_1 &  q_4 

I think this formula is nowhere in this "Main quaternion article". I think it should be included somewhere!

Stamcose (talk) 13:48, 7 August 2008 (UTC)

You've seen quaternions and spatial rotation, right? I think it was originally a section of this article which was split off because the article was getting too long. Anything that would belong in that former section should go in the spun-off article. That article already has a formula for an orthogonal rotation matrix in terms of the components of a unit quaternion, though it doesn't seem to be equivalent to yours (and I don't see how yours can be correct at all, but I may be missing something).
The problem I have with a disambiguation page is that it would essentially be saying "quaternion can mean either this mathematical entity as discussed in this article, or the same mathematical entity as discussed in this other article". I don't think that's how disambiguation pages are supposed to work. This article discusses quaternions in the broadest sense, while the other article discusses a particular application (albeit a very important one). I don't see any way to organize them except to make Quaternion point to this article, which then points to its subarticles. -- BenRG (talk) 16:27, 7 August 2008 (UTC)
Originally there was an Quaternion (disambiguation) that could branch to Quaternion. Standard is the other way around (as far as I understand), the disambiguation is the word itself without paranteses pointing to more specific items. But sure, a pointer at the beginning of your article telling where to go if one only wants the definition and how to compute the rotation matrix would do it as well. Before the users gets scared off by your complete treatment of the Quaternions as an algebra!
Stamcose (talk) 17:57, 7 August 2008 (UTC)
Try looking at Wikipedia:Disambiguation#Primary topic, which explains why Quaternion (disambiguation) is more appropriate than making Quaternion into a disambiguation page.
On another point, I am really very concerned by your opinion of the present article. Is it really that hard to follow? I mean, I'll admit that there are parts which are more appropriate for mathematicians—no disagreement there—but you seem to feel that the whole article is abstruse and illegible. Part of my worry is Wikipedia:Content forking: If the present article is really so bad, then I worry that quaternions and spatial rotation will remodel itself to fill the gap, when the right solution is to fix quaternion. Maybe it would help if you explained what you thought was wrong with the present article. Ozob (talk) 20:32, 7 August 2008 (UTC)