Talk:Quaternion/Archive 3

3672:

conform to the Knowledge standard that an article on a specialized topic should be less complicated than what one would find in a textbook on said topic. I added the Jargon tag and it was removed - it was called a "driveby" - because I hadn't yet come to the talk page and mentioned specifics (in this case I thought it was pretty self-evident). Well, I read the whole article, and I still have little to almost no idea of what a Quaternion is. Now, I'm not a Math major; I've only taken math up to and including Calculus II, but I think I can safely say that if I can't understand it, most people on Knowledge probably won't be able to either. The jargon tag is more than appropriate for this article, which needs a great deal of explanation in English of what's what. That I had to get a reasonably understandable definition of Quaternion from a different article is a quite demonstrative example of what's wrong with this article and how it needs some serious attention from someone who speaks both Math and English.

3568:

triplet of numbers (for Hamilton, vectors were always in 3-dimensional Euclidean space) but is independent of any particular representation or co-ordinate system. Adding and subtracting vectors is straightforward, but Hamilton wanted to develop an algebra of vectors that would allow him to multiply and divide them as well. He defined a quaternion as the ratio of two vectors - in effect, it is an operator that transforms one three-vector into another one. You need six numbers to specifiy two three-vectors, but you have two degrees of freedom in orientating your co-ordinate system, so a quaternion is defined by four numbers. Another way to see this is to think of the transformation of one vector into another as a rotation followed by a stretching or contraction. You need two numbers to determine the direction of the axis of rotation, a third to measure the angle of rotation, and a fourth to measure the amount of stretching or contraction.

1820:)? I don't think you have an intuitive sense of what it is, you only think you do. You seem to be thinking of it as a scaling and a rotation. The scaling part is no problem. But given two linearly independent vectors in 3 dimensions, there are uncountably many rotations that take the direction of one to the direction of the other. One is in the plane of the vectors by an angle equal to the angle between them, another is by 180° around the half-angle vector, and the others interpolate between those extremes. This doesn't happen in two dimensions (where the rotation is unique) or one dimension (where there's no rotation). Which rotation is the right one? Whichever one you pick, ( 2051:

over the break. Seems to me that from his account of the sociology of academic theoretical physics, anybody who thinks very much about Hamilton's views will very quickly find them selves out of a job. According to Smolin, the big thing now is theories based on 10 and 26 dimensional space. Reasoning that leads to the conclusion that vectors are linked to Hamilton's 3+1 quaternion ideas are bound to be unpopular with that crowd. I find it hard to read Tifonov, but as near as I can tell, all he is doing is writing Einstein's ideas about relativity, and mixing in a little bit of Hamilton's gargon, but he is still thinking of a quaternion as some sort of

1728:

what a quaternion is then all the definitions should be listed. The thing is that as we are discovering, there are very different points of view on what a quaternions is, and they all need to be mentioned in the main article, and from there I suppose it might be alright to have a tree structure getting into the different points of view. But we have already gotten into so much trouble with these pesky point of view forks, and it looks like the only way to solve them is starting with the main article on each subject. This principle by the way applies to all of the name spaces that Hamilton and his point of view share with

1460:. Obviously any quaternion can be written in the form (ai+bj+ck)/(di+ej+fk), which in the language of quaternions is a quotient of vectors, but that's putting the cart before the horse. If you start from a geometric understanding of vectors, the idea of a "quotient of vectors" doesn't seem to make sense. (Well, a quotient of 2D vectors does make geometric sense: it's a scaling factor and a rotation angle, which map naturally to the magnitude and argument of a complex number. And a quotient of 1D vectors is a real number. But if that geometric picture extends to three dimensions, I'm not seeing how.) -- 2966:

coordinate system, based on the earth, and then go to the sun, and when you get there check on your watch and see what time it is. This time would be the proper time, not the coordinate time, that the four vectors of relativity are based on. In this system of pairing space and time with quaternions, going someplace at the speed of light, would have a zero in the scalar part of the quaternion, because the proper time traversed by a photon, going from one place to another is always zero. Then make the vector the distance traveled as calculated from the stationary system at the starting point.

1202:

references he gives? But if we add something as a section to the main quaternion page, then this should be referring to notable sources, and most of them go probably back to the early 1900s. A section "Recent developments and research directions" is really misleading. If you indeed find such a direction or effort, in a broader scope, you might contemplate creating a separate page that tracks this research. However, one person's paper(s), if peer-reviewed, wouldn't be enough for a page. I still vote for a line item in the "links" section, for Trifonov's two papers. Thanks, Jens

2946:

quaternions, that represent places where massive particles starting at where you are right now, could not travel to in the future, and could not have traveled from in the past. It would be really tempting try and get back to Hamilton's original view of space as a field of progression by just throwing these "elsewhere quaternions" out, and just think about space as the future and the past sections of the light cone. A quaternion consists of a vector plus a scalar, but there are other choices for how to select these two quantities, besides the standard 20th century method.

1917:(together with other standard algebraic properties, I'd assume) is sufficient to derive the Brougham Bridge law. That's certainly not true because a free algebra satisfies all of those rules without being isomorphic to the quaternions. You need something else. Whatever it is, it needs to be intuitively natural, otherwise the quotient-of-vectors derivation loses its reason for being. But I can't see how you could write down any intuitively natural property of vector quotients since they seem to have no intuitive interpretation to begin with. None that works, anyway. -- 1291:, and this material would be better if it was dumbed down to make it more understandable to people who were not experts in the field. How many numbers are in this rank three tensor. My guess would be that this would be some sort of three dimensional array with numbers in a cube with 16 numbers on each face, for a total of 16 numbers, but that when put in quaternion form most of these numbers would be zero. But the present text is pretty cryptic, for anybody who is not an expert. Cutting straight to the Heart of the matter.... 3025:

things this way. Modern physics has moved entirely to linear algebra: Dot products, cross products, and so on. Remember that quaternions used to be very popular; for a while after Hamilton invented them, they were the standard approach to doing anything with vectors in three dimensions. That they have not survived as such suggests that they're not really well-suited for that purpose. They're much better suited for things with rotations—this is where they're used in computer graphics, for instance.

1255:-because these are very much a part of Hamilton's calculus. Just about any kind of Hamilton's types, from tensor, to versors, to scalars, to vectors, to quaternions can come in real, imaginary and bi flavors, depending if if the coefficients are real, or imaginary or bi- as in having both a real and imaginary part. Along these lines of thinking there has to be a distinction between the vectors i,j and k and the imaginary scalar, which in Hamilton's later writings is represented by H. 3072:

suddenly makes it a lot more interesting to me: how q's combine the dot and cross product in one closed package seems just fundamentally 'right' in a number-theoretical sense - it's very seductive how a spacetime-like construct appears to emerge just from pure maths, compared to the ad-hoc tossing stuff together that happens in the rest of modern physics. I guess many physicists would consider that 'numerology' and maybe it is. But I like being seduced.

31: 970:, but there is a long story about quaternions and relativity, at least from my point of view. A good place to start off, to make the article sound less bias would be with Penrose, there is a good quote from him that using quaternions to represent events in space time is tempting and dangerous, and he gives reasons why it is dangerous. However in a later chapter he explains that quaternions have the proper geometry for 3709:

a course or reading a textbook. If you don't have a solid working understanding of vectors, you have no chance of understanding quaternions, and an article that hand-waves about how they're useful for modeling rotations because that's the only thing normal people can understand will a) give non-experts a false sense of understanding, b) hide all the useful information from people who can actually understand it. —

1004:? From there an article should also mention that the whole idea of a lorentz transform, should actually be called a Fitzgerald-lorentz transforms, because it was a quaternionist from trinity university in Ireland, the same one Hamilton taught at who invented them first. Lorentz actually mentions Fitzgerald in his original article. The history article has an exact quote. There was something called the 2030:

and going some place else, or of moving, as he lays out in his Lectures. In Elements he distills the idea of a vector down into pure mathematics, and from my point of view, part of the reason he does this is that he has already discovered some really strange properties of quaternions, which makes him quit talking in terms of four velocities, because he knew that people would never believe it.

437:(1) you'll simply be working with power series with integral powers of a quaternion; if you chose (2), however, you'll have to clarify "derivative" (because you're in a higher-dimensional vector space). ... Hope this helps; not sure how much could be added to this article. Check out the Conway (2003) book, just as an example. Thanks, Jens 3872:

as you can multiply your factors in any order and get the same result. This relies on commutativity, which does not hold for quaternions, so maybe as you re-order your quaternion factors you get different roots. How this leads to infinity I don't know, but it at least suggests things are a lot more complex when you lose commutativity. --

2055:, and he still has the idea of a coordinate system made of of points, and transforming from on coordinate system to another, with out really digging down to the problem that there might be something essentially wrong with the idea of a coordinate in the first place. If he did that, he would according to Smolin be branded a nut, and the 1733:

great that we have an article or two devoted exclusively to his point of view, since Hamilton had a tendency to work things on in minute detail, and not to have an in depth articles would cause the amount of text on the other points of view be less significantly less than the text devoted to Hamilton in these articles.

1852:, but it won't be the right one. So the quotient can't be an operator on vectors. Then what is it? It's a quaternion, but that's circular—quaternions are what we're trying to model in the first place. I've never seen a presentation of the quotient-of-vector approach that explains what a quotient of vectors actually 2036:

article 213, most of what Hamilton is talking about would have been understandable to Euclid. It is in article 214, where he shows that there has to be an imaginary scalar, and proposes to give this entity meaning as an indicator of geometric impossibility, that he takes a sudden jump into deeper ideas. He says,

855:? Penrose pretends at least to be unaware of this little redefinition, that Cayley pulled off in the 1890's. Not that I am pretending here to fully understand Penrose's position. Just about all great thinkers promote their own ideas, and they have every right to, and Penrose is leading up to his twister theory. 600:

choice as it is (clockwise or counter). Still, it is more than a normal 4 dimensional matrix can handle. A quaternion doesn't obey the distributive law when multiplied thanks to this direction. A * B doesn't necessaryly equal B * A, the quaternion A * B = -A * B or = A * -B depending on the aforementioned angle.

2528:

I would be one of the people involved in that. I suppose I agree with Serre and Bourbaki, in that I think blackboard bold looks ugly in print. (The contrast between the black ink and the whitespace is awful.) Blackboard bold is just a way of writing boldface on a blackboard; if you have the option of

2209:

I was looking at that section with great interest, and also a mild bit of skepticism. The part that I find troubling with these sorts of arguments, is the rash idea that an imaginary scalar, can be substituted for a vector. Algebraically both an imaginary scalar, and a vector have a negative square,

2024:

from dover books translated into english. As I read, and compare I am seeing something really shocking. The first definition in the 2,500 year older version of elements that Hamilton is rewriting starts out with giving the definition of a point. The early approach that Hamilton took, was to define

1645:

Ibenez or something is his name, that is from memory I need to look it up. The thing that really caught my eye about this guys writing is that he uses the term scalar the way Hamilton uses it, as in there are real scalars, and imaginary scalars, and bi scalars(complex numbers), and then goes way out

1630:

which just means the quotient of two vectors that are the same length! In this case, what you have is a versor or a unit quaternion, meaning that the tensor or stretching factor of the quaternion is one. In this case the scalar part of the quaternion is the cosine of the angle between them, and the

1349:

The rank-three structure tensor in Trifonov's paper is just a way of representing the operation of quaternion multiplication. If p=qr then every component of p is equal to a sum over the 16 pairwise products of the components of q and r. That's 16 coefficients for each component of p (of which 12 are

599:

Your effort to define the three dimensional complex number as a matrix by matching properties to functions is noble, but misplaced. A quaternion is more than just a four dimensional vector in most cases, it not only has a 3-D vector and an angle component but a direction for the angle as well, binary

4308:

I would say no and no. Or at least for (1) there is none better than taking those with positive real parts. In practice usually you ignore the problem until you need to deal with it, at which point you insert a check and negate one of the quaternions to fix a discontinuity like you describe. You can

4289:

To give some background: I am using Matlab's Aerospace Toolbox where the scalar part of a quaternion constructed from rotation angles can take any value between -1 and 1. A function I made calculates orientation from rotation angles and should produce something like a sine with the test input. There

3708:

Personally, I'm not a big fan of the approach where we try to explain things in baby terms without any rigor so readers without sufficient background get a false sense of knowing what the article is talking about when they have no actual idea. You can't understand mathematical topics without taking

3024:

I think this sort of approach was followed by a few people right up until the start of the 20th century. You'd have good luck trying to follow the works of Hamilton, Tait, and their intellectual descendants. But a downside of this approach is that you don't seem to really gain anything by looking at

2931:

with each other. So in going from one time and place to another, subtract the start point in space from the end point in space, B - A and put that in the vector part of the quaternion. Subtract the stationary times of leaving A from arrival at B, and put that in the scalar part of the quaternion.

1999:

Hamilton thought that addition and subtraction were acts of synthesis and analysis respectively. He called these ordinal. But he also identified another kind of synthesis and analysis which he called cardinal. Multiplication is cardinal synthesis, and division is cardinal analysis. Any good 21st

1993:

vector algebra should have addition, subtraction, multiplication and division. The really hard part of understanding Hamilton is not learning his ideas, it is unlearning everything you thought you knew about vectors. I know it is hard, well at least it was for me, and I am sure that it is hard for

1732:

and the proponents of the linear algebra exclusive point of view. This means that a main article on vectors, needs to include Hamilton's point of view on them, a main article on tensors needs to include his view, a main article on scalars needs to include his point of view and so on. I think it is

1670:

Pretty fun stuff, and the even better news, is that it is easy to supply five or six good references all saying the same thing and using the exact same notation, that can be hot linked to because the authors have been dead more than 100 years, and the copy rights have expired. But I gotta go to bed

1635:

and then I think you can follow the link to Hardy's really good book. Also I think there is a guy that is cited in the article on using quaternions for rotations, that goes over all this stuff pretty well, and in fact, actually uses the T,U,S,V and K operators, following Hamilton, but he is writing

1020:

That guy at www.quaternions.com who's link you just added, makes the claim to formulate relativity using quaternions as well, so that would be a second source. But that is about as much as I can think of, and we would need other people to contribute if we wanted the article not to sound like it was

933:

Question: Other than V. Trifonov, is there any other research in that direction? I've read a couple of his papers, and find them interesting; maybe a line item in the existing "Links and monographs" would be more appropriate? E.g. "Quaternion use in General Relativity after V. Trifonov: " and then a

813:

I see you already have the U operator. Conceptually and the operators S and V for take the scalar are in the article, but denoted with subscripts. Perhaps it might be helpful to your readers to explain to them that S and V are also operators, and it is permissible to write them as s(q) and v(q).

790:

It is a just a normal sphere. The sphere itself is 2D, it is embedded in the 3D imaginary subspace of the 4D quaternion space. I think what is a bit confusing is that this "regular old" sphere is called 2D. (The 2D sphere is the surface of a 3D ball!) So, Lotu's version without the explicitly stated

4507:

It was probably in to show that you can separate out the individual parts of the quaternion if one can multiply by the basis elements, so in this case they are just functions of four variables in general. This is different from complex numbers where one cannot do anything like this and you have the

3871:

It does say unexpected. It's not a result I'm familiar with, but I suspect the connection is the roots of a polynomial over ℝ or ℚ can be got by factorising, or at least once you have the roots you can write down the factors. This factorisation (and so the set of roots) is unique up to a re-order,

2955:

Hamilton viewed space as a field of progression. He viewed a vector as representing the idea of going from some place to someplace else, something that carried a movable point from A to B. He viewed a scalar as representing progression in time. This suggests that the proper time, as measured by

2217:

In Hamilton's thinking if there are an i,j and k, then there has to be an h also as he demonstrates in article 214 of elements. He does this with a simple argument based on calculating the intersections of a line and a circle. A line can intersect a circle on one, two or zero points. If the line

2105:

is a book Hamilton would have written if he had the time. Hamilton always said he was going to write a basic book on quaternions. In Hamilton's elements he even outlines some basic sections that should be included on first reading. Hardy set out to be absolutely faithful to Hamilton, but writing

1983:

OK so now we are up to the concept that a quaternion defines a plane; and that when a quaternion and a vector in its plane are multiplied the effect is that the vector gets rotated and stretched. But the key point here is that the answer is another vector. You were asking what do you get when you

1616:

This is the way that Hamilton introduces the concept of a quaternion, but at first he does not put a quaternion into quadranomial form, but rather introduces the quaternion as consisting of a versor which has an axis and an angle, and a tensor which does the stretching. An axis takes two angles to

1008:

that I wish I knew more about. The 20's authors that used quaternions for special relativity switched over to tensor calculus when general relativity came out. Oh by the way, Hamilton and Tait sort of invented the word and the idea of a tensor. Tait in particular was working on using quaternions

948:

No, please don't do this, because the general-relativistic part of Trifonov's work (the stuff about the FLRW metric) was nonsense. Only the special-relativistic part made sense, and there wasn't much to that beyond the observation that the scalar part of q is the Lorentz norm. I think everything in

901:

This topic is notable and interesting, but the trouble is that only one person so far ever contributed to it and this gives this particular section the appearance of bias. The way to solve this problem, is not with slash and burn to the ground tactics, but rather, by having more editors contribute

436:

if you want to generalize the integral of a closed path (like above), or over an area or volume in 4 dimensions. For the Taylor series, you can proceed two ways: (1) define a function through its Taylor series, or (2) define the Taylor series through derivatives on a function. When you chose method

3671:

Someone noted on this talk page that the intro needed improvement; indeed, the intro entirely lacked a basic definition of what a Quaternion is! I added a definition, so the intro is somewhat improved, but this article is simply full of jargon that is unexplained. This article certainly does not

3567:

I think that the concept of representing the location of one point in space relative to another point by a triplet of numbers most probably goes back to Descartes. Hamilton had a very geometric viewpoint, and defined the modern concept of a vector as a geometric entity that can be represented by a

2896:

That said, it doesn't mean these metrics are useful. The norm is multiplicative, so it captures some of the algebraic structure of the quaternions. Other metrics won't usually have any relation at all to the algebraic structure of the quaternions, so they'll have less information than the standard

2456:

Ah, it's nice to see edit-warring (even if minor so far) over such a classic debate! To blackboard bold or not, that is the issue. I think I'll just stay out of this one, except to say that if you flip through a variety of texts (particularly those from a couple decades or more ago), you'll find

2235:

which bore little or no relation to actually observable geometric reality, and there will be many editors of this article who advocate exposition of this view to the exclusion of all others. I think it gives the section a rather unfortunate point of view type quality. To add balance we might add

2050:

So as much as I find discussion these things interesting, need to get to work on my Thermodynamics homework, because unfortunately I don't really get any credit for my efforts to understand Hamilton's thinking. One final comment, as far as an area of active research, I re-read Smolin's book again

2029:

which he talks about just a little in article 666 of lectures on quaternions. Hamilton never really tells us why he decided to start over with a different approach, but when he undertook to rewrite Elements, his first definition is not of a point, but of a vector, the idea of starting some place,

1727:

with the current definition, the current definition is a point of view held by great mathematicians, and perhaps we should list their names. But the trouble is that the current text is very, very point of view, and it shouldn't be, if different people think that there are different definitions of

1308:

As I see, already two of V. Trifonov's papers are listed in the "publications" list; one comes right after Adler's seminal "Quaternionic quantum mechanics and quantum fields" - that hurts. But for devoting a whole section on it, no; unless someone can show where this has drawn independent, notable

488:

Thanks so much, Robert and Jheald. The power of Knowledge - leave a note on a topic that you're interested in, but didn't have the chance yet to spend serious time to study. And sure enough, I come back after a short while, and there are two excellent leads straight into the heart of the interest.

451:

The Hestenes and Sobczyk book, "Clifford Algebra to Geometric Calculus" (1984) is also a source looking at how much of Complex Analysis can be carried over to other Clifford Algebras. I think they may well find they can extend the idea of analytic functions, but you'd need to look it up. There's

3071:

I'm intrigued by this idea, TeamQuaternion. The quaternions and their inverse relationship to Minkowski spacetime fascinate me - I suppose largely because of Doug Sweetser's 'Physics with Quaternions' pages. My vector algebra-fu is pretty weak but learning about the history of Hamilton's thinking

2728:

Editors who have been following the section on Functions of a quaternion variable will note that a contributor from Venezuela has noted a restriction to be placed on functions extending a complex-variable function. The concern arises in a function like f(z) = i z which is not easily extended. A

2213:

But what troubles me is the substitution for one type of number, an imaginary scalar, which can be viewed as an indicator of geometric impossibility, for a vector, which is a completely different, and geometrically real type of quantity, carrying both magnitude and direction. In other words some

2005:

So anyway, the first step, is to go back deep into your mind and erase all the ideas you have about what the product of two vectors is, and then Hamilton's point of view just might start to make more sense. But again the product of a quaternion and a vector not in its plane is in general another

2975:

With a mapping based along these lines, placing the proper time as experienced by the object carrying the moving point, in Latin the "vector" in the original way it was used by Hamilton, you could, map all of the quaternions to the "future" and the "past". And this very strange idea of elsewhere

2945:

that are so often depicted in books like "A brief history of time". With this mapping of quaternions to space time, space is divided into three regions, which Hawking and others have called, the future, the past, and the other region, called "elsewhere". "Elsewhere" seems to be extra left over

2732:

The heart of the matter arises in the section viewing H as a union of complex planes where the imaginary i is selected from a sphere of square roots of minus one in H. Once i is selected, so is the antipodal point −i , so a pair of points on the sphere corresponds to a single complex plane. This

1283:

The thing is that I don't think it will be easy to recruit anybody do work on an improving the in depth history of quaternions. You would have to find at leas three people as the article is currently being squatted on by two editors at least in favor if its deletion. The net result is that you

3588:

The introductory section of this article would benefit from a reworking. In particular, the graph is not very helpful, at least not at that stage. Worse, the key equations are missing. I read the first section three or four times and didn't really understand it. (I have a degree in physics

2035:

But I am still struggling with his deeper thoughts. Among them that here has to be both vectors and an imaginary scalar, which he talks about in article 214. There could also be a bit of numerology in this section as well. Euclid had 13 books. Up until book 2 section 13, which happens to be

2019:

In order to make these arguments it seems to me what they did was to first redefine what a quaternion is, and then show that this new and different entity is the same as a matrix. The approach that I am taking right now, however is going farther backwards. I have obtained the older version of

1279:

article and the history article were removed, as was the link to the article on Hamilton's thinking on quaternions. Hence the arrangement were at one time, one could click on the link to history, and from there click on the link to Hamilton's thinking has two removed links. The links from the

3075:

I'm still struggling my way through the Cayley-Dickson construction - realising that since ij=k, we basically don't need k as a separate entity, so i and j are really the only two 'true' imaginaries and the quaternion structure appears as two bits: 00 = the reals, 01 = i, 10 = j, 11 = ij. If I

2965:

In other words, Hamilton's idea of a "vector" was the thing carrying the moving point. So think of starting were you are, like for example the earth, and consider going some place, like the sun, like Hamilton did in his first lecture. Perhaps measure the distance to the sun in the stationary

2239:

Do any of the authors making these kinds of arguments start out by stating that they have read articles 1 to 214 of Hamilton's elements with attention, however disagree with Hamilton's conclusions? Or do they just make their arguments, as if Hamilton's writings don't even exist, and time, and

3551:

I think it's a strange turn in history that a 4-dimensional value was used to represent 3D space. Hamilton wanted something that had division to represent 3D space, but in most things I know you don't need to divide points in 3D... You usually add them or calculate lengths and angles. I can't

605:

It's best to think in algebraic terms and not simplify to matrices until the algebraic terms are fully comprehended. It is trivial to apply your wished for matrix calculations to, just remember that matrix calculations are merely condensed algebraic notations and that your terms don't use the

1201:

receive more attention today. How can you represent that in an encyclopedia? Currently, there's a section "Three-dimensional and four-dimensional rotation groups" on the main quaternion page, maybe there? Or a new section "quaternions and the light cone", talking about Penrose's view and the

1014:

They sort of reinvented quaternions to do quantum mechanics, because they needed them for their spin matrix ideas, and Altman writes pretty extensively on this. At my University library there are only three books on quaternions and he has one of them, so that would be another area of active

2218:

intersects the circle in zero points, evaluating the formula for the intersections, will result in taking the square root of a negative scalar, and the result can't possible be any linear combination of i,j and k, and must be then a distinctly different kind of quantity. Hamilton remarks,

677:

Another note to Nazlfrag: Quaternions may be left-handed or right-handed, that's correct; I assume that this is what you mean with "direction of the angle". However, that is an inner symmetry and does not influence the commutator relations (the "algebraic terms", so to speak). For any two

1514:) can be thought of as an operator, operating on w, but you can't change the order of multiplication as we all know when using Hamilton's definition of multiplication. This operator has two effects, a turning effect, or an act of version, and a stretching effect, or an act of tension. 1350:

zero and the other four are ±1), 64 coefficients in total. You can put those in a rank-three tensor, which has 4 components. There's nothing deep about it, it's just a different notation. I don't think it's worth mentioning here unless it's a much more common notation than I think. --

2994:. If anybody has published about using quaternions in this way second way, I for one would find investigations along these lines very interesting and would like to see them included someplace in Knowledge content. So far I am unaware of any published thinking along these lines. 4275:

I have checked the article and discussion (and archives) but I can't seem to find anything about which quaternion to use for a given rotation. As far as I know, both "q" (angle \theta around vector p) and "-q" (angle -\theta around vector -p) represent the same rotation.

1631:

vector part, is a vector perpendicular to both w and v with a length that is the sine of the angle between them. You get the direction of the vector by some sort of right hand rule, but it is a little late at night for me to remember all this stuff by heart. Click here

1376:

Given this definition one can prove that vectors, and quaternions have to multiply according to the Brougham bridge law, and that vectors, the geometrically real ones at least, have to be three dimensional. Seems an important concept to have been excluded from the main

473:

Due to your interest Domitori, Jens Koeplinger, and Jheald, I looked up the 1973 article on Feuter's quaternion analysis. The summary shows links with theorems in analysis. This topic is rich enough to draw more contributions. At some point it may grow into a separate

4190:

The last line starts: "This makes it possible to divide two quaternions p and q in two different ways." Unless I'm missing a subtlety here, it should more accurately read "This makes it possible to depict division of two quaternions p and q in two different ways."

4309:

check for example that the 4D dot-product between them is positive, and if not negate one. It's impossible to partition the quaternions so all are in one half as it's easy to come up with a rotation that rotates smoothly from one quaternion to −1 × itself.--

3611:

I don't know which it is, but the opening sentence seems to be incorrect. 'quaternions are a noncommutative number system '. The phrase 'a noncommutative number system ' sounds singular. If this is so then the subject and verb should also be singular.

2044:

homebum, of course I knew that this math implies that there are certain 4 velocities that are geometrically impossible, but I am writing in pure math for future generations, and not coming right out and saying it because my generation would never believe

862:

were it belongs in this article and stop using it in the classical article. I thought that perhaps since this talk section has more traffic that some folks with a little bit better diplomatic skills could help out here. We are having trouble reaching a

2461:

has some interesting points regarding blackboard bold...). Hm, ok, I'll say a bit more :-) If you look through a number of math journals (including top journals like Annals, JAMS, Inventiones, etc.), they allow blackboard bold. Anyway... fight on!

2804:

I tried to make the formulas not run over the right edge of the article as it does in my browser. Also, "1 = (1,0,0,0)" etc. were better aligned, though the larger font was not intended. And I made an extra link as well, none of that was useful? --

1079:

a lot of people over a long period of time have thought about the relationship between quaternions and relativity. It was unfortunate that we ran that guy off when he was interested in contributing. On the other hand this is an emotional subject

2985:

I was wondering about, and would be really interested in reading more about if anything that had been written about puting the proper time, in the scalar field of a quaterion. I have been wondering how quaternions might relate to the concept of

2000:

century book on quaternions would probably start out with lies your 20th century math teacher told. The trouble is that there really have not been any good books, that I have found at least from the Hamiltonian view, after the first world war.

3589:

although I am a bit rusty.) Giving up in dispair I finally went on to the main article and saw i2 = j2 = k2 = ijk = − 1 and said "Oh I see!" out loud. This equation is by far the most important bit in the article and MUST be in the intro.

3119:

Also, you speak of "the" dot product. This is a fiction; there is no one single dot product. To define a dot product, or more properly a symmetric positive definite bilinear form, the usual method is to choose a set of basis vectors. Then you

4327:

I knew I read about this dual representation somewhere. And after an hour of searching, I "decided" I read it on Knowledge. Could be that the division was actually according to the sign of the real part. Anyway, thank you for the answer.

215: 1263:

after a failed deletion attempt. The material that I mentioned you were correct once did live in the history section, but the present consensus is that the history of quaternions can contain only historical facts which are approved by

3936:

The introduction gives the impression that Quaternions have been superseded in 3D computer graphics, when in fact using Quaternions to handle rotations is almost always preferable to multiplying matrices together, for two reasons:

1333:? I don't mind mentioning attempts to connect quaternions and quaternion-like objects to the geometry of Minkowski space, I just don't want Trifonov's work mentioned as important new research, because it isn't. Penrose is fine. -- 2189:

If this development trades simplicity for rigor, of course Chapter two of Hamilton's Elements gives a treatment of a quaternion as the quotient of two vectors, with a degree of rigor, which may at first reading seem just a little

4344:

A big advantage of quaternions is you avoid those problems of special values but you replace it with having two equally valid representations. Even if you start with one if you move smoothly around you may end up with the other.

2677: 3853:

I don't see the connection between comutativity and the number of solutions here. To me it looks more like a question of constraints and degrees of freedom. What's the connection? Could someone who understands please explain?

2770:

I'm sure we could find some copyright-free pictures of Quat's for this article, it's a bit dry without some visuals. Nice work though it looks very in-depth! I'll see if I can do a pic myself in UltraFractal and submit it.

3705:

How about: "noncommutative", "number system", "directed line", "vector", "scalar", "tensor". Few or none of those will be comprehensible to someone with only a semester or two of calculus. And that's just from the first

3650:

of quaternions is ..."), but that wouldn't be very helpful for people who don't know what an algebra or ring is. I think "the quaternions form a non-commutative number system" would be better than what we have at present.

2214:

authors ignore the distinction between that which is geometrically real, (a vector), which is being substituted for something, that is both different, and is truly imaginary in a geometric sense, an (imaginary scalar).

2006:

quaternion. Also the product of two vectors is in general a quaternion. Again it is the triple product of a quaternion a vector and the reciprocal of the first quaternion that is a rotation, and can be represented as

2820:

Ah, I see what you mean now! It's odd, I remember the formula wrapping before. I didn't like the way you fixed it, though, because in some cases that technique will lead to incorrect formatting. What do you think now?

825:

Can we all agree that the term norm is a modern term and can only correctly be used in a main article which treats quaternions in a more general way and is not exclusively devoted to the treatment of Hamilton's ideas?

2932:

Then the scalar of the product of the resulting quaternion with itself gives the square of the proper time. The proper time being the time as measured by the massive particle, which traverses the path from A to B.

1573:

is already aware of this, as I have discovered he has been contributing to these discussions for quite some time, but I add this fact for others who might be following along, who might have been confused on this

978:

or moving from one place to another. Penrose would be a good source for the fact that relativistic velocities add like versors, or vector arcs or rotations. You add to versors by multiplying them techically.

771:

No, it's two-dimensional. The article specifies that this is taking place in the three-dimensional space of pure imaginary quaternions, and in that space, this is exactly the standard equation for a 2-sphere.

2306: 2919:. The square root of this quantity, would be a good candidate for a quaternion metric. For this to work, you would have to select the scalar and vector parts of the quaternion using a conventions called 991:, this is also from Penrose, and it is his idea to model velocity space using only some of the quaternions. Basically his idea is to use the quaternions that make up the part of the light cone called the 821:

as some folks like to call it? Fans of modern notation like Cayley's right? Both Hamilton and Cayley used the T operator, I have the documentation. Many readers would find the T operator very useful.

1287:

Excluding Trifonov's ideas from Knowledge, does not really seem like an issue that is worth me wasting a lot of time on. It does seem interesting that he thinks quaternions have something to do with

2429:

There seems to be a change, to have number systems not in blackboard bold, but in bold instead. The text now looks a bit unfamiliar to me, since I've been used to blackboard bold notation only (e.g.

272: 4489:

It does seem oddly irrelevant. Since it seems to have no connection to anything else in that section, I've removed it. Anyone who can give a more complete explanation is welcome to put it back in.

4459: 1094:

Hmm ...... let's see: Currently we have the following pages in Knowledge on quaternions, which highlight the various angles of interest people have (and have had over the decades) in quaternions:

3687:

Well, that's better than your first drive-by tagging, but it's still a pretty vague and generalised complaint that gives other editors next to nothing to work on. Perhap you can start by giving

3906:+ 1 has no roots; over the complex numbers, it has two; and you can never have more than two, no matter what kind of numbers you're looking at, as long as you assume commutativity, i.e., that 3914:. When you don't assume this, then, well, you get the result mentioned in the article, that you can have infinitely many solutions. Proving that there's infinitely many is just a computation. 829:

This is causing a difficulty because some folks are of the opinion that since Cayley actually wrote an article in one of Tait's books on quaternions that this means that Cayley's terminology

3806:

I've come around on this and tend to agree at this point that specific issues should be addressed here but that the article as a whole can do without the jargon template. I also agree that

1984:

multiply a quaternion defined as the quotient of two vectors with a vector that is not in its plane? The result of this act of cardinal synthesis is of course in general another quaternion!

1009:

to represent three dimensional stresses and strains, in solids right about the time that that guy voight or how ever you spell it started writing the same sort of things in matrix notation.

584:

The information in Wolfram Mathworld is different to Knowledge but not correct either, I think... Maybe this problem is related to the convention for matrix multiplication used (pre/post).

4026:

The filename has "plague", but the caption correctly has "plaque". If the word in the caption looks like "plague" to you, this might be because of the font in which the page is displayed.

366: 2335: 1325:

Homebum, you must be misremembering Penrose's paper (which I haven't read) because it's certainly not true that relativistic velocities resemble versors. Relativistic velocities form a (

403: 2178: 1567: 4155:

Since quaternion multiplication is non-commutative, should the multiplication table at the end of 2.1 make it clear which operand is the left operand and which is the right operand?

2040:, as if he had somehow known that I would be skipping ahead to this section, and why. I know I am reading between the lines here, but when I read this, Hamilton kind of said to me 2785:

Oops I thought I was on the fractal wikibooks page about quaternion fractals.. that is confusing how the link looked internal.. oh well.. I will do some work on that page instead.

1068: 2897:

metric. If you could find some other metric which is related to the algebraic structure of the quaternions, then you would have a new and interesting way to study quaternions.

1607:) will have a vector part that is perpendicular to that plane. Unless the angle between the two vectors is a right angle, the quaternion will also have a scalar part as well. 4046:

In the section "noncommutative" the author talks about a "two-dimensional sphere". Sorry but I think you need to look at the wording here. I daren't change it to circle.

3736:

I've changed the lede based on this. I've also removed the jargon template. I don't see much jargon in the article, and I'm not aware of any that's avoidable. If there are

754:

The article mentions a **two dimensional sphere** defined by b2 + c2 + d2 = 1. This looks like are regular old **three dimensional** sphere to me but I want to make sure.

4110:

See the history section, from the paragraph starting "From the mid 1880s". The wording is a little different in the other article but it's describing the same things.--

3973:

I agree that was wrong: quaternions are as popular as they've ever been for the reasons you've given among others. I've rewritten the sentence to better reflect this.--

2375: 983:

Also Penrose has those same old diagrams from the 1920's that Bertrand Russel used in his book ABC relativity, and saying that they are the product of a signature flip.

292: 2355: 312: 1268:

written by fairly partisan writers. According to this consensus view, listing historical facts in chronological order, (since this can be used as a tool to reveal

817:

I would be willing to trade you. Would it be OK to take out the word tensor, and just talk about the T operator. Tell people that you can write T(q) to take the

2708:

I don't see that it makes any difference. Isn't what you've written just a natural equally-valid equivalence of "the angle between the two reflection planes" ?

121: 113: 2184: 1193:

Looking over all these interests, that whole section about Trifonov's work alone is overrepresented. In your long explanation above, you only mention work

2602: 4088:

The article on vector calculus noted that vector calculus evolved out of the study of quaternions. Is this true? If so, should it be mentioned here?

2457:

what you seek. The whole "blackboard bold is wrong" idea seems to be perpetuated by the Bourbaki crowd, particularly Jean-Pierre Serre (the article on

1693:

I see no problem with the definition of quaternion in the current page. I see much value in going into breadth on the vector quotient definition on the

2529:

real bold, then you should use that instead. We mention the blackboard bold notation once in the lead, and I think that's enough for the whole article.

1162:-- Looks a bit out of place; the first link, "Quaternions", redirects to the wrong page (Hamilton's), and most of the rest should be included in the " 2403:

Engineers might like this article about quaternions written by Do-While Jones because it shows you how to use them without learning too much theory:

3810:'s edits definitely are an improvement. I'm glad to at least have initiated a good discussion on the topic. Thanks for the constructive input :) 2068:

are going to get pulled on me. Thanks everyone for putting the time and effort into thinking about how we can improve the article on quaternions.

1110:-- recreates the view Hamilton had on quaternions, and developments throughout his lifetime (1805-1865); this article is "starter", as they call it. 3076:

understand it correctly. This seems to say something very profound about the nature of dimensionality which isn't often explored but ought to be.

2858:

This article suggests that the "norm" or as Hamilton called it the tensor of a quaternion, can be used as a metric, to construct a metric space.

2259:

Just a quick note, on a subject of the definition of a quaternion. I believe that better books on the subject make a clear distinction between

3850:

From the article: "The equation z^2 + 1 = 0, for instance, has infinitely many quaternion solutions z = bi + cj + dk with b^2 + c^2 + d^2 = 1"

678:

non-commutative basis elements of quaternions, they remain non-commutative regardless of left-/right-handedness. Hope this helps! Thanks, Jens

547: 459:

lists some of what can be achieved, to see whether this is the sort of thing you had in mind; but you'll have to look elsewhere for details.

4316: 4290:

are however some jumps and I found out that my output is correct up to the sign. Now I need something to determine whether to use q or -q.

4117: 3980: 3633: 2868: 995:

which are the quaternions if I remember correctly that have both a positive scalar part and when squared also have a positive scalar part.

1812:? Forget quaternions for a moment (since we're trying to develop them from first principles). What's the intuitive geometric meaning of ( 843:

Just how devious was Cayley in promoting his own ideas? Could he have been thinking that redefining the Hamilton operators, so that the

763: 4474: 585: 3615:

The sentence should be either 'quaternions are noncommutative number systems ' or 'A quaternion is a noncommutative number system '.

840:

of a quaternion, is actually the square of what you guys are calling the norm. I believe 'norm' is a modern term that he introduced.

4192: 4095: 4053: 3590: 3553: 2840: 2772: 1632: 1151: 1106: 2272: 3718: 2941:

With this choice, suggested by Einstein, for the vector and scalar parts, all the possible quaternions can be plotted out in the

1694: 1130: 830: 2106:

at a more basic level. Starting at the definition of a quaternion as the quotient of two vectors, in article 21 Hardy gets the

3049:

Please do not delete my talk page edits. Also, it is very hard to reply to you when you extensively modify your initial edits.

2740:

The contributors' asserted restriction on the provisional extension of f(z) = u + i v follows from this string of equalities:

2502:

I'm referring to the beginnings of a brewing battle (check the recent edit history). Neither of us are involved in that. --

2185:

http://books.google.com/books?id=YNE2AAAAMAAJ&printsec=frontcover&dq=Quaternion+vectors+quotient&as_brr=1#PPA32,M1

489:

That is very efficient for me personally, and almost as a side effect, it's publicly available for everyone. Thanks all, Jens

4011:

The title (and I think the filename) of the plaque has the word "plague" instead of "plaque". I don't know how to edit that.

3960: 2225:

The behavior of h is different, because in multiplication it acts just like a scalar. For example i x j = k but j x i = -k.

1326: 2228:

However h x i = i x h, because h, is an imaginary scalar, not a vector. This is because h has magnitude but not direction.

1197:

Trifonov's. In general, you have a focus on physics and Lorentz invariance, and seem to believe that quaternions in physics

2236:

that critics of Hamilton, posit that the 3 + 1 structure he advanced, from their point of view has no cosmic significance.

223: 2084: 1749: 1184: 1178: 921: 3552:

understand how the idea of something so complex, has appeared in history before the idea of a simple triplet of numbers.

4374: 1392:

I've heard people define quaternions as quotients of vectors too, but I've never figured out what they mean by that. If

4368:"However the complications of the quaternion variable still challenge investigators. Consider for example the function 4068:

It doesn't mean a cirle; it means the two-dimensional surface of a sphere. I have changed the wording to clarify this.

2693:

I think this should correspond to a rotation of two times the angle between the vectors in the plane containing them.

1989:

I think that before you can go forwards you need to take a step backwards. Hamilton started out by thinking that any

905:

Can we please now get over the recent episode of mass hysteria, and deleting large sections of text on this subject?

4134:

However the complications of the quaternion variable still challenge investigators. Consider for example the function

2737:. So, rather than a sphere of complex planes in H, there is actually a real projective plane of complex planes in H. 2596:

Now in geometry, two reflections make a rotation, by an angle twice the angle between the two reflection planes, so

92: 1146: 1102:-- subject entry page, should reflect any angle or point of view that has received significant attention over time. 38: 2751:

Thus the given f(z) must satisfy these constraints on u and v before f can be extended in this manner to all of H.

2729:

conservative option is to use power series with real coefficients to generate functions of a quaternion variable.

1134:-- focuses on their applications in 3D space computations (e.g. for computer visualizations and guiding software). 851:

in terms of the product of a quaternion and its conjugate would break the quaternion by making the T operator non

2231:

Of course after around 1905 it became all the rage to promote the point of view that it was reasonable to invent

76: 64: 59: 2430: 1222:

article; this way, the history article would have a bit more breadth beyond what Hamilton thought. Thanks, Jens

730:

In either case, however, the formula for a rotated quaternion should be P' = Q P Q, not the anon's P' = Q Q P.

543: 317: 3877: 3780:'s edits are an improvement, but beyond that there seems to be no consensus on what constitutes "jargon" here. 3629: 3007: 2920: 2872: 2698: 2839:

Looks good now (within the currrent constraints of formula rendering...). Thanks for looking at it again! --

2313: 371: 4478: 2109: 1416:? No, there are infinitely many such rotations. One particular such rotation, for example the one around the 589: 4057: 3996: 3557: 2844: 1029:

Why we might even tell our readers that it is very easy to write a Lorentz invariant in quaternion notation:

2927:, and in it he suggests a method for setting up a series of clocks in a "stationary" system, which are all 2776: 1530: 539: 4196: 4099: 3594: 2924: 2559: 2558:

article, so I suppose you could reasonably interpret that as allowing blackboard bold. I've just asked at

2415: 2195: 1370:

I am confused here, I always thought that the definition of a quaternion was the quotient of two vectors?

1218: 1158: 898:

I took the liberty of reversing the deleting on yet another topic within the subject area of quaternions.

4333: 4298: 4282:

1) Is it possible to divide all quaternions into two sets without the obvious division (scalar part : -->

4160: 4136:- this could be expanded on, as it may not be obvious what the significance of the function mentioned is. 3625: 2956:

the point being carried by the vector might be a good candidate for the scalar component of a quaternion.

3956: 2734: 883: 1038: 2264: 4470: 4091: 4049: 3948: 3836: 3815: 3714: 3677: 3643: 3621: 3132:

a set of basis vectors, and if you make a different choice, then you can get a different dot product.

2864: 2485: 2446: 2411: 2072: 2025:

a vector as the difference between two points. But using that approach he discovers that there is a

1737: 1702: 1314: 1227: 1207: 939: 909: 683: 535: 494: 442: 4329: 4294: 4156: 2260: 2222:, in regards to the obviousness of the fact of the existence of both imaginary scalars and vectors. 4530: 4311: 4112: 4073: 4031: 3975: 3952: 3873: 3785: 3696: 3573: 3003: 2861:

Are there any other metrics that could be used to construct a metric space other than the tensor?

2694: 796: 407:

What is analogy of holomorphizm in the quaternion space? Possibility to expand the function into a

4012: 3992: 3891: 3861: 3647: 3081: 2810: 2790: 1284:

would have to get at least three people interested in working on it in order to add any content.

1170: 1138: 1122: 868: 611: 418: 2554:. But it then goes on to say that a list of things commonly set in boldface can be found at the 1636:

in 2001, and using power point slides. Hamilton I think would be pleased with his presentation.

563:

I'm not 100% sure, but in my opinion the 4x4 matrix representation of the quaternion should be:

1329:) hyperbolic 3-space, totally different from the 3-sphere of versors. Maybe you're thinking of 4542: 4141: 4016: 3656: 2756: 2547: 2389: 2245: 2191: 2080: 1745: 1676: 1655: 1626:

To answer the other part of your question if v and w are unit vectors then what you have is a

1382: 1296: 1085: 917: 791:"two-dimensional" would be less exact, but still correct and less confusing to some people. -- 479: 453: 700:

Note that the matrix representation is not unique; for example Lounesto (2001), p. 72 prefers

210:{\displaystyle f(z)={\frac {1}{2\pi \mathrm {i} }}\oint _{c}{\frac {f(t)}{t-z}}\mathrm {d} t} 2713: 2458: 2360: 2059:

would be used against him. I notice that his ideas have been kicked out of wikipedia again?

1527:

talking about rotating a rigid body, in order to to that as is well known you have to write

1114: 879: 735: 660: 464: 433: 277: 2550:

sort of agrees with me, since it says that we should use boldface for sets of numbers like

2340: 2220:

as would have already occurred to anyone who had read the preceding articles with attention

2038:

as would have already occurred to anyone who had read the preceding articles with attention

297: 3832: 3811: 3769: 3765: 3710: 3673: 2991: 2555: 2481: 2442: 1922: 1698: 1599:

one can then prove the Brougham bridge law. If you think of w and v as forming a plane, (

1465: 1355: 1338: 1310: 1223: 1203: 954: 935: 679: 490: 457: 438: 429: 47: 17: 974:, which is actually a lot closer to Hamilton's original idea about vectors representing 878:

I like these ideas enough that I did a little work on the punctuation of their statement

4513: 4494: 4350: 4260: 4175: 4069: 4027: 3919: 3781: 3745: 3692: 3569: 3532: 3054: 3030: 2902: 2826: 2567: 2384:

Is this still a point of contention, or is this explanation satisfactory for everybody?

1269: 1265: 792: 777: 759: 2915:

The scalar of the product of a quaternion with itself, looks a lot like the square of

1280:

history article to hamilton's notation and the other way around have been maintained.

98: 3857: 3077: 2806: 2786: 2507: 2467: 1729: 864: 607: 414: 408: 3991:

Rotation matrices don't exhibit gimble lock. (Either way, I like the new phrasing.)

3124:

them to be orthonormal. Then the usual formula gives you a dot product. But this is

2672:{\displaystyle r^{\prime \prime }=\sigma _{2}\sigma _{1}\,r\,\sigma _{1}\sigma _{2}} 2406: 4538: 4137: 3652: 2752: 2385: 2381:

in the plane defined by the vectors in the divisor and dividend of the quaternion.

2241: 2076: 1741: 1672: 1651: 1378: 1330: 1292: 1252: 1081: 913: 475: 1646:

on a limb and calls the way people were using the word scalar in the last century

3831:

In retrospect, I think the {{technical}} tag would have been more appropriate.

2942: 2928: 2916: 2709: 1216:

PS: I think the Penrose and a few other mentions above should be added into the

1126:-- includes also Lipschitz quaternions, after concepts of famous mathematicians. 852: 731: 656: 630: 460: 46:

If you wish to start a new discussion or revive an old one, please do so on the

4546: 4517: 4498: 4482: 4354: 4337: 4320: 4302: 4264: 4200: 4179: 4164: 4145: 4121: 4103: 4077: 4061: 4035: 4020: 4000: 3984: 3964: 3923: 3881: 3865: 3840: 3819: 3789: 3749: 3722: 3700: 3681: 3660: 3598: 3577: 3561: 3536: 3085: 3058: 3034: 3011: 2906: 2876: 2830: 2814: 2794: 2780: 2760: 2717: 2702: 2571: 2511: 2489: 2471: 2450: 2419: 2393: 2249: 2199: 2088: 1926: 1753: 1706: 1680: 1659: 1469: 1386: 1359: 1342: 1318: 1300: 1231: 1211: 1089: 958: 943: 925: 887: 872: 800: 781: 739: 687: 664: 615: 593: 581:

pure_imaginary_quaternion_rotated = Rotationmatrix . pure_imaginary_quaternion

551: 498: 483: 468: 446: 422: 1918: 1570: 1523:

Notice also that we are talking about an operator that rotates a vector here,

1479: 1461: 1351: 1334: 1163: 1098: 950: 652: 4509: 4490: 4346: 4256: 4171: 3915: 3807: 3777: 3741: 3528: 3050: 3026: 2898: 2822: 2563: 968:

Here is where I would probably get flamed for original research or synthesis

773: 755: 4467:

Perhaps someone should explain why considering this function is relevant?

3106:? -11? But that doesn't have a representation as a two-bit unsigned number. 2101:

For those of us who don't know Hardy, according to Hardy at least his book

1723:

I don't think we as editors at Knowledge should say that their is anything

3365:. The other dot product writes down their components with respect to the 2987: 2503: 2463: 2883:

Yes indeed. As a topological space, the quaternions are homeomorphic to

2592:

In this section, I think there is a problem in the following statement:

411:, that converges to the values of the function in some finite vicinity? 2744:

u(a,b)+ r v(a,b) = f(a+br) = f(a-br*)=u(a,-b)+ r* v(a,-b) which implies

3764:

It is interesting to note that several of the "jargon" terms cited by

3495:′. This will be different. Try it: Figure out what the dot product of 1633:

Classical_Hamiltonian_quaternions#Division_of_two_non-parallel_Vectors

3932:

Quaternions not superseded by Matrix/Vector math in computer graphics

1245: 2433:

and others). It's not a big deal to me, but it is unfamiliar to see

833:, belongs in an article Titled Classical Hamiltonian Quaternions. 4286:

2) Is there any convention of what are the "correct" quaternions?

1275:

You make a great point about broken links. The links between the

2562:. Maybe we'll end up with a consensus as to which we should use. 2441:

for quaternions. What references use that notation? Thanks, Jens

814:

If you need a post 1901 citation, Jasper Jolly is a great one.

2923:. I have been reading about this method in Einstein's paper on 2210:

at least if one sticks to the definitions offered by Hamilton.

2205:

Some comments on the section functions of a quaternion variable.

2407:

Quaternions quickly transform coordinates without error buildup

2399:

Quaternions quickly transform coordinates without error buildup

527:. And that a side effect of this is all analytic functions of 515:

Suggestion: It's probably worth pointing out that all q, where

3244:

and writes down their components with respect to the basis of

2301:{\displaystyle {\frac {\alpha }{\beta }}\times \beta =\alpha } 25: 3894:

that if your numbers are commutative, then there are at most

1259:

The trouble with the present History article, is that it was

1174:-- about the institution that existed from 1899 through 1913. 1142:-- how one views them from an abstract algebra point of view. 606:

standard rules when it comes to multiplication and division.

523:

is a non-scalar quaternion fall in the same complex plane as

220:

belongs to the 2-dimensional space of linear combinations of

3646:

of quaternions is a non-commutative number system" (or "the

2614: 2611: 2014:

arguments boil down to the idea that everything is a matrix.

432:

for the non-real parts of quaternions; or you could look at

323: 250: 4365:

The functions of a quaternion variable section begins with

4255:. That difference isn't something I'd call a "depiction". 2682:

corresponds to a rotation of 180° in the plane containing σ

1373:

I can cite some pretty high authorities on this subject.

95:

in the quaternion space? Should we insist that the contour

3607:

Bad grammer or my lack of familiarity with the subject...

2027:

little difficulty that he can't really get into in detail

1366:

Definition of quaternion incorrect at worst, POV at best.

2255:

Distinction between plane rotation and conical rotation.

3773: 2240:

quaternions began with their redefinition by Clifford?

1021:

just the opinions or pet ideas of a couple of authors.

368:

means some kind of "quaternion part", in analogy with

267:{\displaystyle \alpha \Re (z)+\beta {\mathcal {Q}}(z)} 4377: 2605: 2363: 2343: 2316: 2275: 2112: 1533: 1041: 934:

couple of links. That would be my vote. Thanks, Jens

374: 320: 300: 280: 226: 124: 101: 4537:

is conjugation, achieved with arithmetic operations.

2480:

I'm not sure which of my edits you're referring to.

2357:

that preforms a turning and stretching operation on

1166:" subject entry page. We should have a look at that. 1150:-- I think I'll soon suggest this to be merged into 4454:{\displaystyle f(q)=-{\frac {1}{2}}(q+iqi+jqj+kqk)} 4132:At the start of this there is a section that reads 1424:

axis? No, no rotation choice function can satisfy (

413:These topics should be mentioned in that section. 4453: 2671: 2369: 2349: 2329: 2300: 2172: 1617:define, so that makes three angles, plus a tensor. 1561: 1062: 397: 360: 306: 286: 266: 209: 107: 3902:polynomial. Over the real numbers, for instance, 4271:two quaternions for each rotation representation 4007:Plague should be plaque in title of the picture. 4186:"Conjugation, the norm, and reciprocal" section 1844:) in general. It will be a rotation that takes 651:seems to be exactly the same matrix we give at 3772:- the same user who added the jargon tag - in 949:his papers was either unoriginal or wrong. -- 578:Rotationmatrix = transpose(conjugate(Q)) . Q 8: 4529:an arithmetic operation. The proposition in 2064:Well off to thermo, because if I don't the 361:{\displaystyle {\mathcal {Q}}(z)=z-\Re (z)} 1506:Notice that the order here is important. ( 4464:which expresses quaternion conjugation." 4396: 4376: 3740:concerns, then we can address them here. 2663: 2653: 2636: 2626: 2610: 2604: 2362: 2342: 2330:{\displaystyle {\frac {\alpha }{\beta }}} 2317: 2315: 2276: 2274: 2143: 2130: 2117: 2111: 1550: 1532: 1054: 1042: 1040: 398:{\displaystyle \mathrm {i} \cdot \Im (z)} 375: 373: 322: 321: 319: 299: 279: 249: 248: 225: 199: 170: 164: 152: 140: 123: 100: 2548:Knowledge:MOSMATH#Common_sets_of_numbers 2173:{\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1} 894:Research directions seems notable topic! 748:Question/Mistake in the Definition : --> 638:be faithfully represented with matrices. 3166:, then another set of basis vectors is 2733:situation is the same as arises in the 2647: 2642: 809:Is anybody offended buy the T operator? 531:have the same formulation as complex. 3691:of unexplained jargon in the article. 3145:For instance, if I have basis vectors 2747:u(a,b)=u(a,-b) and v(a,b) = -v(a,-b). 1562:{\displaystyle q\times r\times q^{-1}} 1154:, because its in the same focus topic. 44:Do not edit the contents of this page. 4206:I'm not sure what you're getting at. 3768:were added to the first paragraph by 1872:, which isn't good enough, and that ( 1000:Remember Bertrand Russel called it a 624:Errm, no. Quaternion multiplication 7: 2925:The Electrodynamics of Moving Bodies 2337:is defined as an operator acting on 836:Also Cayley seems to think that the 91:For example, what is analogy of the 4361:Functions of a quaternion variable 4128:Functions of a quaternion variable 2735:real projective plane#Construction 2584:Quaternions as the even part of Cℓ 1118:-- how they fit into group theory. 1063:{\displaystyle \mathbf {S} .q^{2}} 1015:research, on the last few decades. 383: 376: 346: 230: 200: 153: 85:Functions of a quaternion variable 24: 3944:Less calculations are performed 3098:The bit analogy is incorrect. If 2377:Here the turning, is of course a 2010:to matrix operations. All these 1152:Classical Hamiltonian quaternions 1107:Classical Hamiltonian quaternions 653:Quaternion#Matrix representations 3236:. One dot product takes vectors 1695:classical Hamiltonian quaternion 1131:Quaternions and spatial rotation 1043: 559:4x4 Matrix representation false? 452:now quite a recognised field of 29: 4151:Quaternion multiplication table 4084:Influence upon vector calculus? 2891:is a metric on the quaternions. 2887:, so any metric you can put on 1671:now, got school in the morning! 628:distributive (what it isn't is 89:That section looks incomplete. 4448: 4406: 4387: 4381: 4180:23:14, 26 September 2010 (UTC) 4165:19:47, 26 September 2010 (UTC) 4146:12:49, 22 September 2010 (UTC) 4122:13:10, 22 September 2010 (UTC) 456:, extending Complex analysis. 392: 386: 355: 349: 334: 328: 261: 255: 239: 233: 182: 176: 134: 128: 1: 4104:05:50, 7 September 2010 (UTC) 3924:04:59, 22 December 2009 (UTC) 3882:01:25, 22 December 2009 (UTC) 3866:01:01, 22 December 2009 (UTC) 3820:12:59, 22 December 2009 (UTC) 3790:09:47, 22 December 2009 (UTC) 3750:05:13, 22 December 2009 (UTC) 3723:19:07, 21 December 2009 (UTC) 3701:14:18, 21 December 2009 (UTC) 3682:13:50, 21 December 2009 (UTC) 3509:with respect to the basis of 2854:Quaternions as metric spaces. 1185:Quaternionic projective space 740:18:13, 19 February 2009 (UTC) 688:18:18, 19 February 2009 (UTC) 665:17:46, 19 February 2009 (UTC) 616:17:17, 19 February 2009 (UTC) 594:17:51, 12 February 2009 (UTC) 552:08:50, 12 February 2009 (UTC) 4525:Yes, complex conjugation is 3841:21:09, 16 January 2010 (UTC) 3661:16:46, 16 October 2009 (UTC) 3642:Actually, it should be "the 3599:17:57, 16 October 2009 (UTC) 2847:) 18:44, 26 July 2009 (UTC) 575:This way I get the correct 499:18:35, 20 January 2009 (UTC) 484:02:17, 19 January 2009 (UTC) 469:23:43, 13 January 2009 (UTC) 447:19:35, 12 January 2009 (UTC) 423:04:36, 12 January 2009 (UTC) 4279:My question is as follows: 4265:23:45, 4 January 2011 (UTC) 4201:17:41, 4 January 2011 (UTC) 3537:00:32, 15 August 2009 (UTC) 3086:05:13, 14 August 2009 (UTC) 1955:; the question is, what's ( 1880:) is a rotation around the 1788:; the question is, what's ( 858:Can we please put the term 4562: 4518:21:56, 31 March 2011 (UTC) 4508:Cauchy–Riemann equations. 4499:10:30, 31 March 2011 (UTC) 4483:15:45, 29 March 2011 (UTC) 4042:"a two-dimensional sphere" 4001:20:38, 28 March 2010 (UTC) 3985:18:25, 28 March 2010 (UTC) 3965:17:45, 28 March 2010 (UTC) 3890:It's a consequence of the 3578:10:50, 2 August 2009 (UTC) 3562:09:53, 2 August 2009 (UTC) 3059:00:46, 8 August 2009 (UTC) 3035:22:21, 7 August 2009 (UTC) 3012:23:41, 7 August 2009 (UTC) 2560:WT:MOSMATH#Blackboard_bold 2394:22:24, 11 April 2009 (UTC) 2250:22:10, 11 April 2009 (UTC) 1408:? The rotation that takes 1400:are unit vectors, what is 1179:Quaternion-Kähler manifold 1147:The vector of a quaternion 888:13:33, 13 March 2009 (UTC) 873:04:25, 12 March 2009 (UTC) 801:00:31, 11 March 2009 (UTC) 782:22:01, 10 March 2009 (UTC) 766:19:54, 10 March 2009 (UTC) 4547:00:59, 1 April 2011 (UTC) 4355:11:36, 2 March 2011 (UTC) 4338:07:15, 2 March 2011 (UTC) 4321:15:45, 1 March 2011 (UTC) 4303:15:27, 1 March 2011 (UTC) 4170:An excellent idea. Done! 4078:08:17, 22 June 2010 (UTC) 4062:21:41, 21 June 2010 (UTC) 4036:08:17, 22 June 2010 (UTC) 3513:s. Is it zero, i.e., are 2907:17:27, 30 July 2009 (UTC) 2877:10:20, 30 July 2009 (UTC) 2831:22:00, 16 July 2009 (UTC) 2815:11:26, 16 July 2009 (UTC) 2795:05:39, 19 June 2009 (UTC) 2781:05:37, 19 June 2009 (UTC) 2761:21:58, 18 June 2009 (UTC) 2437:for the complex numbers, 2200:21:21, 6 April 2009 (UTC) 2089:18:50, 6 April 2009 (UTC) 1927:15:30, 6 April 2009 (UTC) 1754:17:50, 6 April 2009 (UTC) 1707:12:39, 6 April 2009 (UTC) 1681:04:51, 6 April 2009 (UTC) 1660:04:51, 6 April 2009 (UTC) 1478:By jove you have got it, 1470:23:23, 5 April 2009 (UTC) 1387:20:58, 5 April 2009 (UTC) 1360:23:35, 5 April 2009 (UTC) 1343:22:48, 5 April 2009 (UTC) 1319:12:46, 6 April 2009 (UTC) 1301:20:52, 5 April 2009 (UTC) 1232:14:07, 5 April 2009 (UTC) 1212:13:58, 5 April 2009 (UTC) 1090:02:52, 5 April 2009 (UTC) 959:22:48, 5 April 2009 (UTC) 944:18:17, 4 April 2009 (UTC) 926:15:59, 4 April 2009 (UTC) 93:Cauchy's integral formula 3941:Gimble lock is prevented 2921:Einstein synchronisation 2718:15:10, 8 June 2009 (UTC) 2703:12:03, 8 June 2009 (UTC) 2572:14:45, 2 June 2009 (UTC) 2512:00:35, 2 June 2009 (UTC) 2490:23:57, 1 June 2009 (UTC) 2472:21:30, 1 June 2009 (UTC) 2451:21:13, 1 June 2009 (UTC) 2420:15:14, 12 May 2009 (UTC) 2066:levers of academic power 2057:levers of academic power 1309:attention. Thanks, Jens 1272:bias) was unacceptable. 989:H+ is really interesting 765:_Non-commutativity": --> 274:with real coefficients 4021:22:48, 4 May 2010 (UTC) 2425:Blackboard bold vs bold 2370:{\displaystyle \alpha } 2097:quotient of two vectors 1888:axis, which isn't true. 287:{\displaystyle \alpha } 4455: 3898:solutions to a degree 2673: 2371: 2351: 2350:{\displaystyle \beta } 2331: 2302: 2267:. In the expression: 2174: 1856:. They just say that ( 1563: 1219:History of quaternions 1159:History of quaternions 1064: 1006:bi-quaternion disaster 428:You would need to use 399: 362: 308: 307:{\displaystyle \beta } 288: 268: 211: 109: 4456: 3584:Intro needs some work 2674: 2372: 2352: 2332: 2303: 2175: 1994:other people as well. 1971:is not coplanar with 1804:is not coplanar with 1564: 1065: 831:Cayley (1890), pg 146 400: 363: 309: 289: 269: 212: 110: 42:of past discussions. 4375: 3689:one specific example 3453:, and then computes 3326:, and then computes 3128:canonical: You must 2766:Images would be nice 2603: 2361: 2341: 2314: 2273: 2110: 1531: 1039: 1002:Fitzgerald transform 511:Commutative subrings 372: 318: 298: 278: 224: 122: 99: 4531:quaternion variable 4210:is not the same as 3102:= 11, then what is 2180:law by article 29. 1697:page. Thanks, Jens 1261:burnt to the ground 634:), and quaternions 115:of integration in 4451: 3892:division algorithm 2724:Conjugate symmetry 2669: 2648: 2643: 2367: 2347: 2327: 2298: 2170: 1559: 1289:rank three tensors 1171:Quaternion Society 1139:Quaternion algebra 1123:Hurwitz quaternion 1060: 647:The anon's matrix 395: 358: 304: 284: 264: 207: 105: 4473:comment added by 4404: 4314: 4115: 4094:comment added by 4052:comment added by 3978: 3968: 3951:comment added by 3846:Noncommutativity? 3638: 3636:) 16 October 2009 3624:comment added by 2867:comment added by 2800:Formulas too long 2325: 2284: 2092: 2075:comment added by 1757: 1740:comment added by 1188:, and a few more. 1027:Lorentz invariant 929: 912:comment added by 750:Non-commutativity 555: 538:comment added by 454:Clifford analysis 197: 158: 108:{\displaystyle c} 82: 81: 54: 53: 48:current talk page 4553: 4485: 4460: 4458: 4457: 4452: 4405: 4397: 4310: 4248:is not equal to 4247: 4226: 4111: 4106: 4064: 3974: 3967: 3945: 3637: 3618: 2879: 2678: 2676: 2675: 2670: 2668: 2667: 2658: 2657: 2641: 2640: 2631: 2630: 2618: 2617: 2459:Nicolas_Bourbaki 2376: 2374: 2373: 2368: 2356: 2354: 2353: 2348: 2336: 2334: 2333: 2328: 2326: 2318: 2307: 2305: 2304: 2299: 2285: 2277: 2265:conical rotation 2183:Please see link: 2179: 2177: 2176: 2171: 2148: 2147: 2135: 2134: 2122: 2121: 2091: 2069: 1756: 1734: 1568: 1566: 1565: 1560: 1558: 1557: 1115:Quaternion group 1069: 1067: 1066: 1061: 1059: 1058: 1046: 928: 906: 902:to the subject. 554: 540:Marc B. Reynolds 532: 434:Clifford algebra 404: 402: 401: 396: 379: 367: 365: 364: 359: 327: 326: 313: 311: 310: 305: 293: 291: 290: 285: 273: 271: 270: 265: 254: 253: 216: 214: 213: 208: 203: 198: 196: 185: 171: 169: 168: 159: 157: 156: 141: 114: 112: 111: 106: 73: 56: 55: 33: 32: 26: 4561: 4560: 4556: 4555: 4554: 4552: 4551: 4550: 4468: 4373: 4372: 4363: 4319: 4273: 4228: 4218: 4217:; for example, 4188: 4153: 4130: 4120: 4089: 4086: 4047: 4044: 4009: 3993:—Ben FrantzDale 3983: 3946: 3934: 3848: 3669: 3619: 3609: 3586: 3549: 3526: 3519: 3508: 3501: 3494: 3487: 3480: 3473: 3466: 3459: 3452: 3445: 3438: 3431: 3424: 3417: 3410: 3403: 3396: 3389: 3382: 3375: 3364: 3358: 3351: 3345: 3338: 3332: 3325: 3319: 3312: 3306: 3299: 3293: 3286: 3280: 3273: 3267: 3260: 3254: 3235: 3228: 3221: 3214: 3207: 3200: 3193: 3186: 3179: 3172: 3165: 3158: 3151: 2992:Proper velocity 2862: 2856: 2802: 2768: 2726: 2689: 2685: 2659: 2649: 2632: 2622: 2606: 2601: 2600: 2590: 2587: 2556:blackboard bold 2427: 2401: 2359: 2358: 2339: 2338: 2312: 2311: 2310:The quaternion 2271: 2270: 2257: 2207: 2139: 2126: 2113: 2108: 2107: 2070: 1901:You said that ( 1836:) won't equal ( 1735: 1650:. Great stuff! 1628:radial quotient 1546: 1529: 1528: 1368: 1270:Historiographys 1266:Historiographys 1050: 1037: 1036: 907: 896: 811: 752: 716: 570: 561: 533: 513: 430:vector calculus 370: 369: 316: 315: 296: 295: 276: 275: 222: 221: 186: 172: 160: 145: 120: 119: 97: 96: 87: 69: 30: 22: 21: 20: 18:Talk:Quaternion 12: 11: 5: 4559: 4557: 4523: 4522: 4521: 4520: 4502: 4501: 4462: 4461: 4450: 4447: 4444: 4441: 4438: 4435: 4432: 4429: 4426: 4423: 4420: 4417: 4414: 4411: 4408: 4403: 4400: 4395: 4392: 4389: 4386: 4383: 4380: 4362: 4359: 4358: 4357: 4341: 4340: 4324: 4323: 4315: 4312:JohnBlackburne 4272: 4269: 4268: 4267: 4187: 4184: 4183: 4182: 4152: 4149: 4129: 4126: 4125: 4124: 4116: 4113:JohnBlackburne 4085: 4082: 4081: 4080: 4043: 4040: 4039: 4038: 4008: 4005: 4004: 4003: 3988: 3987: 3979: 3976:JohnBlackburne 3970: 3969: 3942: 3933: 3930: 3929: 3928: 3927: 3926: 3885: 3884: 3874:JohnBlackburne 3847: 3844: 3829: 3828: 3827: 3826: 3825: 3824: 3823: 3822: 3797: 3796: 3795: 3794: 3793: 3792: 3757: 3756: 3755: 3754: 3753: 3752: 3729: 3728: 3727: 3726: 3668: 3665: 3664: 3663: 3626:134.187.128.99 3608: 3605: 3603: 3585: 3582: 3581: 3580: 3548: 3545: 3544: 3543: 3542: 3541: 3540: 3539: 3524: 3517: 3506: 3499: 3492: 3485: 3478: 3471: 3464: 3457: 3450: 3443: 3436: 3429: 3422: 3415: 3408: 3401: 3394: 3387: 3380: 3373: 3362: 3356: 3349: 3343: 3336: 3330: 3323: 3317: 3310: 3304: 3297: 3291: 3284: 3278: 3271: 3265: 3258: 3252: 3233: 3226: 3219: 3212: 3205: 3198: 3191: 3184: 3177: 3170: 3163: 3156: 3149: 3138: 3137: 3136: 3135: 3134: 3133: 3112: 3111: 3110: 3109: 3108: 3107: 3091: 3090: 3089: 3088: 3073: 3066: 3065: 3064: 3063: 3062: 3061: 3042: 3041: 3040: 3039: 3038: 3037: 3017: 3016: 3015: 3014: 3004:TeamQuaternion 2998: 2997: 2996: 2995: 2980: 2979: 2978: 2977: 2976:would go away. 2970: 2969: 2968: 2967: 2960: 2959: 2958: 2957: 2950: 2949: 2948: 2947: 2936: 2935: 2934: 2933: 2910: 2909: 2893: 2892: 2869:76.191.171.210 2855: 2852: 2851: 2850: 2849: 2848: 2834: 2833: 2801: 2798: 2767: 2764: 2749: 2748: 2745: 2725: 2722: 2721: 2720: 2695:Enisbayramoglu 2691: 2687: 2683: 2680: 2679: 2666: 2662: 2656: 2652: 2646: 2639: 2635: 2629: 2625: 2621: 2616: 2613: 2609: 2594: 2589: 2585: 2582: 2581: 2580: 2579: 2578: 2577: 2576: 2575: 2574: 2537: 2536: 2535: 2534: 2533: 2532: 2531: 2530: 2519: 2518: 2517: 2516: 2515: 2514: 2495: 2494: 2493: 2492: 2475: 2474: 2426: 2423: 2400: 2397: 2379:plane rotation 2366: 2346: 2324: 2321: 2297: 2294: 2291: 2288: 2283: 2280: 2261:plane rotation 2256: 2253: 2206: 2203: 2169: 2166: 2163: 2160: 2157: 2154: 2151: 2146: 2142: 2138: 2133: 2129: 2125: 2120: 2116: 2094: 2093: 2061: 2060: 2047: 2046: 2032: 2031: 2016: 2015: 2002: 2001: 1996: 1995: 1986: 1985: 1934: 1933: 1932: 1931: 1930: 1929: 1894: 1893: 1892: 1891: 1890: 1889: 1765: 1764: 1763: 1762: 1761: 1760: 1759: 1758: 1714: 1713: 1712: 1711: 1710: 1709: 1686: 1685: 1684: 1683: 1665: 1664: 1663: 1662: 1640: 1639: 1638: 1637: 1621: 1620: 1619: 1618: 1611: 1610: 1609: 1608: 1578: 1577: 1576: 1575: 1556: 1553: 1549: 1545: 1542: 1539: 1536: 1518: 1517: 1516: 1515: 1501: 1500: 1499: 1498: 1473: 1472: 1367: 1364: 1363: 1362: 1346: 1345: 1323: 1322: 1321: 1257: 1256: 1249: 1248: 1241:Don't forget: 1239: 1238: 1237: 1236: 1235: 1234: 1191: 1190: 1189: 1175: 1167: 1155: 1143: 1135: 1127: 1119: 1111: 1103: 1073: 1072: 1071: 1070: 1057: 1053: 1049: 1045: 1031: 1030: 1023: 1022: 1017: 1016: 1011: 1010: 997: 996: 985: 984: 980: 979: 972:velocity space 964: 963: 962: 961: 895: 892: 891: 890: 810: 807: 806: 805: 804: 803: 785: 784: 764:_Remarks_: --> 751: 746: 745: 744: 743: 742: 725: 724: 723: 722: 714: 713: 712: 711: 710: 704: 703: 702: 701: 695: 694: 693: 692: 691: 690: 670: 669: 668: 667: 642: 641: 640: 639: 619: 618: 602: 601: 568: 560: 557: 512: 509: 508: 507: 506: 505: 504: 503: 502: 501: 412: 406: 394: 391: 388: 385: 382: 378: 357: 354: 351: 348: 345: 342: 339: 336: 333: 330: 325: 303: 283: 263: 260: 257: 252: 247: 244: 241: 238: 235: 232: 229: 218: 217: 206: 202: 195: 192: 189: 184: 181: 178: 175: 167: 163: 155: 151: 148: 144: 139: 136: 133: 130: 127: 104: 90: 86: 83: 80: 79: 74: 67: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 4558: 4549: 4548: 4544: 4540: 4536: 4532: 4528: 4519: 4515: 4511: 4506: 4505: 4504: 4503: 4500: 4496: 4492: 4488: 4487: 4486: 4484: 4480: 4476: 4475:142.103.4.114 4472: 4465: 4445: 4442: 4439: 4436: 4433: 4430: 4427: 4424: 4421: 4418: 4415: 4412: 4409: 4401: 4398: 4393: 4390: 4384: 4378: 4371: 4370: 4369: 4366: 4360: 4356: 4352: 4348: 4343: 4342: 4339: 4335: 4331: 4326: 4325: 4322: 4318: 4313: 4307: 4306: 4305: 4304: 4300: 4296: 4291: 4287: 4284: 4280: 4277: 4270: 4266: 4262: 4258: 4254: 4251: 4246: 4243: 4239: 4235: 4231: 4225: 4221: 4216: 4213: 4209: 4205: 4204: 4203: 4202: 4198: 4194: 4185: 4181: 4177: 4173: 4169: 4168: 4167: 4166: 4162: 4158: 4150: 4148: 4147: 4143: 4139: 4135: 4127: 4123: 4119: 4114: 4109: 4108: 4107: 4105: 4101: 4097: 4093: 4083: 4079: 4075: 4071: 4067: 4066: 4065: 4063: 4059: 4055: 4051: 4041: 4037: 4033: 4029: 4025: 4024: 4023: 4022: 4018: 4014: 4006: 4002: 3998: 3994: 3990: 3989: 3986: 3982: 3977: 3972: 3971: 3966: 3962: 3958: 3954: 3950: 3943: 3940: 3939: 3938: 3931: 3925: 3921: 3917: 3913: 3909: 3905: 3901: 3897: 3893: 3889: 3888: 3887: 3886: 3883: 3879: 3875: 3870: 3869: 3868: 3867: 3863: 3859: 3855: 3851: 3845: 3843: 3842: 3838: 3834: 3821: 3817: 3813: 3809: 3805: 3804: 3803: 3802: 3801: 3800: 3799: 3798: 3791: 3787: 3783: 3779: 3775: 3771: 3767: 3763: 3762: 3761: 3760: 3759: 3758: 3751: 3747: 3743: 3739: 3735: 3734: 3733: 3732: 3731: 3730: 3725: 3724: 3720: 3716: 3712: 3704: 3703: 3702: 3698: 3694: 3690: 3686: 3685: 3684: 3683: 3679: 3675: 3666: 3662: 3658: 3654: 3649: 3645: 3641: 3640: 3639: 3635: 3631: 3627: 3623: 3616: 3613: 3606: 3604: 3601: 3600: 3596: 3592: 3583: 3579: 3575: 3571: 3566: 3565: 3564: 3563: 3559: 3555: 3546: 3538: 3534: 3530: 3523: 3516: 3512: 3505: 3498: 3491: 3484: 3477: 3470: 3463: 3456: 3449: 3442: 3435: 3428: 3421: 3414: 3407: 3400: 3393: 3386: 3379: 3372: 3368: 3361: 3355: 3348: 3342: 3335: 3329: 3322: 3316: 3309: 3303: 3296: 3290: 3283: 3277: 3270: 3264: 3257: 3251: 3247: 3243: 3239: 3232: 3225: 3218: 3211: 3204: 3197: 3190: 3183: 3176: 3169: 3162: 3155: 3148: 3144: 3143: 3142: 3141: 3140: 3139: 3131: 3127: 3123: 3118: 3117: 3116: 3115: 3114: 3113: 3105: 3101: 3097: 3096: 3095: 3094: 3093: 3092: 3087: 3083: 3079: 3074: 3070: 3069: 3068: 3067: 3060: 3056: 3052: 3048: 3047: 3046: 3045: 3044: 3043: 3036: 3032: 3028: 3023: 3022: 3021: 3020: 3019: 3018: 3013: 3009: 3005: 3002: 3001: 3000: 2999: 2993: 2989: 2984: 2983: 2982: 2981: 2974: 2973: 2972: 2971: 2964: 2963: 2962: 2961: 2954: 2953: 2952: 2951: 2944: 2940: 2939: 2938: 2937: 2930: 2926: 2922: 2918: 2914: 2913: 2912: 2911: 2908: 2904: 2900: 2895: 2894: 2890: 2886: 2882: 2881: 2880: 2878: 2874: 2870: 2866: 2859: 2853: 2846: 2842: 2838: 2837: 2836: 2835: 2832: 2828: 2824: 2819: 2818: 2817: 2816: 2812: 2808: 2799: 2797: 2796: 2792: 2788: 2783: 2782: 2778: 2774: 2765: 2763: 2762: 2758: 2754: 2746: 2743: 2742: 2741: 2738: 2736: 2730: 2723: 2719: 2715: 2711: 2707: 2706: 2705: 2704: 2700: 2696: 2690: 2664: 2660: 2654: 2650: 2644: 2637: 2633: 2627: 2623: 2619: 2607: 2599: 2598: 2597: 2593: 2583: 2573: 2569: 2565: 2561: 2557: 2553: 2549: 2545: 2544: 2543: 2542: 2541: 2540: 2539: 2538: 2527: 2526: 2525: 2524: 2523: 2522: 2521: 2520: 2513: 2509: 2505: 2501: 2500: 2499: 2498: 2497: 2496: 2491: 2487: 2483: 2479: 2478: 2477: 2476: 2473: 2469: 2465: 2460: 2455: 2454: 2453: 2452: 2448: 2444: 2440: 2436: 2432: 2424: 2422: 2421: 2417: 2413: 2409: 2408: 2404: 2398: 2396: 2395: 2391: 2387: 2382: 2380: 2364: 2344: 2322: 2319: 2308: 2295: 2292: 2289: 2286: 2281: 2278: 2268: 2266: 2262: 2254: 2252: 2251: 2247: 2243: 2237: 2234: 2229: 2226: 2223: 2221: 2215: 2211: 2204: 2202: 2201: 2197: 2193: 2187: 2186: 2181: 2167: 2164: 2161: 2158: 2155: 2152: 2149: 2144: 2140: 2136: 2131: 2127: 2123: 2118: 2114: 2104: 2099: 2098: 2090: 2086: 2082: 2078: 2074: 2067: 2063: 2062: 2058: 2054: 2049: 2048: 2043: 2039: 2034: 2033: 2028: 2023: 2018: 2017: 2013: 2009: 2004: 2003: 1998: 1997: 1992: 1988: 1987: 1982: 1981: 1980: 1978: 1974: 1970: 1966: 1962: 1958: 1954: 1950: 1946: 1942: 1938: 1937:Good question 1928: 1924: 1920: 1916: 1912: 1908: 1904: 1900: 1899: 1898: 1897: 1896: 1895: 1887: 1883: 1879: 1875: 1871: 1867: 1863: 1859: 1855: 1851: 1847: 1843: 1839: 1835: 1831: 1827: 1823: 1819: 1815: 1811: 1807: 1803: 1799: 1795: 1791: 1787: 1783: 1779: 1775: 1771: 1770: 1769: 1768: 1767: 1766: 1755: 1751: 1747: 1743: 1739: 1731: 1726: 1722: 1721: 1720: 1719: 1718: 1717: 1716: 1715: 1708: 1704: 1700: 1696: 1692: 1691: 1690: 1689: 1688: 1687: 1682: 1678: 1674: 1669: 1668: 1667: 1666: 1661: 1657: 1653: 1649: 1644: 1643: 1642: 1641: 1634: 1629: 1625: 1624: 1623: 1622: 1615: 1614: 1613: 1612: 1606: 1602: 1598: 1594: 1590: 1586: 1582: 1581: 1580: 1579: 1572: 1554: 1551: 1547: 1543: 1540: 1537: 1534: 1526: 1522: 1521: 1520: 1519: 1513: 1509: 1505: 1504: 1503: 1502: 1497: 1493: 1489: 1485: 1481: 1477: 1476: 1475: 1474: 1471: 1467: 1463: 1459: 1455: 1451: 1447: 1443: 1439: 1435: 1431: 1427: 1423: 1419: 1415: 1411: 1407: 1403: 1399: 1395: 1391: 1390: 1389: 1388: 1384: 1380: 1374: 1371: 1365: 1361: 1357: 1353: 1348: 1347: 1344: 1340: 1336: 1332: 1331:biquaternions 1328: 1324: 1320: 1316: 1312: 1307: 1306: 1305: 1304: 1303: 1302: 1298: 1294: 1290: 1285: 1281: 1278: 1273: 1271: 1267: 1262: 1254: 1251: 1250: 1247: 1244: 1243: 1242: 1233: 1229: 1225: 1221: 1220: 1215: 1214: 1213: 1209: 1205: 1200: 1196: 1192: 1187: 1186: 1181: 1180: 1176: 1173: 1172: 1168: 1165: 1161: 1160: 1156: 1153: 1149: 1148: 1144: 1141: 1140: 1136: 1133: 1132: 1128: 1125: 1124: 1120: 1117: 1116: 1112: 1109: 1108: 1104: 1101: 1100: 1096: 1095: 1093: 1092: 1091: 1087: 1083: 1078: 1075: 1074: 1055: 1051: 1047: 1035: 1034: 1033: 1032: 1028: 1025: 1024: 1019: 1018: 1013: 1012: 1007: 1003: 999: 998: 994: 990: 987: 986: 982: 981: 977: 973: 969: 966: 965: 960: 956: 952: 947: 946: 945: 941: 937: 932: 931: 930: 927: 923: 919: 915: 911: 903: 899: 893: 889: 885: 881: 877: 876: 875: 874: 870: 866: 861: 856: 854: 850: 847:operator was 846: 841: 839: 834: 832: 827: 823: 820: 815: 808: 802: 798: 794: 789: 788: 787: 786: 783: 779: 775: 770: 769: 768: 767: 761: 757: 749:Remarks : --> 747: 741: 737: 733: 729: 728: 727: 726: 720: 719: 718: 717: 708: 707: 706: 705: 699: 698: 697: 696: 689: 685: 681: 676: 675: 674: 673: 672: 671: 666: 662: 658: 654: 650: 646: 645: 644: 643: 637: 633: 632: 627: 623: 622: 621: 620: 617: 613: 609: 604: 603: 598: 597: 596: 595: 591: 587: 586:141.76.62.164 582: 579: 576: 573: 567: 564: 558: 556: 553: 549: 545: 541: 537: 530: 526: 522: 518: 510: 500: 496: 492: 487: 486: 485: 481: 477: 472: 471: 470: 466: 462: 458: 455: 450: 449: 448: 444: 440: 435: 431: 427: 426: 425: 424: 420: 416: 410: 409:Taylor series 389: 380: 352: 343: 340: 337: 331: 301: 281: 258: 245: 242: 236: 227: 204: 193: 190: 187: 179: 173: 165: 161: 149: 146: 142: 137: 131: 125: 118: 117: 116: 102: 94: 84: 78: 75: 72: 68: 66: 63: 61: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 4534: 4526: 4524: 4466: 4463: 4367: 4364: 4292: 4288: 4285: 4281: 4278: 4274: 4252: 4249: 4244: 4241: 4237: 4233: 4229: 4223: 4219: 4214: 4211: 4207: 4193:140.232.0.70 4189: 4154: 4133: 4131: 4096:65.50.39.118 4087: 4054:82.69.22.206 4045: 4010: 3935: 3911: 3907: 3903: 3899: 3895: 3856: 3852: 3849: 3830: 3737: 3707: 3688: 3670: 3617: 3614: 3610: 3602: 3591:78.86.229.20 3587: 3554:81.83.41.159 3550: 3527:orthogonal? 3521: 3514: 3510: 3503: 3496: 3489: 3482: 3475: 3468: 3461: 3454: 3447: 3440: 3433: 3426: 3419: 3412: 3405: 3398: 3391: 3384: 3377: 3370: 3366: 3359: 3353: 3346: 3340: 3333: 3327: 3320: 3314: 3307: 3301: 3294: 3288: 3281: 3275: 3268: 3262: 3255: 3249: 3245: 3241: 3237: 3230: 3223: 3216: 3209: 3202: 3195: 3188: 3181: 3174: 3167: 3160: 3153: 3146: 3129: 3125: 3121: 3103: 3099: 2888: 2884: 2860: 2857: 2841:91.86.51.227 2803: 2784: 2769: 2750: 2739: 2731: 2727: 2692: 2681: 2595: 2591: 2551: 2438: 2434: 2428: 2410: 2405: 2402: 2383: 2378: 2309: 2269: 2258: 2238: 2232: 2230: 2227: 2224: 2219: 2216: 2212: 2208: 2192:Robotics lab 2188: 2182: 2102: 2100: 2096: 2095: 2065: 2056: 2052: 2041: 2037: 2026: 2021: 2011: 2007: 1990: 1976: 1972: 1968: 1964: 1960: 1956: 1952: 1948: 1944: 1940: 1936: 1935: 1914: 1910: 1906: 1902: 1885: 1881: 1877: 1873: 1869: 1865: 1861: 1857: 1853: 1849: 1845: 1841: 1837: 1833: 1829: 1825: 1821: 1817: 1813: 1809: 1805: 1801: 1797: 1793: 1789: 1785: 1781: 1777: 1773: 1724: 1647: 1627: 1604: 1600: 1596: 1592: 1588: 1584: 1524: 1511: 1507: 1495: 1491: 1487: 1483: 1457: 1453: 1449: 1445: 1441: 1437: 1433: 1429: 1425: 1421: 1417: 1413: 1409: 1405: 1401: 1397: 1393: 1375: 1372: 1369: 1327:compactified 1288: 1286: 1282: 1276: 1274: 1260: 1258: 1253:biquaternion 1240: 1217: 1198: 1194: 1183: 1177: 1169: 1157: 1145: 1137: 1129: 1121: 1113: 1105: 1097: 1076: 1026: 1005: 1001: 992: 988: 975: 971: 967: 904: 900: 897: 859: 857: 848: 844: 842: 837: 835: 828: 824: 818: 816: 812: 753: 648: 635: 629: 625: 583: 580: 577: 574: 571: 565: 562: 528: 524: 520: 519:is real and 516: 514: 219: 88: 70: 43: 37: 4533:shows that 4469:—Preceding 4090:—Preceding 4048:—Preceding 3947:—Preceding 3620:—Preceding 2943:light cones 2929:synchronous 2917:proper time 2863:—Preceding 2773:210.5.33.33 2071:—Preceding 1939:Of course ( 1772:Of course ( 1736:—Preceding 1277:quaternions 1080:apparently. 908:—Preceding 880:Taits Wrath 853:holomorphic 631:commutative 534:—Preceding 36:This is an 3833:Spiral5800 3812:Spiral5800 3776:. I think 3770:Spiral5800 3766:Simetrical 3711:Simetrical 3706:paragraph. 3674:Spiral5800 3369:s, say as 3248:s, say as 2482:Koeplinger 2443:Koeplinger 2412:Matagamasi 2012:isomorphic 2008:isomorphic 1991:reasonable 1699:Koeplinger 1648:in correct 1569:Of course 1482:! .......( 1311:Koeplinger 1224:Koeplinger 1204:Koeplinger 1195:other than 1164:Quaternion 1099:Quaternion 936:Koeplinger 863:consensus. 680:Koeplinger 566:Q:matrix( 491:Koeplinger 439:Koeplinger 4330:Dedekmraz 4295:Dedekmraz 4157:Fizzbowen 4070:Gandalf61 4028:Gandalf61 3782:Gandalf61 3774:this edit 3693:Gandalf61 3570:Gandalf61 2546:In fact, 793:GluonBall 715:, , , 709:Q:matrix( 569:, , , 77:Archive 4 71:Archive 3 65:Archive 2 60:Archive 1 4471:unsigned 4092:unsigned 4050:unsigned 3961:contribs 3953:Joebeard 3949:unsigned 3858:Sukisuki 3738:specific 3719:contribs 3634:contribs 3622:unsigned 3078:Natecull 2988:rapidity 2865:unsigned 2807:StevenDH 2787:Danwills 2190:tedious. 2103:Elements 2085:contribs 2073:unsigned 2022:Elements 1750:contribs 1738:unsigned 1448:for all 1377:article? 922:contribs 910:unsigned 865:Hobojaks 608:Nazlfrag 548:contribs 536:unsigned 474:article. 314:? Here, 4539:Rgdboer 4293:Thanks 4138:Autarch 4013:Bah2222 3653:Zundark 3644:algebra 3547:Strange 3122:declare 2753:Rgdboer 2386:Homebum 2242:Homebum 2077:Homebum 1742:Homebum 1673:Homebum 1652:Homebum 1583:Given ( 1379:Homebum 1293:Homebum 1082:Homebum 976:vection 914:Homebum 849:defined 476:Rgdboer 39:archive 3667:Jargon 3130:choose 2710:Jheald 2431:Conway 2263:, and 2233:spaces 2053:matrix 1967:where 1800:where 1730:Cayley 1574:point. 1246:versor 1199:should 1077:So yes 993:future 732:Jheald 657:Jheald 461:Jheald 4317:deeds 4227:, so 4118:deeds 3981:deeds 2686:and σ 1919:BenRG 1725:wrong 1571:BenRG 1480:BenRG 1462:BenRG 1352:BenRG 1335:BenRG 951:BenRG 16:< 4543:talk 4514:talk 4510:Dmcq 4495:talk 4491:Ozob 4479:talk 4351:talk 4347:Dmcq 4334:talk 4299:talk 4283:0) 4261:talk 4257:Ozob 4197:talk 4176:talk 4172:Ozob 4161:talk 4142:talk 4100:talk 4074:talk 4058:talk 4032:talk 4017:talk 3997:talk 3957:talk 3920:talk 3916:Ozob 3878:talk 3862:talk 3837:talk 3816:talk 3808:Ozob 3786:talk 3778:Ozob 3746:talk 3742:Ozob 3715:talk 3697:talk 3678:talk 3657:talk 3648:ring 3630:talk 3595:talk 3574:talk 3558:talk 3533:talk 3529:Ozob 3520:and 3502:and 3481:′ + 3467:′ + 3411:and 3287:and 3240:and 3082:talk 3055:talk 3051:Ozob 3031:talk 3027:Ozob 3008:talk 2990:and 2903:talk 2899:Ozob 2873:talk 2845:talk 2827:talk 2823:Ozob 2811:talk 2791:talk 2777:talk 2757:talk 2714:talk 2699:talk 2568:talk 2564:Ozob 2508:talk 2486:talk 2468:talk 2447:talk 2416:talk 2390:talk 2246:talk 2196:talk 2081:talk 1975:and 1923:talk 1808:and 1746:talk 1703:talk 1677:talk 1656:talk 1591:) x 1490:) x 1466:talk 1440:) = 1396:and 1383:talk 1356:talk 1339:talk 1315:talk 1297:talk 1228:talk 1208:talk 1086:talk 955:talk 940:talk 918:talk 884:talk 869:talk 860:norm 838:norm 819:norm 797:talk 778:talk 774:Ozob 760:talk 756:Lotu 736:talk 684:talk 661:talk 612:talk 590:talk 544:talk 495:talk 480:talk 465:talk 443:talk 419:talk 415:dima 294:and 4527:not 4240:= − 4232:= − 4222:= − 3126:not 2588:(R) 2586:3,0 2504:C S 2464:C S 2045:it. 2042:duh 1848:to 1828:) ( 1525:not 1412:to 655:. 636:can 4545:) 4516:) 4497:) 4481:) 4394:− 4353:) 4336:) 4301:) 4263:) 4238:ij 4236:= 4234:ji 4230:ji 4208:pq 4199:) 4178:) 4163:) 4144:) 4102:) 4076:) 4060:) 4034:) 4019:) 3999:) 3963:) 3959:• 3922:) 3912:ba 3910:= 3908:ab 3880:) 3864:) 3839:) 3818:) 3788:) 3748:) 3721:) 3717:• 3699:) 3680:) 3659:) 3651:-- 3632:• 3597:) 3576:) 3560:) 3535:) 3439:+ 3425:+ 3397:+ 3383:+ 3352:+ 3339:+ 3313:+ 3300:+ 3274:+ 3261:+ 3229:+ 3222:+ 3215:= 3208:, 3201:- 3194:= 3187:, 3180:+ 3173:= 3159:, 3152:, 3104:ji 3100:ij 3084:) 3057:) 3033:) 3010:) 2905:) 2875:) 2829:) 2813:) 2793:) 2779:) 2759:) 2716:) 2701:) 2661:σ 2651:σ 2634:σ 2624:σ 2615:′ 2612:′ 2570:) 2510:) 2488:) 2470:) 2462:-- 2449:) 2418:) 2392:) 2365:α 2345:β 2323:β 2320:α 2296:α 2290:β 2287:× 2282:β 2279:α 2248:) 2198:) 2165:− 2087:) 2083:• 1979:? 1963:) 1959:/ 1951:= 1947:) 1943:/ 1925:) 1913:= 1909:) 1905:/ 1884:× 1876:/ 1868:= 1864:) 1860:/ 1854:is 1840:/ 1832:/ 1824:/ 1816:/ 1796:) 1792:/ 1784:= 1780:) 1776:/ 1752:) 1748:• 1705:) 1679:) 1658:) 1595:= 1552:− 1544:× 1538:× 1494:= 1468:) 1456:, 1452:, 1432:)( 1385:) 1358:) 1341:) 1317:) 1299:) 1230:) 1210:) 1182:, 1088:) 957:) 942:) 924:) 920:• 886:) 871:) 799:) 780:) 762:) 738:) 686:) 663:) 626:is 614:) 592:) 572:) 550:) 546:• 497:) 482:) 467:) 445:) 421:) 384:ℑ 381:⋅ 347:ℜ 344:− 302:β 282:α 246:β 231:ℜ 228:α 191:− 162:∮ 150:π 4541:( 4535:f 4512:( 4493:( 4477:( 4449:) 4446:k 4443:q 4440:k 4437:+ 4434:j 4431:q 4428:j 4425:+ 4422:i 4419:q 4416:i 4413:+ 4410:q 4407:( 4402:2 4399:1 4391:= 4388:) 4385:q 4382:( 4379:f 4349:( 4332:( 4297:( 4259:( 4253:j 4250:i 4245:j 4242:i 4224:i 4220:i 4215:p 4212:q 4195:( 4174:( 4159:( 4140:( 4098:( 4072:( 4056:( 4030:( 4015:( 3995:( 3955:( 3918:( 3904:z 3900:n 3896:n 3876:( 3860:( 3835:( 3814:( 3784:( 3744:( 3713:( 3695:( 3676:( 3655:( 3628:( 3593:( 3572:( 3556:( 3531:( 3525:3 3522:e 3518:1 3515:e 3511:f 3507:3 3504:e 3500:1 3497:e 3493:3 3490:w 3488:′ 3486:3 3483:v 3479:2 3476:w 3474:′ 3472:2 3469:v 3465:1 3462:w 3460:′ 3458:1 3455:v 3451:3 3448:f 3446:′ 3444:3 3441:w 3437:2 3434:f 3432:′ 3430:2 3427:w 3423:1 3420:f 3418:′ 3416:1 3413:w 3409:3 3406:f 3404:′ 3402:3 3399:v 3395:2 3392:f 3390:′ 3388:2 3385:v 3381:1 3378:f 3376:′ 3374:1 3371:v 3367:f 3363:3 3360:w 3357:3 3354:v 3350:2 3347:w 3344:2 3341:v 3337:1 3334:w 3331:1 3328:v 3324:3 3321:e 3318:3 3315:w 3311:2 3308:e 3305:2 3302:w 3298:1 3295:e 3292:1 3289:w 3285:3 3282:e 3279:3 3276:v 3272:2 3269:e 3266:2 3263:v 3259:1 3256:e 3253:1 3250:v 3246:e 3242:w 3238:v 3234:3 3231:e 3227:2 3224:e 3220:1 3217:e 3213:3 3210:f 3206:2 3203:e 3199:1 3196:e 3192:2 3189:f 3185:2 3182:e 3178:1 3175:e 3171:1 3168:f 3164:3 3161:e 3157:2 3154:e 3150:1 3147:e 3080:( 3053:( 3029:( 3006:( 2901:( 2889:R 2885:R 2871:( 2843:( 2825:( 2809:( 2789:( 2775:( 2755:( 2712:( 2697:( 2688:2 2684:1 2665:2 2655:1 2645:r 2638:1 2628:2 2620:= 2608:r 2566:( 2552:R 2506:( 2484:( 2466:( 2445:( 2439:H 2435:C 2414:( 2388:( 2293:= 2244:( 2194:( 2168:1 2162:= 2159:k 2156:j 2153:i 2150:= 2145:2 2141:k 2137:= 2132:2 2128:j 2124:= 2119:2 2115:i 2079:( 1977:w 1973:v 1969:x 1965:x 1961:w 1957:v 1953:v 1949:w 1945:w 1941:v 1921:( 1915:v 1911:w 1907:w 1903:v 1886:w 1882:v 1878:w 1874:v 1870:v 1866:w 1862:w 1858:v 1850:v 1846:x 1842:x 1838:v 1834:x 1830:w 1826:w 1822:v 1818:w 1814:v 1810:w 1806:v 1802:x 1798:x 1794:w 1790:v 1786:v 1782:w 1778:w 1774:v 1744:( 1701:( 1675:( 1654:( 1605:w 1603:/ 1601:v 1597:v 1593:w 1589:w 1587:/ 1585:v 1555:1 1548:q 1541:r 1535:q 1512:w 1510:/ 1508:v 1496:v 1492:w 1488:w 1486:/ 1484:v 1464:( 1458:x 1454:w 1450:v 1446:x 1444:/ 1442:v 1438:x 1436:/ 1434:w 1430:w 1428:/ 1426:v 1422:w 1420:× 1418:v 1414:v 1410:w 1406:w 1404:/ 1402:v 1398:w 1394:v 1381:( 1354:( 1337:( 1313:( 1295:( 1226:( 1206:( 1084:( 1056:2 1052:q 1048:. 1044:S 953:( 938:( 916:( 882:( 867:( 845:T 795:( 776:( 758:( 734:( 721:) 682:( 659:( 649:Q 610:( 588:( 542:( 529:q 525:q 521:q 517:r 493:( 478:( 463:( 441:( 417:( 405:. 393:) 390:z 387:( 377:i 356:) 353:z 350:( 341:z 338:= 335:) 332:z 329:( 324:Q 262:) 259:z 256:( 251:Q 243:+ 240:) 237:z 234:( 205:t 201:d 194:z 188:t 183:) 180:t 177:( 174:f 166:c 154:i 147:2 143:1 138:= 135:) 132:z 129:( 126:f 103:c 50:.

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index