Talk:History of quaternions

Article creation

Hi, I have taken the liberty of copying and pasting some material from Classical_Hamiltonian_Quaternions into this empty space for an article. I don't pretend that this material constitutes any where near what is needed for an in depth article about the history of quaternions, however I do think there is some valuable material here.

I have a suggesting for a description of the article.

It should be a history of the who and when of quaternions. Issues of historical notation and methodology should go into either classical_hamiltonian_quaternions for things written in the notation of 19th century authors, up until 1901, when Hamilton's second and final volume of elements of quaternions was written.

Math and modern quaternions already contains a short history of quaternions.

The material in this article originally started out in that article but it was suggested that the material be split off from the main article.

It still needs a lot of work. —Preceding unsigned comment added by Hobojaks (talk • contribs) 22:28, 27 January 2008 (UTC)

Don't know if this article will ever be comprehensible by non-mathematicians. But just in case, should point out that Special Relativity supports the quaternion idea specifically, rather than just alluding to it. —Preceding unsigned comment added by 65.217.188.20 (talk • contribs) 14:39, 12 July 2008

Red text

Wikipedia just doesn't do multi-coloured text. It's not it's style.

So I've removed the <font> tags. Where we really need to underline that something is modern notation, we can say so as and when. Jheald (talk) 09:58, 13 September 2008 (UTC)

I have now copy-edited the text a bit, and I think it should now always be clear from the context, without highlighting, when old notation is being used and when new notation is being used. Jheald (talk) 11:43, 13 September 2008 (UTC)

Original research

Please be aware of WP's policies WP:No original research and its subsection WP:No original syntheses.

I have serious concerns about the latest additions to this article, particularly the line of thinking that Special Relativity and General Relativity in some way "vindicate" Quaternions.

This is a pretty out-there claim to make, and to stay in the article it needs to be line-by-line sourced -- and in particular, sourced to articles which actually explicitly make that claim, not just articles from which the wiki-editor thinks it can be inferred. Jheald (talk) 21:11, 14 September 2008 (UTC)

"Euclidean" / Lorentz invariant

I should add that the description of the proponents of the vector notation as "Euclideans" seems inappropriate to me, as well as OR.

Line removed thank you for pointing that out.

Hobojaks (talk) 03:09, 15 September 2008 (UTC)

The truth is that both quaternions and vectors describe a Euclidean world; they just use different notation to do it. Jheald (talk) 21:17, 14 September 2008 (UTC)

(ec) Both quaternions and 3-vectors describe the 3D Newtonian/Euclidean world. Neither of them describe the Minkowski of Special Relativity, and neither of them contain the Lorentz transformations. The vector paradigm gets there comparatively straightforwardly by going to 4-vectors, with a (+,+,+,-) metric; and Lorentz transformations can then be expressed with a 4x4 matrix.

Actually the scalar of the product of two quaternions is the Lorentz invariant.

The FitGerald-Lorentz transforms are based on the idea of dividing the square root of the scalar part of the square of a quaternion.

In other words, 1/sqrt(Sq²)

• Yes, that may match the expression for the norm of a 4-vector. But the "FitzGerald-Lorentz transforms" are not "based" on it, at least not in any sense with a meaning for quaternions, because there's no way to represent such transformations with quaternions. Jheald (talk) 14:28, 15 September 2008 (UTC)

I think that some degree of confusion might have come from the fact that the word quaternion like the word vector has been somewhat redefined. For example the tensor of a quaternion, a quantity that preforms an act of tension, stretching or shrinking it, is sometimes called the euclidean norm. This is a serious redefinition of classical thinking on the subject. Riemann died just one year after Hamilton. They were both interested in approaches to higher dimensional space. His first lecture in 1954 was just a little while after Hamilton's book lectures on quaternions came out.

Both made significant contributions to the notion of four dimensional space, but technically Hamilton came first.

Yes, analogues of quaternions do exist for flat hyperbolic spaces; but to see what's going on and why, it's really easiest to jump to the relevant Clifford algebra Cl(3,1), because Clifford algebras offer a systematic way of thinking about vectors and quaternion-like things together in flat spaces with metrics with any signature of signs, and then look for the even part of that. Jheald (talk) 21:37, 14 September 2008 (UTC)

OR (continued)

You may have a good point here, not in saying that any of this is not something that I can source, but in that this history is controversial.

I can source a lot of the material from the history given by Lee Smolin in the Trouble with Physics, I have it underlined and ready to go.

According to a quick search on Google books, that book doesn't contain the word "quaternion" or even the word "Hamilton" even once.

Can I underline what I wrote above: controversial claims need to be sourced to publications which actually explicitly make those claims. That does not seem to be the case here. Jheald (talk) 21:42, 14 September 2008 (UTC)

The other thing is that I am adding in a great deal of material in a small amount of time here, so I may in good faith have made some factual errors. If you could please pick out what you think needs to be sourced, or possibly if I have allowed my own point of view to filter into the article I think this would be unfortunate.

The controversy between Tait and the Gibbs Heavyside crew is well documented, I realize that the 1977 to present section has some statements that are very controversial and need to have opposing views presented.

First of all, before we even consider WP:NPOV, we need to consider WP:Reliable Sources. Statements which can't be verified to reliable sources shouldn't be there at all. Jheald (talk) 21:47, 14 September 2008 (UTC)

The fact that Einstein came to believe that quantum mechanics was not true is well documented.

That's as may be; he certainly was dubious as to whether it was the final answer. But it has nothing to do with the issue of quaternions as against vectors. Jheald (talk) 21:47, 14 September 2008 (UTC)

Modern string theory has found many solutions to Einstein's equations, and is the major topic of Lee Smolin's book.

That is also somewhat questionable, not least by Mr Smolin. But again, it has nothing to do with the issue of quaternions as against vectors. Jheald (talk) 21:47, 14 September 2008 (UTC)

The solution to the problem of unifying quantum mechanics and general relativity is still an open ended question, and quaternions have not been vindicated as the one and only solution. If the current text gives that impression I am sorry for my lack of writing skills.

Quaternions have not been vindicated as the one and only solution. This statement fundamentally misunderstands the issue.

There is no difference in physical content between the world described by quaternions and the world described by vectors. They're both just a mathematical language for describing flat 3D space.

Here is where I think that you might have some miss conceptions. First of all every quaternion can be deconstructed not only into a vector and a scalar, but also into a tensor and a versor. The versor represents a very curved notion, it can be represented graphically as an great circle arc on a unit sphere, where as the tensor part of the quaternion when deconstructed in this way scales the sphere larger or smaller this effecting the curvature of the arc, and Hamilton showed that each point in space and time have these characteristics. What determines if the space is 'flat' or not, in the Riemann sense is the question of if time is proceeding at the same rate at every point in space at every point in time. If in some places and times the clocks are moving slower than in others, which is the case in general relativity then you get a gravity effect, at least that was what Bertrand Russel posited. The fact that the vector part of a quaternion is for ever coupled to a time coordinate, combined with the fact that the product of a vector with itself, it seemed to some thinkers at least made it very different from the Euclidean notion of a distance.

Yes, quaternions offer the two-sided representations for rotations, which is useful, and is relevant to understanding spin. But that has been completely assimilated in physics since the 1930s, and is completely integrated, in Group theory or Clifford algebra analyses -- which are comprehensive in a way quaternions are not, because quaternions don't even describe space-time, never mind relativistic quantum mechanics or general relativity.

I think that your points about group theory and Clifford algebra are important in an article about the history of quaternions, but the time line I think is important. It will come up again later near the end of the 20th century, at least in the version of history given by Smolin. As the title suggests, the trouble with physics, is that there are way to many unified field theories, and no way to falsify them. Hence Lie groups and Clifford algebras are an early example of a problem which has only gotten worse with time. So dating exactly when the trouble first began is an important part of the history of quaternions. In fact in some respects, since before quaternions there was only algebra and geometry, this big bang creating a whole universe of algebras seems to have the quaternions at or near the singularity.

Also this article sites peer reviewed articles which state that quaternions have applications in general and special relativity, but I am sure that this is just one view, and your view is important as well, and should be included in the article if you have a source that you can site on the subject.

Also I am inclined to agree with you that from the mid 1930s there began a period where the subject of quaternions was almost completely ignored. At one point there was a time period that I mention in just one paragraph, where as far as quaternions were concerned nothing happened, but since this article sites modern sources that the period where it was thought that quaternions did not have uses in relativity ended, it might be relent to at least mention that nothing happened in these times, quaternion wise, but that changes were taking place.

But well hey gotta go to sleep, got school tomorrow, but thanks for all of the valuable input.

But the point is that none of this is an issue about quaternions vs vectors, because both describe exactly the same physics. Vectors won, because they were more direct and more straightforward, and could do everything that was being asked of quaternions could in a simpler, more transparent, easier to understand way. Jheald (talk) 22:06, 14 September 2008 (UTC)

The FitzGerald transform part, I can document, I have been looking at Lorentz's 1895 and 1904 articles. The fact that there was great hopes for quaternions before 1933 can also be well documented. Hobojaks (talk) 21:33, 14 September 2008 (UTC)

Bring it on. You write:

Two years later in 1889[9]George Francis FitzGerald an Irish Professor at Trinity College,Dubland and a devoted follower of the Hamiltonian camp, well trained in quaternions easily predicted the solution that moving objects were foreshortened in the direction of travel. In the 20th century this theory which flows naturally from Hamilton's point of view was experimentally verified again and again.

The FitzGerald contraction prediction was not inspired by quaternions. Quaternions did not "predict the solution that moving objects were foreshortened in the direction of travel." The theory does not "flow naturally from Hamilton's point of view". And I doubt the prediction was "easy".

Let's clear all this rubbish out, and stick to what is actually in the texts, that you can verify. Jheald (talk) 22:19, 14 September 2008 (UTC)

Fitzgerald contraction.

Let me take another stab at it here. These are facts.

(1)Fitzgerald introduced the Fitzgerald transform, now called the Lorentz transform. I have copies of Lorentz's original papers in which Lorentz admits this.

• Fair enough, but by itself that has nothing to do with quaternions.

(2)Fitzgerald was a professor at the trinity school of Dubland, the same school that Hamilton taught at. Again well documented fact. I am willing to allow that Fitzgerald was a great thinker in his own right, not a camp follower. Every time you listen to a radio, or talk on a cell phone, thank Fitzgerald. He was the one who first suggested the possibility of making man made electromagnet waves, and then Hertz following his suggestions made the first radio waves.

• Again fair enough, but by itself it has nothing to do with quaternions.

(3)There is a natural flow from Hamilton's thinking to Lorentz's in several respects.

(a)The idea that space has to be four dimensional, that space and time are intrinsically linked. Again this is pretty much undeniable.

• Can you point to a WP:Reliable Source that states that the line of thought that led either FitzGerald or Lorentz to the Lorentz transformations was in any way connected with thoughts of quaternions?

(b)The scalar of the product of two quaternions is the Lorentz invariant

• Maybe. But the trouble is that in all other respects, quaternions simply don't have enough structure to represent space-time co-ordinates. Correct me if I'm wrong, but it would seem that there is no easy way to represent space-time boosts (Lorentz transformations) - quaternion rotations don't have enough structure to do it. It needs a bigger algebra - something like Cl(1,3); or Cl⁺(1,3), if you're prepared to accept muddling up vectors and pseudovectors in the way Quaternions do.

(c)Hamilton 'formulated the wave equation using quaternions in a particularly elegant fashion', that is a quote from Tate. The wave equation that says that light always moves at the speed of light.

• But does Hamilton's formulation imply Special Relativity? Answer: No. Because Hamilton's quaternions are rooted in a fixed reference frame for 3D. They don't allow you to do a Lorentz transformation and switch from one relative frame to another, because the algebra simply isn't big enough to contain such a transformation.

The wave equation that says that light always moves at the speed of light. But because Hamilton's quaternions don't let you change from one relative frame to another, you can't express the idea that "such a transformation leaves the speed of light invariant" -- because you can't express the idea of such a transformation. Jheald (talk) 11:36, 15 September 2008 (UTC)

The the other thing I am willing to retract is that Russel was a camp follower if you think my early wording implied this.

Let me give it another try! Thanks for the encouragement.

Also when you multiply several quaternions you can bend, rotate or subject space to acts of tension and compression just about any way you want.

• You can rotate space, and you can magnify space. You can't reflect space; and it's misleading to say that you can "bend" space - the space stays flat. Jheald (talk) 11:39, 15 September 2008 (UTC)

The point that I am getting at, the one that Smolin brings out is about the importance to modern physics of the geometry of space. The Einstein school of thinking, saw a space of four dimensions as an array of real numbers, much like the 'rehabilitated quaternion' you are doing a fine job of explaining.

But there is another line of thinking that needs to be traced in the history. Yes Hamilton's ideas sat dormant during a long period of the mid 20th century. The were only accessible in musty old books sitting on library shelves collecting dust. Then that changed again, and my vision for this article is to document that transition.

• Not really. The one important aspect of quaternions that got left out in the vector treatment - the half angle formulae for rotations - came back into physics with a vengeance with Dirac's theory of the electron (1928). But it's an equation in 4D space-time, not 3D space, so although the half-angle two-sided rotations are there, they can't be described with quaternions. Instead they were described with Dirac spinors. And the Dirac spinors were then understood in terms of Spin groups (for their group properties) and Clifford Algebras (for their overall algebraic setting). These two concepts are more powerful and more general, and assimilate everything you can do with quaternions.

Nowadays, even when quaternions are being revived for 3D rotations in computer games or satellite control systems, the world has moved on; we now recognise those quaternions as simply one particular setting for the more general tools. Jheald (talk) 11:52, 15 September 2008 (UTC)

Lee Smolin does mention Octonians in his book, but the main topic is the trouble that science is having with all these theories based on higher dimensions. It is a good read, and has some important material very relevant to the article.

The important point I think is to point out that there are certain properties of four dimensional space.

Hamilton proves that there is only one reasonable way to multiply vectors, and when you do it that way the answer you get is another quaternion, and further more the scalar part of the answer is the lorentz invarient. Hamilton's equations further demand that if light is a wave that it must travel at a certain speed. Hamilton set up the equations, suggesting that there would be abundant thing in nature that acted that way, light just happened to be one of them.

• What Hamilton proved was that the Quaternions were a division algebra. You could always divide one quaternion by another and get a definite answer, also a quaternion. There were no idempotents in the algebra, and no nilpotents apart from zero.

But it turns out that sometimes you want to have idempotents and nilpotents. Sometimes they are physically meaningful. So, yes, in Clifford algebras you can almost all the time divide one Clifford number by another and get a definite answer, just like you can with Quaternions. But, just as the proof requires, there are some elements of the algebra that break it.

And it turns out that those "breakages" are just what you need to describe inevitable physical singularities of some parts of the co-ordinate system -- such as the light-cone in space-time.

Hamilton's quaternions don't have that property, but then they don't describe space-time. Jheald (talk) 12:06, 15 September 2008 (UTC)

What Lee Smolin lets me document is the motive. General relativity can't be reconciled with quantum mechanics.

What I can prove as well is that to a large extent Einstein had to reinvent some of Hamilton's thinking, but I believe he just might have missed something, that was proved by Frobenus 12 years after Hamilton's death. Hamilton just showed that any other way would be an 'absurdity', I know right were to dig out that quote, from lectures on quaternions. Frobenus proved that it was not only absurd but impossible to do it any other way, now overlooked.

• See previous comment. The fact that other algebras have spaces of elements which don't obey Frobenius's requirements turns out to be a very useful and important aspect, with real physical meaning, of why they can describe space-time -- and why Hamilton's quaternions don't. Jheald (talk) 12:09, 15 September 2008 (UTC)

Now in 2007 some people are saying that quaternions satisfy the Einstein equations, and what I think is interesting to do is to trace the idea of the development of quaternions over the last century and a half.

• Quaternions don't in any normal sense satisfy the Einstein equations.

Quaternions are an important part of bigger algebraic systems, and there are times when they can turn up. But it's usually more revealing to try to see what's going on in terms of those bigger systems, and through group theory, rather than thinking there's any kind of in-line connection with what Hamilton was trying to do. Jheald (talk) 12:30, 15 September 2008 (UTC)

Let me make a few changes and see if I can get things a little more to your liking? —Preceding unsigned comment added by Hobojaks (talk • contribs) 01:44, 15 September 2008 (UTC)

• Gut it. Wikipedia is not for expressing your own syntheses. Stick to reporting and summarising what modern WP:Reliable Sources have to say. Jheald (talk) 12:30, 15 September 2008 (UTC)

Minkowski Space and Time 1908

I have a copy of Minkowski's paper front of me right now. Interestingly it does not have any matrix algebra in it at all. He died just one year later.

I suppose that there must have been some other articles he wrote, but his metric in terms of a matrix is absent from this particular paper.

The first metric tensor I can find in my collection of articles titled "principle of relativity" is Einstein's article

The foundation of the General Theory of Relativity in 1916.

In this article Einstein gives a metric tensor with a trace of (-,-,-,+), which is essentially the scalar part of Hamilton's product. It seems to me that at this point a critical line of thinking had changed among the German Physics community, in that the Scalar part of a quaternion had been ripped away from the complete whole and viewed as an entity in its own right.

• It didn't have anything to do with quaternions. Quaternions don't represent 4-vectors, because there's no appropriate way to make them transform like 4-vectors. Jheald (talk) 12:37, 15 September 2008 (UTC)

Also I believe that Einstein may have selected the methods he did because he was working inside the German speaking community where Riemann's ideas were better known than Hamilton's. In any case the story goes that a friend told him about some math he could use. In this case the fate of the quaternion in the mid century was an accident of history.

If Einstein was unaware of quaternions, either way, he came up with the same form for the scalar of the product that Hamilton did, yet I think that it is important to record in the history, that Riemann's four dimensional ideas were very different from Hamilton's.

Riemann had the idea of extending the Euclidean distance formula, called in flat space the Euclidean inner product.

Essentially Hamilton's original idea of a quaternion, is different from other spaces because it deliberately does not define an 'inner product'. I know that some notion of a rehabilitated quaternion for some reason seems to dominate wikipedia these days, can't resist the urge to define one, but in doing so, it seems to me, as it did to Tait, that you have lost an important idea.

In saying that the product or the quotient of two vectors is a quaternion, the idea of distance independent of time has been abolished. And Hamilton set about proving this relentlessly, with proof after proof, and example, page after page, and in my estimation leaving no possibility in the minds of his readers that there is any possible alternative.

• You're horribly confused here. Hamilton's scalar was just that - a scalar. Not a time co-ordinate. Hamilton may have flirted with the idea of calling it a time co-ordinate, but it is simply not used as a time co-ordinate. Jheald (talk) 13:09, 15 September 2008 (UTC)

Hamilton's idea was a classical vector plus a scalar, three Geometrically real spacial coordinates and one real time coordinate. He proved that there was really no reasonable alternative. He and his followers believed that they have overthrown Euclid. He proved his ideas with some very long winded at times, yet also relentless logic. Riemann's idea was different, and more an extension of Euclid's idea than an overthrow of it.

Riemann's space had time and space as being much more interchangeable. Riemann's idea was an array of four or more identical numbers, not the sum of two distinctly different kinds of numbers. Each of the four dimensions had equal parity. Hence the notion of a curved space, that time could curve space because all four dimensions were so much alike that they could curve into one another.

Hamilton at the time of his death was working on extending the idea of his Del operator. Rocketing at least in my opinion towards an alternate formulation of general relativity, based on time changing due to gravity fields. He never got there, but the evolution of this idea can be traced at least to 1925, in Bertrand Russell's book. The idea of the rate of change of time with respect to time, being the root source of gravity.

Then the war came and the lights went out and people forgot for the most part about quaternions, and in the USA at least there were a lot of German speaking scientists developing the atomic bomb, but that physics developments were very much secrete.

But a very important point that Lee Smolin makes is that during this dark age, some things that Hamilton and his cohorts had proven through cold logic had become lost. Smolin documents the quest to look for solutions in an ever higher number of dimensions.

• Smolin doesn't even mention Hamilton or quaternions. To put the words "some things that Hamilton and his cohorts had proven through cold logic had become lost." into Smolin's mouth is simply and utterly without foundation. Jheald (talk) 12:41, 15 September 2008 (UTC)

Lee Smolin has some interesting thoughts on the culture of 'modern' physics that are very relevant.

I don't think that anyone can find a good classical source that makes the claim that Quaternions are Euclidean, the whole title of Hamilton's book Elements is based on the notion that he is completely and totally overthrowing Euclidean Geometry. Has anybody making that claim taken the time to read it?

A lot of things can be deduced from Hamilton's vector multiplication and division formulas, and they were considered axioms of a new geometric calculus. From them a great deal can be deduced, like the intrinsic four dimensional nature of space, the Lorentz invariant, that once it was shown that light was a wave that the quaternion wave equation required that light always travel at the speed of light, that higher dimensional spaces, other than octonians are an absurdity at least as a candidate for the actual geometry of space and time.

• No, it is Maxwell's equations that require that light always travel at the speed of light. The fact that the equations could be written using quaternions is neither here not there. Quaternions were simply a way of writing them down; just as Heaviside and Gibbs could write them using vectors. The important point is that the Quaternion formulation could not even simply represent Galilean transformations, never mind Lorentz transformations. It is not quaternions that require that light always travel at the speed of light. If anything, Quaternions serve to obfusticate the fact that Maxwell's equations do not transform appropriately under Galilean transformations, not point the way to something new. Jheald (talk) 13:00, 15 September 2008 (UTC)

• Saying "the intrinsic four dimensional nature of space" or "the Lorentz invariant" can be deduced from Hamilton's quaternions is drivel. The quaternions are simply one mathematical structure among many, many others. Saying that there are features of the quaternion structure which must be reflected in the physical world automatically provokes the question "Why?" And with good reason. The places where division breaks down in some other algebras turn out to be just what you need to describe the geometry of higher dimensional projective spaces, and the geometry of space-time. Jheald (talk) 13:00, 15 September 2008 (UTC)

And now since according to the view of some, quaternions are an important solution to the Einstein equations, evidently Hamilton in a way predicted a set of differential equations of which quaternions would be a solution, 75 years before the equations were discovered. For over 100 years now, Quantum Mechanics, which does not work with the general theory has been using math according to Lee Smolin which is pretty much based on Euclidean thinking from 2000 BC.

• In case you hadn't noticed, if you mean "based on flat space", then Quaternions in the way Hamilton used them are wholly and entirely Euclidean too. Jheald (talk) 13:00, 15 September 2008 (UTC)

Questions

So a few questions, when did this idea of a metric with a trace of plus two actually get into our thinking. It was not in 1916 with Einstein gives a trace of minus two, and Minkowski died in 1909, after 1905 when it was learned that space had to be at least four dimensional, but before it was well understood, that space could not be a flat four dimensional Riemann space.

In the 1920's people still had high hopes for quaternions, was it the blunder of the bi-quaternion that did them in for a while? Hamilton had been transforming four dimensional space since the 1840's but did not really have an application for his math.

Since the quantum mechanics of the time was incompatible with quantum mechanics, did thinking along quaternion lines go astray, and abandon its essential axioms, in an effort to gain popularity with the quantum physics crowd, only to fade into obscurity until the dawn of the computer age?

Why did it take so long to show that quaternions were a solution to the Einstein equations?

A "history of" article should be based on published histories

What Wikipedia prizes above all is sourced content, based on the best, most authoritative sources.

An article on "History of Quaternions" ought to be based on published histories by historians of science. Historians of science have certainly looked at the story, as a fascinating case study of a "paradigm shift", and of mathematical views in conflict.

It is unfortunate, therefore, that our article appears not to be based on the researches of those careful historians; and in fact they don't even get a single citation at the moment. This should be changed. The article should much more closely review what professional scientific historians have had to say about this controversy.

I have had a brief chance to look at Crowe's book. It is very informative. His thesis seems to put give Hamilton a great deal of credit for the development of so called modern vector analysis. However it also seems that a new chapter of the history of quaternions may be written in this century. Also I think if historical events can be gleaned from primary sources this might be better than just relying on the a few narratives.

Hobojaks (talk) 04:01, 16 September 2008 (UTC)

See WP:PRIMARY. Primary sources should be sourced for what they explicitly say. But remarks about their significance, or how they fit in to the flow of history, need to be sourced to secondary sources. Jheald (talk) 07:56, 16 September 2008 (UTC)

For example the fact that in 1916 that Einstein's original paper was based on a metric with a trace of (-,-,-,+), thus reproducing the same operation as Hamilton's scalar of the product operation, might be admissible.

It should be mentioned only if sources can be identified that explicitly make this connection and present it as significant. Jheald (talk) 07:56, 16 September 2008 (UTC)

Note also, (1) as I've said, even if quaternions could reproduce the norm from Einstein's papers, they don't represent the transformations in any particularly natural or compelling way. A lot has been learnt about relativity in the last 100 years, and it is not described with quaternions. Dirac matrices, yes. Quaternions, no.

Secondly, Sq² isn't even a particularly natural quaternion operation (unlike, say S(Vq)²).

Here's what Roger Penrose had to say in The Road to Reality (2004), ch. 11, p.201. (And Penrose, inventor of twistors and spin networks is probably as well qualified as anyone to make an assessment.)

The temptation is strong to take this t to represent the time, so that our quaternions would describe a four-dimensional space-time, rather than just space. We might think that this would be highly appropriate, from our 20th-century perspective, since a four-dimensional spacetime is central to modern relativity theory. But it turns out that quaternions are not really appropriate for the description of spacetime, largely for the reason that the 'quaternionically natural' quadratic form $q{\bar {q}}=t^{2}+u^{2}+v^{2}+w^{2}$ has the 'incorrect signature' for relativity theory (a matter we shall be coming to later). Of course, Hamilton did not know about relativity, since he lived in the wrong century for that.

The chapter then goes on to discuss the geometrical understanding of quaternions in 3D, particularly how they relate to rotations; and how in higher dimensionalities (including 4D) one can generalise from quaternions to Clifford algebras; and identify Grassmann algebras (exterior algebras) contained within them. Jheald (talk) 08:55, 16 September 2008 (UTC)

1967 seems to be a very long time ago, and there have been a lot of developments since then. Also a good chronology should provide some context, as to what was going on at the time.

I think it would be unfortunate to have the history of quaternions forever stuck in the past, when it seems like most of the developments, were either before the mid 1930's or in just the last few years, Crowe's history, seems to be based on ideas that are basically before the computer age.

Also what you are proposing takes a lot of work, so I hope that you are not pointing blame at the few individuals who have put in their best efforts at recoding the important events we have been able to find.

Did you have any specific concerns that the historical facts given were not factually correct?

Anyway, when I logged on today, I had planned to do my part, but the source I planned to cite was Lee Smolin, and his fresh 2007 perspective. Smolin does not use the term quaternion specifically, instead talking about a more generalized idea of space consisting of three spacial dimensions, and one time dimension.

Hobojaks (talk) 04:01, 16 September 2008 (UTC)

But this article is about the history of quaternions, not the history of "space consisting of three spatial dimensions and one time dimension". You would need to find citations specifically saying that quaternions are being used to understand space-time in the 21st century. And I don't think you will, because the modern renaissance of quaternions is to execute transformations in 3D, not 4D.

We don't need to be giving a history of modern physics since 1905, because that's not where quaternions are now being used. It may be interesting, but it's not relevant. Jheald (talk) 07:56, 16 September 2008 (UTC)

Any good history would it seems to me, would trace the development of quaternions in the context of the general development of four dimensional geometry. Hamilton proved a lot of very important theorems about these types of spaces.

One standard work appears to be:

• Michael J. Crowe (1967), A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (Notre Dame Press 1967; Dover Publications 1994)

(though note the caveats of William Waterhouse (review, 1972, pp 387-388) that a full story might include more on early vector calculus (Stokes), exterior algebra (Grassmann), and differential forms (Cartan); to which might also be included Clifford Algebra (Clifford).

yes, and since you brought it up, why not go in and get some good references to the chapter on the struggle for survival. Crowe does a good job on documenting that this struggle took place, but does not really do a very good job of explaining the issues that were being debated. Perhaps the best sources for this might be primary sources.

Also

• Simon L. Altmann (1986), Rotations, Quaternions, and Double Groups (OUP, 1986; Dover Publications 2005)

including his discussion of Rodrigues, and other forerunners of Hamilton.

There's also a sketch in

• E. T. Bell, The Development of Mathematics pp 198-211 (1945; Dover, 1992)

which gives a useful overview, putting things in their place

No doubt there are others.

These are key WP:Reliable Sources we should be basing our presentation on; and per WP:NOR, we ought to be sticking closely to the story they tell. Jheald (talk) 08:08, 15 September 2008 (UTC)

Not perhaps primarily about history, but this paper seems quite a useful reference (and it's online!), exploring in quite an accessible way the difference between using the imaginary part of quaternions to represent vectors, as against pseudovectors; and how much awareness there was of this in the nineteenth century:

Cibelle Celestino Silva and Roberto de Andrade Martins (September 2002), Polar and axial vectors versus quaternions, Am. J. Phys. 70 (9), 958-963

Jheald (talk) 09:10, 15 September 2008 (UTC)

Euclideanists?

One objecting that I don't have to much trouble with agreeing to is the term Euclideanist. That was just something that I made up for lack of a better term and don't mind changing it. The central idea is that there are two ideas at war here.

What is very well documented and a great source for this is Crowe is that at the end of the 19th century there was a knock down drag out fight, between two factions. In fact Crowe devotes a chapter, titled the struggle for existence, to the subject.

This came right before the time when there was a major revolution in physics, with the rise of quantum mechanics, as well as further developments in geometry, like general and special relativity. Smolin 2006 and Bertrand Russel 1925, provide great sources of the idea that the revolution of general relativity is a revolution in geometry.

Quantum mechanics and relativity (in the opinion of plenty of people that it would be easy to site sources from) still can't be unified. If I have connected a few dots on my own by seeing a connection between the turn of the century debate over notation, and the 20th century fundamental split between quantum mechanics, and relativity sorry about that, but to tell the truth I don't think I have made some break through historical interpretation here. Quantum mechanics, and relativity use different..... let me skip a complex explanation here about the difference, but just assert that it seems to me, that there might be a connection between these two debates, in that they are over the same issue in a different context. In other words, the argument between Gibbs and Tait has spilled over into the next century in many profound ways, so that an idea can be traced as it develops from then to now. Smolin in his book the trouble with physics, who is really interested in history after 1970, may have some incite for us, in part one of his book, titled the unfinished revolution, where he devotes an entire chapter to The world as geometry

But hey I gotta get my homework done, so I can't really get into a point by point rebutal here. —Preceding unsigned comment added by Hobojaks (talk • contribs) 20:29, 20 September 2008 (UTC)

Penrose on quaternions

JHeald wrote: Secondly, $Sq^{2}$ isn't even a particularly natural quaternion operation (unlike, say $S(Vq)^{2})$.

An interesting identity is that $S(Vq)^{2})=(Vq)^{2})$ In other words in Hamilton's notation as JHeald is of course aware, but which I point out for the benefit of other readers. This was something that 19 century thinkers thought very important.

Interestingly and just to show that great minds think alike, I think that the Euclidean norm that Euclid actually taught 1000's of years ago was a lot like the square root of the square of the vector part of a quaternion, differing only in that Euclid's square always had a positive value.

But Euclid's thinking did not connect space with time.

JHeald wrote:

Here's what Roger Penrose had to say in The Road to Reality (2004), ch. 11, p.201. (And Penrose, inventor of twistors and spin networks is probably as well qualified as anyone to make an assessment.)

The temptation is strong to take this t to represent the time, so that our quaternions would describe a four-dimensional space-time, rather than just space. We might think that this would be highly appropriate, from our 20th-century perspective, since a four-dimensional spacetime is central to modern relativity theory. But it turns out that quaternions are not really appropriate for the description of spacetime, largely for the reason that the 'quaternionically natural' quadratic form q \bar{q} = t^2 + u^2 + v^2 + w^2 has the 'incorrect signature' for relativity theory (a matter we shall be coming to later). Of course, Hamilton did not know about relativity, since he lived in the wrong century for that.

The chapter then goes on to discuss the geometrical understanding of quaternions in 3D, particularly how they relate to rotations; and how in higher dimensionalities (including 4D) one can generalise from quaternions to Clifford algebras; and identify Grassmann algebras (exterior algebras) contained within them. Jheald (talk) 08:55, 16 September 2008 (UTC)

As far as the scalar part of the square of a quaternion not being natural, I think that this notion must be one from the 20th century, and not be reflective of Hamilton's thinking on the subject. The operations of squaring, and taking the scalar part, are some of the most basic and fundamental ones in his calculus.

To me it seems that the problem all along was like trying to put a square peg into a round hole. If you think of classical quaternions as the square peg, and all these advanced algebras as the round hole, in order to get quaternions to fit into the theory, they had to be pounded quite a bit.

One of the worst effects of this pounding, was the notion that the tensor of a quaternion somehow corresponded to the notion of the Euclidean Norm of some real four space. So I would think that Penrose's comment really applies to some modern quaternion that is a problematic textbook case, in advanced algebra, more than it did to the original 19th century idea.

But you have a great advantage over me, JHeald, having read Penrose's book. I actually had thumbed through this book a little bit at the books store, and almost bought a copy. I was looking at the part about SU3, but really only thumbed through it for a few seconds, and instead picked up a book with all those old papers from 1900-1918 translated into English, including all of Einstein's papers on special and general relativity. I stayed up all night reading it, and the next morning did not do as well as I could have on my engineering dynamics test.

Penrose's book will be on my shopping list for next trip. After I have had a look at it I will be able to comment more intelligently rather than just speculating. Thank you for your outstanding comments, but I need to get to class now!

130.86.76.114 (talk) 15:49, 24 September 2008 (UTC)

Having read Penrose on quaternions

I have added some material about Penrose's book on quaternions, thanks for recommending it. Actually I was thinking about buying and reading it before this discussion even came up.

One section that I found particularly enlightening, was on page 246 and 247, were Penrose explains that quaternions make a great representation of velocity space.

I found figure 18.11 particularly enlightening. I had compared diagrams from Bertrand Russels 1925 book ABC of relativity with Minkowski's 1908 article on space and time and always though there was a connection there but did not really understand it as well.

Jheald, could you please check my typings for factual correctness, I have tried to do my best to render Penrose's thinking on the subject, hopefully I have it close to correct.

It is tempting for me to go back now and rework a section I tried to type on night on the application of versors to special relativity that got deleted for being OR now that I have a more recent source on the subject.

Hobojaks (talk) 03:16, 9 November 2008 (UTC)

Penroses Quaternically natural form

I have to agree with Penrose that the notion of the Quaternion Dot Product which is discussed on the main page, in my opinion is problematic. The idea of extending quaternions with a dot product was what Tate thought of as a Hermiphroditical Monstrosity. So this is a problem that people much smarter than me have been objection to for several centurys

But you have to take the tern natural in the context of several centuries of thought here. I think Penrose is borrowing from Minkowski's 1908 article. That is what Minkowski calls his mystic formula that the square root of -1 second is equal to the distance that light travels in a second, or 300,000 km. Penrose calls complex numbers Magical and Mysterious and calls real numbers natural at the start of the book, but I think you have to keep reading Jheald!

In my copy of Penrose's book on page 201 I have underlined the phrase Quaternically natural and scrawled in the margin, see page 1035.

In the first paragraph of page 1035, Penrose explains that while some may view real numbers as being natural and complex numbers as magical and mysterious, that it may in fact be that the complex numbers are more god given.

Hence according to Penrose complex number quantities like the square root of scalar part of the square of a quaternion${\sqrt {(S.q^{2})}}$ may , while seeming unnatural to some in fact be not only magical and mysterious but also ordained by god.

Hence while the dot product may seem most natural to some, including some of the authors of the main quaternion page, who are representing a long held view, that they share with giants like Euclid, Gibbs and Heavyside, scalar part of the square of a quaternion, which is a Lorentz invariant, when used as an element of velocity space as Penrose suggests in 18.1 -18.4 of his book.

The last sentence two sentences of the paragraph in question on page 201, explain that Hamilton did not know about special relativity, in fact it was in the year of his death that it was being discovered that light was electromagnetic, and traveled an a constant speed.

Penrose then explains in the last sentence of the paragraph in question on page 201 that he is opening a whole can of worms here. And gives a whole list of sections that deal with this problem, the last one being in chapter 32, the chapter before the one where he introduces twister theory.

You may have noticed that a twister seems to have a quaternion part?

Since a twister consists of an array of four complex numbers, for example you could take the real number from the first complex number and the imaginary parts from the remaining three and have a quaternion, hence a quaternion can be viewed as a special type of twister? Or a twister as a generalization of a quaternion?

Hobojaks (talk) 03:17, 9 November 2008 (UTC)

I think you should be careful about inferences, something that could be done but is not documented is original research. Citations are key. If you are certain all the citations mention quaternions and what's mentioned in close proximity you'll be on reasonable ground. 'You may have noticed' is not good grounds for including things in a history. Dmcq (talk) 16:12, 2 March 2009 (UTC)

Merging section 5

Section 5 is some old material that was cut and pasted out of the Classical Quaternions article because it really did not belong there.

There is an important section in the main article about the relationship between R^3 and the vector part of a quaternion. When the content of section 5 was written that section in the main article did not exist yet, in fact the idea that a quaternion had both a vector and a scalar part did not really exist in the main article. So we have come a long way.

Some of section 5 the who and when part, needs to be in the history article.

The "what" the technical details of how the vector part of a quaternion and a vector in R^3 and the relationship between the scalar product of two R^3 vectors, and the scalar of the product of two right quaternions, that is two quaternions which have a zero scalar part, and between the vector of the product of two right quaternions and the vector product of two vectors in R^3 needs a clear explanation in the main article. Since many readers will already know about dot and cross products this information will draw a line between what they know and what they don't know.

Can someone please help me with Merge tags???? So that it reflects the suggested merging between what and what. —Preceding unsigned comment added by Hobojaks (talk • contribs) 19:00, 1 March 2009 (UTC)

Merge section 6

Section 6 has some really great information. All of it belongs somewhere!

There are some really technical parts in this section that in my opinion belong in the main article on quaternions.

But when did this modern synthesis happen. What were important dates? Who had he ideas?

I don't know all the history, but I don't think that the names Clifford and Grassmann even appear in the current article. Who in the 20th century was responsible for the synthesis? When did these ideas begin to be understood.

Needs to be in the history article. Hobojaks (talk) 19:06, 1 March 2009 (UTC)

No, this contains Original Synthesis. Septentrionalis PMAnderson 23:40, 4 March 2009 (UTC)

Redirect

I have, per the AfD, redirected this to Quaternion. The previous text is here. I am prepared to consider arguments that any part of it is a neutral, sourced, statement of the facts, and worth adding to the article on quaternions. Please note, however, that one editor's own interpretation of primary sources, like Gibbs, is not enough; see WP:SYNTH; we need the history as presented in reliable secondary sources on the history of physics, phrased neutrally.

Among the worst, and least supported, claims here are

• The classical vector of a quaternion was multiplied by the square root of minus one and then again by negative one, and installed into modern vector analysis. The computational power of the classical quaternion vector product was exported into the new notation as the new cross and dot products. I think I can tell what the author of this thinks he is saying by multiplied by the square root of minus one and then again by negative one, but it is not true; and the power of dot and cross products is much greater than quaternions ever had, as a special case of Grassmann algebra.

• I am not a great lover of the text of this article, but that statement is actually technically correct. (I think it's lifted directly from a paper, maybe by Altmann). The square root of minus 1 is because vectors square to 1, rather than -1; the factor of -1 because the vectors conventionally form a right handed co-ordinate system rather a left handed one. This can be seen explicitly in Clifford algebra, which incorporates both.
  • Then let it be sourced (or preferably rewritten to avoid the appearance of absurdity). Septentrionalis PMAnderson 04:48, 5 March 2009 (UTC)

• The "classical quaternion vector product" can be seen as a special case of the Clifford product. Clifford algebra incorporates Grassmann algebra in a significantly more powerful system. Jheald (talk) 02:17, 5 March 2009 (UTC)

• To provide a vastly oversimplified, short introduction to what motivated these debates consider that in the new notation that i · i =+1, j · j =+1 and k · k =+1. So apparently i,j,k in the modern vector notational system represent three new square roots of positive one.

In the new notational system i, j, and k also apparently represented square roots of zero, since i × i = 0 , j × j = 0, k × k = 0. The new notation system was then based on numbers that were the square root of both zero and positive one. Advocates of the classical quaternion system liked the older idea of a single vector product with a unit vector multiplied by itself being negative one better.

It is mistaken kindness to describe this as oversimplified; it is the babble of someone who doesn't understand what an algebra is.

• I think this was a criticism that was actually raised, historically. An advantage of the quaternion product was that the quaternions were a division algebra. And, as just mentioned, the Clifford product goes from strength to strength. Clifford algebras are very nearly division algebras, and this makes Clifford calculus much neater than vector calculus, with extraordinarily compact expressions for many of the most fundamental equations of physics. Jheald (talk) 02:17, 5 March 2009 (UTC)
  • Yes, before Whitehead, lots of people didn't understand what an algebra was, arguably including Hamilton himself. Septentrionalis PMAnderson 04:48, 5 March 2009 (UTC)

• 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. The first and last claims here are wrong.

• Are they? Clifford analysis goes from strength to strength, and does indeed seem to be the natural step up from complex analysis. Lie algebras have indeed become a standard tool for analysing symmetries that can be related to infinitessimal transformations. Jheald (talk) 02:17, 5 March 2009 (UTC)
  • Both Clifford and Lie algebras are powerful things; but not the ones described in this sentence. In particular, the standard Lie algebras consist of infinitesmal transformations of spacetimes. Septentrionalis PMAnderson 05:02, 5 March 2009 (UTC)

• Representations of rotations by quaternions are more compact and faster to compute than representations by matrices. Since quaternions can be fully and faithfully represented as a class of 4x4 matrices, this is absurd.

• Read Quaternions_and_spatial_rotation#Quaternions_versus_other_representations_of_rotations and learn. Jheald (talk) 02:17, 5 March 2009 (UTC)
  • Both, again, being versions of (three dimensional) projective space, which uses 4x4 matrices; the statement could be saved by inserting orthogonal 3-dimensional, but it isn't there.Septentrionalis PMAnderson 04:48, 5 March 2009 (UTC)
    • Typically, you don't use the matrix representation to calculate quaternion rotations in performance applications. You use the quaternion multiplication formulas directly, which makes for a lot fewer operations. Some chips now even have quaternion multiplication hard-wired. Jheald (talk) 10:23, 5 March 2009 (UTC)

• Minkowski space has an indefinite bilinear product, with a signature of +1; Quaternions have a definite product, with a signature of +4. This major difference between quaternions may be alluded to in the painful bafflegab about Descartes and imaginary numbers under Sign of distance squared, but not distinctly.

Any such argument should cite sections and subsections of the former article by name, not by number. I can guess that Section 6, above, means "20th-century extensions" as my skin shows, but numbering can vary from skin to skin. Septentrionalis PMAnderson 23:40, 4 March 2009 (UTC)

• I have undone the redirect. The clear sense of the AFD was keep and fix, rewriting from the ground up if necessary. Jheald (talk) 02:21, 5 March 2009 (UTC)

I just undid the redirect again. As you said, the result was Keep, not redirect. They are moving against consensus. Dream Focus 02:28, 5 March 2009 (UTC)

The question of whether to insert a redirect or not is not decided by an AfD, which only determines whether or not to remove the article and its edit history. Redirects, merges, and othe alterations are normal editing. Septentrionalis PMAnderson 03:52, 5 March 2009 (UTC)

A redirect is the same is deleting it, since the article is no longer there. The vote was keep. Dream Focus 10:01, 5 March 2009 (UTC)

That is a complete misunderstanding of our deletion policy. Please read the summary box at the beginning of WP:AfD: For problems that do not require deletion, including duplicate articles, articles needing improvement, pages needing redirects... Deletion is a specific and drastic action, requiring admin intervention, which gets rid of an article and its edit history. Septentrionalis PMAnderson 15:26, 5 March 2009 (UTC)

The "burden of evidence lies with the editor who adds or restores material". You do not do that, so I reverted it back to a redirect, for the time being. Instead of this edit warring, help here to work on how to arrive at a good article. Or provide directly a (new) article which is verifiable, NOR and NPOV. -- Crowsnest (talk) 14:15, 5 March 2009 (UTC)

To make myself clear: I will not (re)write the article. But I am willing to assist. -- Crowsnest (talk) 14:43, 5 March 2009 (UTC)

Now, do either of you have any defense for this ignorant, illiterate, and tendentious content? If so, it belongs in the section above. Septentrionalis PMAnderson 03:52, 5 March 2009 (UTC)

So much for the defenses. The AfD came to three conclusions:

1. It is worth having an article on the history of quaternions (I agree)
2. Therefore this article should not be deleted, even if it "needs to be burnt to the ground and rebuilt"
3. This version of the article needs to be burnt to the ground.

If Jheald wants to rebuild a history of quaternions, in a neutral voice, great; he will find a list of sources in the AfD. Septentrionalis PMAnderson 04:57, 5 March 2009 (UTC)

Reliable sources for building the "History of quaternions"

Further information: § A "history of" article should be based on published histories

Crowsnest (talk) 13:50, 5 March 2009 (UTC)

Regarding sources: the old version uses twice a link to another WP article as a reference (source), but one can not use WP for its own verification. Further many references are made to "Vector Analysis" by Gibbs & Wilson (1901), who do not say anything about quaternions (as far as I can see). So that leaves only the first 4 references (which lack page numbers). -- Crowsnest (talk) 05:11, 5 March 2009 (UTC)

The references to Gibbs are part of the tirade against vector analysis. Since Gibbs would not agree, this is Original Synthesis from a primary source. Septentrionalis PMAnderson 05:23, 5 March 2009 (UTC)

I agree. Gibbs & Wilson (1901) say only something about quaternions in the introduction of their book. On p. xi:

"Notwithstanding the efforts which have been made during more than half a century to introduce Quaternions into physics the fact remains that they have not found wide favor. On the other hand there has been a growing tendency especially in the last decade towards the adoption of some form of Vector Analysis."

And p. xii:

"It has been asserted by some that Quaternions, Vector Analysis, and all such algebras are of little value for investigating questions in mathematical physics. Whether this assertion shall prove true or not, one may still maintain that vectors are to mathematical physics what invariants are to geometry. As every geometer must be thoroughly conversant with the ideas of invariants, so every student of physics should be able to think in terms of vectors."

Crowsnest (talk) 05:56, 5 March 2009 (UTC)

Besides the first reference (to Crowe?, without page numbers given), the remaining few (leaving Gibbs & Wilson out, which are unrelated to what they are supposed to back up) are primary sources with respect to the present history article: things written by the subjects of this history. They are of course valuable to be included, but a history article needs to be primarily based on reliable secondary sources about the history. So at this moment there appears to be only one reference (if page numbers are added) to a reliable secondary in the article. -- Crowsnest (talk) 18:20, 5 March 2009 (UTC)

Just a copy from the AfD page: some relevant sources w.r.t. the history of quaternions I found (just by googling):

• Conway & Smith, "On Quaternions and Octonions", 2003, ISBN 1568811349
• Crowe, "A History of Vector Analysis", Dover, 1994, ISBN 0486679101
• van der Waerden, Mathematics Magazine 49(5), 1976, 227-234

But there must be (many) more. -- Crowsnest (talk) 05:19, 5 March 2009 (UTC)

Elements of the History of Mathematics by Nicolas Bourbaki Dover 1994 ISBN 3540647678 has a couple of pages on it. Dmcq (talk) 12:57, 5 March 2009 (UTC)

• Bell, "The Development of Mathematics", 1992, ISBN 0486272397 -- Crowsnest (talk) 13:41, 5 March 2009 (UTC)

Conway and Smith does not appear to have anything on the history of quaternions, although it is a very good book; Eric Temple Bell is severely dated and notoriously unreliable. Septentrionalis PMAnderson 15:29, 5 March 2009 (UTC)

See: Conway and Smith, pp. 6-10. Which is mainly a (very) long quote from an article by John Baez, which of course should be used as the ref for this. -- Crowsnest (talk) 15:46, 5 March 2009 (UTC)

Baez' paper "The Octonions", quoted above, is available on the web at various places including here

pablo_hablo. 16:26, 5 March 2009 (UTC)

Just for fun, Heaviside on Vectors Versus Quaternions -- Crowsnest (talk) 10:28, 9 March 2009 (UTC)

Wikiquette alert

Wikiquette alert here - seems only polite to mention it seeing as the complainer has not bothered. pablo_hablo. 16:51, 5 March 2009 (UTC)

Administrator noticeboards incident

http://en.wikipedia.org/wiki/Wikipedia:Administrators%27_noticeboard/Incidents#AFD_consensus_was_keep.2C_but_some_editors_keep_deleting_and_redirecting_instead I was told to take the issue there, so did so. And I forgot to post a link previously. Everyone is encouraged to participated, and state their opinions, of course. Dream Focus 17:56, 5 March 2009 (UTC)

Any contributions to make as far as cleaning up/rewriting go? pablo_hablo. 20:02, 5 March 2009 (UTC)

Dream Focus, you apparently have no interest or knowledge to work on these topics. Your "intervention" here is singularly unhelpful. People not interested in discussing or involving themselves in the normal editorial process of editing an article should not be policing an article. The eventual result of what happens is not governed by the AFD. In other words, nobody made you Wikipedia Cop, Dream Focus. You don't have any special privilege here to go around enforcing decisions made in AFD. I imagine somehow I or some others have gotten your goat. Your spamming of WQA and AN/I are indicative of that. But I think you should let it go. --C S (talk) 02:16, 6 March 2009 (UTC)

There's no reason at all that the article should not be redirected to a well-written history section while discussion is underway on creating this "better article" alluded to by AFD participants. Nobody that is advocating the redirect really has any personal grudge against the topic "history of quaternions", as you seem to imagine. If there is a good "history of quaternion" article written, it will undoubtedly stay, and indeed, the people you've been edit-warring with are working on such an aritlce right now. And what have you been doing to help this? Nothing. (excerpt from my comment at AN/I [1]) --C S (talk) 02:33, 6 March 2009 (UTC)

You seriously need to drop this and respect the AFD consensus. Wikipedia is not a battleground, sometimes decisions are made that you might not agree with. Deal with it and move on. Use your time to improve the article instead. --neon white talk 04:39, 6 March 2009 (UTC)

(Apparently the above comment is addressed to Dream Focus). I wholehearted agree with this. The consensus was to burn this article down and start anew. That's what is being done. --C S (talk) 05:03, 6 March 2009 (UTC)

Strange, from Neon White's contributions, it really seems like s/he was commenting to me! Well, not that I don't appreciate being chastised, but comments like Neon White's are out of line. I would suggest Neon White point out to Dream Focus that distracting people here with antics on WQA, AN/I (and who knows where else since the AN/I thread didn't go his/her way), is not "improving the article". Indeed, Pmanderson has asked for my help on working on this article, but every time I look at it, there's somebody trying to stir up drama. --C S (talk) 05:14, 6 March 2009 (UTC)

No, my advice is good and you need to follow it. Continually denying the afd consensus to a disruptive level is only going to lead to blocks. The ANI alert was recommended by volunteers at the WQA board because it was decided that was the best place to deal with it. It was the correct course of action considering the edit warring on this article. It should not have been necessary but sadly do the poor behaviour of a few editors it was. --neon white talk 20:46, 6 March 2009 (UTC)

Your "help" here is not needed, so get off your high horse. I know people like you get their jollies from acting like the police, but you aren't helping anything. Go away. Shoo. --C S (talk) 01:56, 7 March 2009 (UTC)

The article is being improved, and it isn't redirected any more. Can we call this thread closed? — Carl (CBM · talk) 02:06, 7 March 2009 (UTC)

I'm more than willing to call it closed. Until the next time Neon White feels the need to emphasize that people must behave and do his/her bidding. Then I might feel compelled to respond in kind. Not very saintly of me, I admit. I don't feel a need to silently tolerate fools. --C S (talk) 02:10, 7 March 2009 (UTC)

It would be nice to have a cronology!

By this I mean a list of important dates, listed in order.

At one time I tried to start one, but I didn't contribute very much material to it.

What I mean is a list of events, in order of their date?

The Birth and death of Hamilton, should be included, and the date of the first discovery of quaternions. Perhaps the dates of the births and deaths of Clifford, Grassman, and Gibbs should be included. That might help to give readers a better idea of how the idea of quaternions and extensions and alternatives to it evolved, in a linear time like manner.

Perhaps, we can postpone a discussion of what should go into a time line, till after we agree that there should at least be a time line.

• I don't think there should be; the birth and death dates of the protagonists should be in parentheses (if needed, that's one of the things links are for). Please note that Conway's account of the history begins with Euler. Septentrionalis PMAnderson 23:28, 5 March 2009 (UTC)

Baez

I have added a note on the mathematical uses of quaternions, and the couple of sentences from Conway/Smith themselves on quaternions.

That leaves what they say on octonions, if we want octonions here, and John Baez's paper, which is probably the source of what we now say insofar as it was taken from Quaternion#history. I encourage others to attack Baez, since the source is readily available. Septentrionalis PMAnderson 19:48, 6 March 2009 (UTC)

Can you multiply triples?

Currently the article states, "for many years Hamilton had known how to add and multiply triples of numbers. But he had been stuck on the problem of division: He did not know how to take the quotient of two points in space." However, according to Baez's octonions articles, and indeed, this is a very famous story, up until the month of his discovery of the quaternion relations, he did not know how to multiply triples. Indeed, the story goes that his two boys would ask him at breakfast every day, "Can you multiply triples, daddy?" And his answer would be "no" until that fateful day. --C S (talk) 02:13, 7 March 2009 (UTC)

That's Bell, and he may be right, especially if Baez supports him. Let me see if I can find a life of Hamilton. Septentrionalis PMAnderson 18:30, 7 March 2009 (UTC)

See here

I'm not going to edit this article myself; I'll leave it to those with a better understanding of the subject. pablo_hablo. 12:22, 8 March 2009 (UTC)

Can this important concept have a place in this article some where?

Hamilton's concept of a tensor was a one-dimensional quantity, quite distinct from the modern 
sense of tensor, derived from Bernhard Riemann. 

It is a single important concept that I cut out of the soon to be deleted article on classical notation. It is more about historical people, who invented what, and also contrasting notation other than classical quaternion notation with other types of notation, which runs out side of the topic of the streamlined version of the article. If you try to compare classical notation with other kinds of notation you end up with an article as much about modern notation as an article on what the article is supposed to be about.

Not one to destroy important content, if no one else puts it in I plan on doing it, if it gets deleted, oh well, —Preceding unsigned comment added by Hobojaks (talk • contribs)

It's not clear what you're talking about. I'd ask you to explain it but I suspect the request would just lead to your own original research. Instead, please give a reference I can check for this concept of Hamilton of "tensor as a one-dimensional quantity". --C S (talk) 08:41, 9 March 2009 (UTC)

Hobojaks misunderstands (as he misunderstands the likelihood of deletion): This is a purely verbal point. There have been two mathematical definitions of tensor: Hamilton's, as the square root of the norm of a quaternion, now obsolete; and the one in current use, which was coined by a German physicist in 1898. If there were any reason to include that here, the OED would do as a source; but it can be left where it was, as an effort to explain one confusion in presenting Hamilton's system to the historically minded. Septentrionalis PMAnderson 20:32, 9 March 2009 (UTC)

If your only reason for deleting this statement is that you think I typed it you would be wrong. I think it is an important concept for readers to understand, because it causes a lot of confusion, when they try to read an article on the subject of quaternion notation crippled by the arrogant vandalism of its analysis. Perhaps the difference is so self evident as to not require a source, however I decided to wash my hands of this issue a long time ago and just focus on the notation itself. The concept was first vandalized out the article on notation, and then thrown back in as an after thought by someone who instantly recognized that with out the articles analysis, somehow considered as original research at the time, the notation section became very confusing, or dare I say, completely incomprehensible.

If this was an accidental error, I apologize in advance for thinking otherwise. You are correct in stating Caylay's definition, but this is not correct in an article about Hamilton's definition.

Gotta go to school!Hobojaks (talk) 17:53, 12 March 2009 (UTC)

The result of this operation is a number which represents its magnitude, the "stretching factor", the amount by which the application of the quaternion lengthens a quantity; specifically, the tensor is defined as the square root of the norm — this is a one-dimensional quantity, quite distinct from the modern sense of tensor, coined by Woldemar Voigt in 1898 to express the work of Riemann and Ricci. As a square root, tensors cannot be negative, and the only quaternion to have a zero tensor is the zero quaternion. Since tensors are numbers, they can be added, multiplied, and divided. The tensor of the product of two quaternions is the product of their tensors; the tensor of a quotient (of non-zero quaternions) is the quotient of their tensors; but the tensor of the sum of two quaternions ranges between the sum of their tensors (for parallel quaternions) and the difference (for anti-parallel ones) .

The text above is very good, most of it belongs in a well sourced history article, where people will actually see it not is some article filled with cryptic terms that no one really cares about. I have tried to rewrite it, so that it will be understandable to someone, with out a lot of the irrelevant stuff from the notation article. Below is better because it makes the text coherent. Still needs a little work.Taits Wrath (talk) 20:43, 12 March 2009 (UTC)

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The tensor was defined by Hamilton who died in 1865 as the square root of the norm of a quaternion. — this is a one-dimensional quantity, quite distinct from the modern sense of tensor, first coined 32 years later by Woldemar Voigt in 1898 to express the work of Riemann and Ricci. Riemann died in 1866 coincidentally the same year as the publication of Hamilton's life work on quaternions, in a book called Elements of Quaternions.

As a square root, tensors cannot be negative, and the only quaternion to have a zero tensor is the zero quaternion.

Since Hamilton's tensors are numbers, they can be added, multiplied, and divided. The tensor of the product of two quaternions is the product of their tensors; the tensor of a quotient (of non-zero quaternions) is the quotient of their tensors; but the tensor of the sum of two quaternions ranges between the sum of their tensors (for parallel quaternions) and the difference (for anti-parallel ones). Taits Wrath (talk) 20:43, 12 March 2009 (UTC)

Moving Evolution of notation this out of talk space and into the article. It still needs some work, but we are pretty much working an a very raw starter article.Taits Wrath (talk) 22:45, 13 March 2009 (UTC)

While I cannot attest the exact factual truth of the mathematics, perhaps this text is at least good enough for a somewhat non-technical history article?Taits Wrath (talk) 18:11, 13 March 2009 (UTC)

Double quaternion is key concept.

It will matter less to readers studying Hamilton's ideas who made up names for concepts, and matter more the history of the concepts.

I have looked at that half page before when it used to be at cornell, but I suspect that there has been enough interest in the subject that wikipedia was driving to much traffic over to cornell so they shut us down.

Hamilton did speak before the Royal Irish what ever they called it, on the subject of octonains, he did suggest a name for them to the society, and he did say that Graves thought of them first.

His idea was for the basis vectors to be i,j,k,l,m,n,o

The key concept here is that there are two possible double quaternions. One is the bi-quaternion, that Hamilton did a lot of work on, that the other one is what ever you want to call it, but not much work has ever really been done on it.

To construct the bi-quaternion in Hamilton's alter work suggested the letter h be used.

h is just the plain old imaginary from scalar algebra. It is both commutative and associative. l,m,n and o are not commutative or associative, but I believe that it is possible to construct a double quaternion in two ways so that it does not have a geometrically imaginary part. This means that all the coefficients except 1,i,j and k are zero. Also looks like to me that given l one could construct m n and o as li,lj and lk.

Here is the thing, as far as I can tell there really are no credible sources on non-associative double quaternions, since I hate the idea of the norm, I would be inclined to construct one that wasn't a normed division algebra.

Also it is pure fiction to suggest that Bi-quaternions can't be divided, Hamilton showed that they can, unless that is tensor a new word some of us are learning has meanings in Hamilton's context, is zero.

I don't really understand the reasoning behind saying that the biquaternons are not a normed division algebra, other than the fact that it is possible for them to have an imaginary tensor.

So well, attribute the term double quaternion to Penrose if you must, we can verify that and that will let us explain a key concept, that is needed for anyone reading this history because they are tracing the development of mathematical ideas back to their origina. —Preceding unsigned comment added by Taits Wrath (talk • contribs) 11:37, 14 March 2009 (UTC)

I don't generally work on this page but

A large section of text was recently deleted from classical hamiltonian quaternions and the fact that clue bot and then even a live person accused me of vandalism got me to thinking. Even with out this text of this section, that article is 63K long, so this article seems the logical place to put this text.

So here is some old text written by a beginning user who does not exist any more, pasted into this article one more time. Actually there might actually be some good material in there some place, but it definitely needs a major rewrite. I notice that the slash and burn, and then build back up approach to this article basically left it devoid of content.

Some history in this section is that when the article tensor of a quaternion was voted as deleted and merged part of the content of the article to be merged contained content regarding the concept that Hamilton uses the word tensor in a different context from other users. Given the laborious nature of the task at hand, that content was merged back in as an early version of the main article. Hence some of this material has made the long trip from classical hamiltonian quaternions to the articles tensor of a quaternion and the vector of a quaternion.

There are some important concepts, but it needs to be reworded to remove both the essay and OR problems. I could eventually help with sourcing but right now, most of us interested in working on the subject of quaternions are focused on bickering over other issues, so I would have to come back to this subject at some later date after other chores were completed. —Preceding unsigned comment added by Robotics lab (talk • contribs) 20:36, 15 April 2009 (UTC)

I have no idea or opinion on your motivation for deleting this material somewhere else, but it certainly does not belong in this history article. This is about the history of quaternions, not about how to read historical texts on quaternions. The original location, classical hamiltonian quaternions, or a new article, seem better suited (if this essay is appropriate material for Wikipedia). -- Crowsnest (talk) 21:31, 15 April 2009 (UTC)

Gauss

Gauss made a contribution to quaternions.

Hamilton might have been unaware of Gauss's work.

Gauss's work of 1819 was not published until 1900.

Familton

The reference to the thesis of Johannes C. Familton is a continuous narrative of the development of quaternions and related representations of rotation. The reference was moved here from another article for which it was less appropriate. The value of the thesis is its survey of algebraic rotation representations spanning several decades. Such a broad survey is rare and might be compared to History of Lorentz transformations in its scope. However, Familton's thesis for a Department of Mathematical Education fails to meet standards expected in research mathematics. Examples include a deficient definition of simple groups (pages 77,8), and assertion that Pauli matrices form an algebra isomorphic to quaternions (page 80). Some of the historical sketch appears to be parody, and for the insightful reader of this article, such commentary can be taken as humorous. — Rgdboer (talk) 19:06, 30 March 2018 (UTC)