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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
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)
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)
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.
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.
The fact that Einstein came to believe that quantum mechanics was not true is well documented.
Modern string theory has found many solutions to Einstein's equations, and is the major topic of Lee Smolin's book.
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.
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)
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.
(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.
(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.
(b)The scalar of the product of two quaternions is the Lorentz invariant
(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.
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.
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.
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 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.
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.
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)
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.
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.
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.
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.
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.
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?
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.
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:
Also
There's also a sketch in
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)
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)
JHeald wrote: Secondly, isn't even a particularly natural quaternion operation (unlike, say .
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)
130.86.76.114 ( talk) 15:49, 24 September 2008 (UTC)
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)
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 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)
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)
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)
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
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)
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:
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)
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)
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 here - seems only polite to mention it seeing as the complainer has not bothered. pablo hablo. 16:51, 5 March 2009 (UTC)
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)
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)
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)
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 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)
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)
See here [1]
direct quotation from Hamilton's - letter to his son Archibald 5 August 1865
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)
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)
The result of this operation is a number which represents its magnitude, [1] the "stretching factor" [2], the amount by which the application of the quaternion lengthens a quantity; specifically, the tensor is defined citation needed as the square root of the norm failed verification [3] — 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. [4] As a square root, tensors cannot be negative citation needed, and the only quaternion to have a zero tensor is the zero quaternion citation needed. 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)
The tensor was defined by Hamilton who died in 1865 as the square root of the norm of a quaternion. [5] — 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. [6] 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)
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)
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)
Gauss made a contribution to quaternions.
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)
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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
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)
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)
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.
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.
The fact that Einstein came to believe that quantum mechanics was not true is well documented.
Modern string theory has found many solutions to Einstein's equations, and is the major topic of Lee Smolin's book.
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.
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)
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.
(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.
(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.
(b)The scalar of the product of two quaternions is the Lorentz invariant
(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.
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.
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.
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 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.
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.
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)
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.
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.
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.
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.
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.
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?
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.
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:
Also
There's also a sketch in
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)
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)
JHeald wrote: Secondly, isn't even a particularly natural quaternion operation (unlike, say .
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)
130.86.76.114 ( talk) 15:49, 24 September 2008 (UTC)
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)
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 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)
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)
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)
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
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)
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:
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)
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)
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 here - seems only polite to mention it seeing as the complainer has not bothered. pablo hablo. 16:51, 5 March 2009 (UTC)
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)
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)
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)
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 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)
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)
See here [1]
direct quotation from Hamilton's - letter to his son Archibald 5 August 1865
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)
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)
The result of this operation is a number which represents its magnitude, [1] the "stretching factor" [2], the amount by which the application of the quaternion lengthens a quantity; specifically, the tensor is defined citation needed as the square root of the norm failed verification [3] — 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. [4] As a square root, tensors cannot be negative citation needed, and the only quaternion to have a zero tensor is the zero quaternion citation needed. 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)
The tensor was defined by Hamilton who died in 1865 as the square root of the norm of a quaternion. [5] — 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. [6] 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)
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)
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)
Gauss made a contribution to quaternions.
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)