Talk:Quaternion/Archive 4

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First image showing Quaternion unit products is confusing

I can't make any sense out of the first image at the top of the page. Quaternions can be mysterious if you don't understand them, and I would hope that any images in this article would be as insanely clear as possible. --Almo2001 (talk) 19:20, 11 May 2011 (UTC)

Difference between quaternions and vectors.

To me it is not quite clear what the difference is between quaternions and 3 dimensional vectors. It seems as if you could define any n dimensional number system, you could say a number of the form ai+bj+ck+dl+em could describe certain properties of a 5 dimensional space. Do quaternions really 'exist' or are they definitions of an alternative way to describe things you could describe with vectors? Complex numbers have a meaning in the sense that they can be square roots of negative numbers, you can solve certain equations with them that you couldn't solve with real numbers.

Do quaternions have comparable arithmetic, or are they just an alternative way of describing 3 dimensional vectors? Maybe it is good of this article addresses this and explains the differences and the exact meaning of quaternions. — Preceding unsigned comment added by 83.160.61.76 (talk) 13:19, 12 April 2012 (UTC)

Quaternions are four dimensional vectors (technically, they form a vector space of dimension 4 over the real numbers), but they are more than this because they also come equipped with operations of multiplication and division. This means they are not just vectors, they are a division algebra. Gandalf61 (talk) 13:29, 12 April 2012 (UTC)

There isn't really a "difference", nor are they the same. Vectors are elements of a vector space, and quaternions happen to be a vector space. They are also more than a vector space because they form a division ring. Most vector spaces don't carry an associative multiplication, as the quaternions do.

You could say the same for the real n-by-n matrix algebra. It is an n² dimensional vector space, each matix itself a vector. There isn't a "difference" between matrices and vectors, but the algebra does have a particular multiplication (matrix multiplication) specified. Rschwieb (talk) 16:37, 12 April 2012 (UTC)

You both seem to have missed to OP's point about representing 3D geometric vectors. Quaternions are a pretty natural algebra for 3D vectors, not 4D (even though pairs can represent a rotation in 4D). Though being a very "real" algebra in its own right, quaternions can be used for 3D vectors much like in "standard" vector calculus, except that rotations are more natural. — Quondum ^☏ 16:59, 12 April 2012 (UTC)

Wow, I can't wait to spring that back on you when you attempt to answer a vague post.

I have to admit, I cannot understand the original question. It looks like "what is the difference between apples and fruits?" Help the 2/3 of point-missing posters understand what interpretation makes them confusingly similar to vectors. Rschwieb (talk) 00:07, 13 April 2012 (UTC)

Okay, feel free, I guess I deserve it.

If you consider the section Quaternion#Quaternions and the geometry of R3, you'll see some motivation for what I say. Historically, Hamilton used quarternions for and and was motivated by manipulation of 3D geometric vectors; it was a competing (and out-competed) alternative to Gibbs's vector operations. This is behind the naming of the "vector part" of a quaternion. Reading between the lines, it seems to me that the OP intended this interpretation, and if I'm correct, the response highlighting only the 4D vector space quality is almost certain to confuse the asker. — Quondum ^☏ 12:45, 17 April 2012 (UTC)

I'm positive you interpreted it correctly because I trust your familiarity with these this topic. I suppose I have to blame myself for not finding time to read this article all the way through, yet. I'll direct future questions on the topic to your talkpage.

I did just see something that excites me though:

He concluded by stating "I firmly believe that quaternions can supply a shortcut for pure mathematicians who wish to familiarize themselves with certain aspects of theoretical physics."

^^ exactly what I'm looking for! Rschwieb (talk) 13:14, 17 April 2012 (UTC)

Using the vector part of a quaternion to hold a 3D vector doesn't give you anything much. Where they really shine is in the way they can handle a rotation about a point and that's why they are used in 3D game engines for instance. That's what the last sentence in the lead is referring to though the article, see Quaternions and spatial rotation about this. Dmcq (talk) 16:52, 17 April 2012 (UTC)

Pronunciation

I know this is a trivial point, but how is the word 'quaternion' pronounced? In English the vowel 'a' following 'qu' can have at least six pronunciations: short a as in 'quack'; the 'ah' sound as in the traditional British pronunciation of 'qualm'; the 'o' sound, as in 'quaff' or 'quash'; the 'aw' sound as in 'quarter'; the long 'a' sound as in 'quake'; and the indeterminate vowel sound as in 'equatorial'. (I'm aware that some or all of these may vary as between British and American pronunciation and indeed between regions within Britain and America.) Do mathematicians have a uniform pronunciation of the word, and if so what?86.183.202.149 (talk) 16:21, 11 March 2013 (UTC)

Since it is clearly using the same root as "quarter" is, I would imagine most English speakers prounounce the "a" in the same way. That is the case for all the US speakers that I know of. Rschwieb (talk) 17:10, 11 March 2013 (UTC)

Incidentally, I am not aware of any sound difference between "qualm" "quaff" and "quash". They all sound like the same vowel to me. Rschwieb (talk) 17:12, 11 March 2013 (UTC)

"qualm" rhymes with "calm"; "quaff" rhymes with "cough" Gandalf61 (talk) 17:36, 11 March 2013 (UTC)

Maybe this differs as you cross the Atlantic, but I've always assumed "quaternion" had the short 'o' sounds as in "quaff"/"cough"; fairly sure I've heard it pronounced that way in the UK. Gandalf61 (talk) 17:36, 11 March 2013 (UTC)

According to the OED:

Quaternion: Brit. /kwəˈtəːnɪən/ , U.S. /kwəˈtərniən/ , /kwɑˈtɛrniən/
Quaff: Brit. /kwɒf/ , U.S. /kwɑf/
Qualm: Brit. /kwɑːm/ , U.S. /kwɑ(l)m/ , /kwɔ(l)m/

Presumably, see International Phonetic Alphabet and International Phonetic Alphabet chart for English dialects to find out how to turn these glyphs into actual sounds. Jheald (talk) 17:58, 11 March 2013 (UTC)

(Incidentally, per the OED entry quaternion is first recorded in Wycliffe's Bible of 1384, where Peter in the Acts of the Apostles is put into the hands of four quaternions of soldiers -- ie four squads of four men each). Jheald (talk) 17:58, 11 March 2013 (UTC)

@Gandalf61 OK, maybe it is the difference between "all" and "off" :) Without that example, I would have a hard time hearing they are different.

@Jheald I think what you've listed has identified a second pronunciation that sounds familiar to me: "kwuh". With "kwah", these seem like the most familiar US pronunciations. I think regional dialect might also cause "kwatt". But never, as far as I can tell, "kway". Rschwieb (talk) 18:21, 11 March 2013 (UTC)

-1 in the multiplication table

In the multiplication, should -1 be included? I guess it is sort of self explanatory, but 1 is even more simple. TheKing44 (talk) 18:29, 2 August 2013 (UTC)

Error?

I've never made an edit (except for the occasional spelling fix) so not sure of protocol.

The section "Three-dimensional and four-dimensional rotation groups" refers to the 3-sphere as a three dimensional sphere, it isn't, the 3-sphere is four dimensional (its hypersurface has 3 dimensions)

Ds1392 (talk) 14:25, 20 October 2013 (UTC)

The 3-sphere or sphere of dimension three is a manifold of dimension 3 that may be embedded as an hypersurface in the Euclidean space of dimension 4. This embedding is realized by defining the 3-sphere as the zero set of the equation

x^{2}+y^{2}+z^{2}+t^{2}-1=0.

Thus the article is correct, although somehow too technical.

There is no protocol for editing. You have just to edit. However, if your edit is wrong or does not follows Wikipedia rules and policies, it is likely that it will be quickly reverted. D.Lazard (talk) 14:48, 20 October 2013 (UTC)

Point taken :D Perhaps the wording could be adjusted a smidge to make that clearer? I guess it's difficult to satisfy both the requirement that wikipedia be readable by a general audience (where intuitively, an n-dimensional object is one that can be embedded in Rⁿ) and the requirement that the information be accurate (an n sphere is an n-dimensional manifold.) If I can think of a way to improve the phrasing that isn't too wordy, I'll make the edit. Ds1392 (talk) 01:19, 21 October 2013 (UTC)

I don't think it can be clarified without properly distinguishing what "dimensions" are being discussed. It has (geometric) dimension 4 when embedded in R⁴, but has (topological) dimension 3. Rschwieb (talk) 13:23, 21 October 2013 (UTC)

No, an n-sphere always has dimension n. It can be embedded in a larger dimensional space, but that does not change its dimension. See manifold. Ozob (talk) 14:07, 21 October 2013 (UTC)

I agree with Ozob. The problem may come that for many people the distinction between a sphere and a ball is unclear: The sphere of dimension n is the boundary of the ball of dimension n+1. The surface of Earth is roughly a 2-sphere, while Earth in the whole is roughly a 3-ball. The lead of Sphere deserve to be edited to emphasize this distinction. D.Lazard (talk) 14:28, 21 October 2013 (UTC)

@Ozob (cc @D.Lazard): I'm saying that there are two subsets of humans: those who think of dimension in the topological way and those thinking of it in the geometric way. I'm pretty sure most laypeople carry the geometric dimension learned in grade school through 2-d an 3-d geometry. So, for example, they will report that the 2-sphere is a "three dimensional object," even if it is just the surface of a ball. I've seen this misconception cleared up a handful of times in graduate and undergraduate setting, so it is even common among non-laypersons.

Anyhow in summary, you and I know it has an intrinsic dimension that doesn't depend on where it's embedded, but stubbornly pretending that everyone else will understand it that way if we say so is an invitation for misunderstanding. Rschwieb (talk) 15:00, 21 October 2013 (UTC)

Nobody would say that a line is anything more than one dimensional or that a plane is anything more than two dimensional. I agree that some people are confused about the precise meaning of dimension, but the standard definition is both not too surprising and used universally within mathematics. I don't see any reason why this article should equivocate on this point. Ozob (talk) 18:44, 21 October 2013 (UTC)

I've changed it to "3-sphere S³". In this instance, using a less familiar term might lead to less confusion, for the reason that it does not as readily trigger an unintended interpretation. — Quondum 00:41, 22 October 2013 (UTC)

This change works for me. I was tripped up even though I should know better. Quondum's point re "less familiar" terms is quite valid; It's probably a good idea to avoid phrases with a natural language interpretation as much as possible because it's hard to avoid the reflexive interpretation. In everyday speech I refer to the 3-ball/2-sphere-in-R³ as a "three dimensional sphere" (formally correct or not this is how natural language is, and natural language "got there first" so to speak.) If I'm referring to the manifold I'll explicitly use the term "3-sphere" to avoid ambiguity. My 2c anyway :) Ds1392 (talk) 12:13, 23 October 2013 (UTC)

regarding basis shown in Matrix representations

Even though the section says that there are at least two ways, should'nt it be explicitly said that the basis made up of four 4x4 matrices shown in the example are not unique and that other matrices which have the same properties can be used to represent i,j and k. also how many such bases can be possible? a trivial case is a basis which is made of the transpose (equivalent to choosing a basis of -i, -j and -k) or basis where matrices corresponding to i, j and k are cyclicaly shifted. does another basis which cannot be made up by doing these two operations exist? Does the basis have to be made up of 0, 1 and -1? Cplusplusboy (talk) 13:22, 20 January 2012 (UTC)

These questions make decent research projects, but they will not be appropriate for the article (unless there is some very nice citable result). (Ordered) bases of the type you described will correspond naturally to the ring automorphisms of H. Rschwieb (talk) 13:58, 20 January 2012 (UTC)

Arbitrary 4 × 4 real matrix without Jordan blocks with same eigenvalues (namely, {

i

,

i

, −

i

, −

i

} ) is eligible to represent the quaternion

i

. You may construct real 4-dimensional quaternions' representations by algebraic conjugation:

X \to U -1 X U

where

X

is a canonical representation and

U

is an arbitrary reversible 4 × 4 real matrix chosen for this particular representation. This is actually nothing but a (two-side) intertwiner, or simply a change of basis, and is considered the same in the representation theory. Incnis Mrsi (talk) 14:30, 20 January 2012 (UTC)

(Note to OP: the conjugation described here produces an automorphism of H. Rschwieb (talk) 15:12, 20 January 2012 (UTC)

It is an automorphism of ℍ only if

U

belongs to SO(4). I guess that it is also sufficient (the 3-sphere of unit quaternions in the canonical representation seems to be the same as left-isoclinic rotations), but am not completely sure. Moreover, as we discuss representations by arbitrary matrices,

U

does not even have to be orthogonal, this means that

U

⁻¹

X

U

not necessary is a canonical representation of any quaternion. Incnis Mrsi (talk) 16:20, 20 January 2012 (UTC)

Oh. I've never heard of a reversible matrix, so I was guessing it meant special orthogonal. Rschwieb (talk) 20:25, 20 January 2012 (UTC)

Having considered the group of matrices that may be U, this does not directly say the obvious things about the resulting representation. For example, the first matrix always remains the identity matrix. Next, it would seem to me that the remainder of the basis matrices obey a linear transformation law, which, unlike U, has only three dimensions: the symmetry group of S²? — Quondum ^☏✎ 05:18, 21 January 2012 (UTC)

Ahem. Perhaps we can keep this to language accessible to those who do not already know the answer to the original question? Cplusplusboy may have a point that "There are at least two ways of representing quaternions as matrices" may be so weak a statement as to be misleading, and should at least be rephrased. There are an infinite number of ways (for example derivable from each of those representations via 3-dimensional rotations and reflections of the (i,j,k) basis on a 4×4 real matrix representation alone (the ring automorphism group being isomorphic to O(3), I guess). So perhaps it would be reasonable to change this to "There are many ways of representing quaternions as matrices" – even without citations. Those given just happen to be two of the "neat" ways. — Quondum ^☏✎ 14:50, 20 January 2012 (UTC)

Hehe, I'm not very familiar in this math and so didn't want to edit the article myself. I was just comparing an example given in a book on quaternions and found that the bases it showed differed from wikipedia's. Since I was under the impression that the basis was unique, I tried to do the check of ijk=-1 property on both bases and found that both were right and wanted to confirm the fact. Should this talk be removed as the confusion is resolved? I didn't see anything about that in the guidelines. Cplusplusboy (talk) 16:36, 21 January 2012 (UTC)

I've edited the article in an attempt to address the initial problem; we'll see what others make of it. No, we leave the discussion as is; there are tight constraints on any editing of prior comments; it'll be removed in due course by the archiving process. See Wikipedia:Talk page guidelines#Editing comments. — Quondum ^☏✎ 06:35, 22 January 2012 (UTC)

Please, in an article on mathematics, be more precise. For instance: The sentence ,Using 4x4 real matrices ...' is clear, since a,b,c,d must be real numbers, but this should be stated there, even if this is tedious. But in the 2-dimensional matrix representation some lines above, nothing is clear: Are the a,b,c,d real numbers as well or complex numbers? Obviously complex!? And why there are different letters for a,b,c,d in the text and in the matrix representation? And the same for i. Is this the same complex unit as in the text line before. The same question some blocks before in the determinant - please state whether these a,b,c,d are real or complex numbers. — Preceding unsigned comment added by 130.133.155.70 (talk) 13:17, 18 July 2014 (UTC)

Matrix vector product

The following text was removed from Matrix representation:

The multiplication of two quaternions

ab=c

can be represented by matrix vector multiplication:

{\begin{bmatrix}a_{0}&-a_{1}&-a_{2}&-a_{3}\\a_{1}&a_{0}&-a_{3}&a_{2}\\a_{2}&a_{3}&a_{0}&-a_{1}\\a_{3}&-a_{2}&a_{1}&a_{0}\end{bmatrix}}{\begin{bmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{bmatrix}}:={\begin{bmatrix}c_{0}\\c_{1}\\c_{2}\\c_{3}\end{bmatrix}}

If we define

B(a)\equiv {\begin{bmatrix}a_{0}&-a_{1}&-a_{2}&-a_{3}\\a_{1}&a_{0}&-a_{3}&a_{2}\\a_{2}&a_{3}&a_{0}&-a_{1}\\a_{3}&-a_{2}&a_{1}&a_{0}\end{bmatrix}}

and a, b, and c are real column vectors constructed from quaternions, we can rewrite the multiplication as

B(a)b=c

or

B(a)B(b)=B(c)

.

We can also define

A(a)\equiv {\begin{bmatrix}a_{0}&a_{1}&a_{2}&a_{3}\\-a_{1}&a_{0}&-a_{3}&a_{2}\\-a_{2}&a_{3}&a_{0}&-a_{1}\\-a_{3}&-a_{2}&a_{1}&a_{0}\end{bmatrix}}

.

The A and B matrix constructions have the following basic properties.

A(a)b=B(b)a^{*}

A(a)B(b)=B(b)A(a)

Two matrices must be multiplied to represent the quaternion product. The text removed today was unreferenced and made a false assertion.Rgdboer (talk) 20:51, 18 July 2014 (UTC)

While it would be easy enough to correct this, the matrix representations as they stand in the article are sufficient. Also not being referenced makes it look like the OR it probably is. I agree with the removal. —Quondum 21:28, 18 July 2014 (UTC)

But this looks very useful. Why would you multiply the whole matrix; that's four times the amount of work???

Think it was just transposed incorrectly:

{\begin{bmatrix}a_{0}&a_{1}&a_{2}&a_{3}\\-a_{1}&a_{0}&-a_{3}&a_{2}\\-a_{2}&a_{3}&a_{0}&-a_{1}\\-a_{3}&-a_{2}&a_{1}&a_{0}\end{bmatrix}}{\begin{bmatrix}b_{0}&b_{1}&b_{2}&b_{3}\\-b_{1}&b_{0}&-b_{3}&b_{2}\\-b_{2}&b_{3}&b_{0}&-b_{1}\\-b_{3}&-b_{2}&b_{1}&b_{0}\end{bmatrix}}:={\begin{bmatrix}c_{0}&c_{1}&c_{2}&c_{3}\\-c_{1}&c_{0}&-c_{3}&c_{2}\\-c_{2}&c_{3}&c_{0}&-c_{1}\\-c_{3}&-c_{2}&c_{1}&c_{0}\end{bmatrix}}

{\begin{bmatrix}a_{0}&a_{1}&a_{2}&a_{3}\end{bmatrix}}{\begin{bmatrix}b_{0}&b_{1}&b_{2}&b_{3}\\-b_{1}&b_{0}&-b_{3}&b_{2}\\-b_{2}&b_{3}&b_{0}&-b_{1}\\-b_{3}&-b_{2}&b_{1}&b_{0}\end{bmatrix}}:={\begin{bmatrix}c_{0}&c_{1}&c_{2}&c_{3}\end{bmatrix}}

{\begin{bmatrix}a_{0}&-a_{1}&-a_{2}&-a_{3}\\a_{1}&a_{0}&a_{3}&-a_{2}\\a_{2}&-a_{3}&a_{0}&a_{1}\\a_{3}&a_{2}&-a_{1}&a_{0}\end{bmatrix}}{\begin{bmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{bmatrix}}:={\begin{bmatrix}c_{0}\\c_{1}\\c_{2}\\c_{3}\end{bmatrix}}

213.205.240.129 (talk) 13:06, 12 September 2014 (UTC)

Type of isomorphism is unclear

In the section Quaternion#Matrix_representations, the following sentence occurs:

Restricted to unit quaternions, this representation provides an isomorphism between S³ and SU(2).

This sentence has the problem that technically it is ill-defined, or more correctly, since both these objects are only in the same category as sets, this only says that they have the same cardinality. I expect that most people will find a natural interpretation as an isomorphism of topologies and/or as a congruence of geometric objects in Euclidean 4-space. Given that the representation is given as the basis of the isomorphism, the geometric interpretation may be intended, but is inappropriate (we would not normally call a linear mapping between representations an isomorphism in the algebraic context). However, S³ regarded as a topological object or a geometric object, H regarded as a ring and SU(2) regarded as a group leaves room for confusion of what isomorphism is meant. Could someone with more knowledge in the area please qualify this to clarify what is meant? Perhaps leave S³ out of it altogether, and simply state that there is a group isomorphism between the multiplicative group of unit quaternions and SU(2)? —Quondum 13:56, 19 July 2014 (UTC)

Where to even begin.... there's lots of confusion all around, it should be clarified.

(lower-case) su(2) is an algebra, and specifically a Lie algebra.
the quaternions are an algebra, too. (an algebra is a vector space endowed with multiplication, usually a non-commutative multiplication)
the structure constants of su(2) are equal to those of H, except that they are multiplied by an extra factor of ${\sqrt {-1}}$ . Thus, the generators of su(2) when squared, give you +1, instead of -1 when you square the generators of H.

(upper case) SU(2) is a Lie group it corresponds to the fundamental representation of the algebra. Give a point ${\vec {\theta }}$ in the lie algebra su(2), you get the corresponding group element $U=exp(i{\vec {\theta }}\cdot {\vec {\sigma }})$ , which is a 2x2 unitary matrix. Here exp is the exponential map used to convert dirction vectors into geodesics. The ${\vec {\sigma }}$ are the Pauli matricies.

You can do exactly the same thing as above, using +1, i,j,k instead of using the identity matrix plus the pauli matricies. You get exactly the same thing (except for an extra confusing factor of ${\sqrt {-1}}$ that floats around and makes thing randomly confusing.

The 3x3 matrix group representation of su(2) is call the rotation group SO(3). The explicit mapping is this. Let ${\vec {v}}$ be a 3D vector. Let R be a 3x3 rotation matrix. Then, $R\cdot {\vec {v}}=U^{\dagger }{\vec {v}}\cdot {\vec {\sigma }}U$ where U is same as above. By contrast, R is given by $exp({\vec {L}}\cdot {\vec {\theta }}/2)$ where ${\vec {L}}$ are the generators of angular momentum i.e. the purely real 3x3 matrixes that generate SO(3) rotations. anyway, its the same theta in U and R.

(upper case) SU(2) is a manifold that is topologically isomorphic to S_3

(upper case) SO(3) is a topological manifold that is covered twice over (double cover) by SU(2).

The complex projective plane CP(2) is topologically isomorphics to SU(2)

The last bit is made use of in quantum mechaics, where a spinor is a 2D complex vector of unit length (thus its projective) and is spun around by elements of SU(2).

The resulting manifold is called the Bloch sphere.

The metric on SU(2) aka S_3 aka CP(2) is called the ... crap, I don't remember the name. Oh right Fubini-Study metric. Its basically just the standard metrix on the sphere, but it looks interesting when you write it out for the typcal notation used in QM and in algebraic geometry which each have thier unique notiations (alg. geom studies complex projective spaces).

One of the things that confuses people is the relationship between SU(2) and su(2) because both use 2x2 complex matrices. They're not the same tho, because SU(2) is a group, su(2) is an algebra. Likewise, there is a similar confusion for quaternions: There is the algebra H and there is the group H and they both use 1, i, j, k to understand how these differ, it helps to keep su(2) vs SU(2) firmly rooted in mind.

Wait -- there's more... if you allow the vector ${\vec {v}}$ to be complex, then you get representations of the group SL(2,C) which have SO(3,1) as a representation -- this is the group of special relativity. which is why the outer product of two relativistic spinors is a spin-1 boson. Add the sqrt(-1) and you can say the same with quaternions, if you wanted to. You could write out Einstein's equations for general relativity using quaternions, if you wanted to. This is because the quaternions are $sqrt{-1}$ times the usual generators of sl(2,C). People have actually done this: its vaguely instructive to see those eqns as SL(2,C) instead of text-book standard SO(3,1).

The outer product of two quaternions gives the Runge-Lenz vector: it describes the orbital mechanics of planetary systems (planets orbiting a sun) and the conserved qauantities are given by SO(4) (not just SO(3)).

The restriction of SL(2,C) to real numbers gives SL(2,R) which is a hyperbolic manifold important in number theory and seems to have somethhing to do with the Riemann hypothesis. This is one of the many Siren's calls that string theorists are unable to resist. (well, string world sheets are Riemann surfaces which leads to conformal field theory and AdS/CFT correspondance which gives monstrous moonshine. Heh. So there.

In short, when you really start fucking with it, you find all of these low-dimensional concepts are isomorphic or homomorphic to each other, which makes for a very rich playground of things related to each other.

Anyway, I clearly had too much fun writing the above. Thanks for posing the question. 67.198.37.16 (talk) 01:39, 13 February 2015 (UTC)

I see the question served its purpose: Ozob addressed it with this edit. Yes, the connections are varied and deep, and fun if you live with this stuff. —Quondum 03:52, 13 February 2015 (UTC)

Terminology: "scalar part" and "vector part"

I find this terminology inept and ugly. Why not stick with real and imaginary parts?

Is that terminology standard in the literature (the maths literature, not the physics one)? Not if I trust the few papers I've read today, but I could be wrong. If it is standard, then I guess some could argue that Wikipedia should continue to spread this ugliness. Otherwise it would be a shame that Wikipedia helps set up or propagate an ill-suited standard for no reason.--Seub (talk) 05:59, 11 March 2015 (UTC)

Real and imaginary parts is used in complex numbers where the parts are equal in dimension. scalar and vector parts emphasises that they are not equally sized, but a 1-dimensional scalar part and a 3-dimensional vector part. And they are vectors in a very real sense too; vector operations such as the cross product, dot product scalar product arise from quaternion product by considering how the 'vectors' in them multiply.--JohnBlackburne^words_deeds 12:41, 11 March 2015 (UTC)

I think that there are arguments both ways. Doesn't the scalar/vector terminology date to Hamilton, whereas real/imaginary and other variants tend to be more contemporary? I am not particularly a fan of the use of "vector part", because it does not seem to generalize directly to hypercomplex numbers or Clifford algebras, and would be destined to fade. John is correct that it does relate directly to vector algebra, but only in exactly 3 dimensions. IMO, it would be more appropriate to speak of the "scalar" and "nonscalar" parts. At least "real" can "imaginary" do fit as a generalization of the use of the concepts from complex numbers, and I've seen the term "imaginary" used to refer to any nonreal component that squares to a real number, such as in split-complex numbers. For the purposes of this article, perhaps we could switch to "real" and "nonreal" (just a suggestion)? —Quondum 17:54, 11 March 2015 (UTC)

At the level of this article vector algebra is done in 3D. E.g. you have the cross product, defined only in three dimensions, as a key operation. This and the dot product show how it is a 'vector' part as both operations arise directly from the quaternion product if restricted to products only of the vector parts. At least that's how I learned it. I later learned how to generalise it in various ways, which leads to other ways to think of the non-scalar parts, as imaginaries, as bivectors . But vectors I think is most usual at a less advanced level.--JohnBlackburne^words_deeds 00:21, 12 March 2015 (UTC)

Just because quaternions can be related to vectors and rotations in Euclidean 3-space doesn't mean that it is their nature or purpose, it's just a use you can make of them. So I disagree with the argument that the imaginary part of a quaternion is a vector in a "very real sense". And the "scalar part" term is no more clever imho, why would you want to think of it as a scalar? It is just a quaternion that is identified as a real. I recall that scalars are what you name the elements of the base field of a vector space, when you think of them acting on vectors by multiplication. If you work with vector spaces over the quaternions, then the quaternions themselves are all scalars. I also recall that in (not really) modern mathematics, vectors are not just vectors in the plane and the 3-space. If you define quaternions as a 4-dimensional algebra over the reals, then all quaternions are "vectors". Anyway, my recommendation is "real" and "imaginary", I see no drawback in that terminology. I also think it's the more common usage in modern mathematics (we could survey the arXiv a little), but I'm not certain of that.--Seub (talk) 10:17, 12 March 2015 (UTC)

It is not the role of Wikipedia to decide if a terminology is correct or convenient, nor to promote a new terminology. The true question is "what is the terminology that is used in most textbooks?" If several terminologies are commonly used, they have to be all described. For coherence, one has to be used in the article body, either the most common, or another one, if some reliable sources say explicitly that they have to be preferred for some good reasons. D.Lazard (talk) 11:04, 12 March 2015 (UTC)

Seub: The quaternions form a vector space over the real numbers, and in that sense, it can be thought of as "the scalar part" (quaterion multiplication by elements of this subspace is equivalent to multiplication by a (real) scalar). But I agree with D.Lazard: we should be determining the dominant modern terminology (or similar criteria as given), and sticking to that. —Quondum 18:26, 12 March 2015 (UTC)

What D. Lazard says is sensible. However the influence of Wikipedia cannot be denied, if Wikipedia says "vector part" then a lot of people will think this is the "right" terminology and are going to follow that standard. Anyway, to the point: when you google search quaternion "real part" "imaginary part", you get 14,200 results against 8,440 for quaternion "scalar part" "vector part". If you add the keyword arXiv then the results are 18,200 against 1,270 (!). From the little I have seen, the "scalar part" and "vector part" terminology is most used in physics or engineering textbooks.--Seub (talk) 05:44, 13 March 2015 (UTC)

You are not going against what has been said; we need to somehow determine the appropriate terminology to use. Google results that are of similar magnitude are not useful, though, so we'd need to be more careful, surveying several contemporary texts more closely if we wanted to reach a conclusion. I'm not opposed to the replacementg if these terms. —Quondum 15:03, 13 March 2015 (UTC)

That careful survey was not done when the terminology "scalar part" and "vector part" was chosen, was it. The Google results I indicated suggest that "real part" and "imaginary part" are more commonly used, especially in contemporary mathematical papers. I suggest that we go for that terminology unless someone can argue otherwise. — Preceding unsigned comment added by Seub (talk • contribs) 17:01, 13 March 2015 (UTC)

Deleted "spammy" reference

The reference 6 (Alam, Mohammed Shah. "Comparative Study of Quaternions and Mixed Numbers". Journal of Theoretics 3 (6). ISSN 1529-3548) is rehashing the obvious that's available in any college level algebra textbook. The Journal of Theoretics is a journal of questionable repute, and the reference isn't adding anything new. We would be much better off referencing Hamilton's own work: Hamilton, W. R. Elements of Quaternions. London: Longmans, Green, 1866. — Preceding unsigned comment added by Metlin (talk • contribs) 03:31, 6 April 2015 (UTC)

Regarding section on Multiplication of basis elements

We should include further proofs rather than just say. "All the other possible products can be determined by similar methods". Remember although this may seem obvious to those familiar with quaternions, the article should aim at those who aren't. Even just one poof where one of the sides is negative for example to prove ji =-k

ijk = -1 Multiply across by ij

iijjk= -1ij

(substituting -1 for the squares of i and j we get..)

(-1)(-1)k = -ij

k=-ij

Feel free to correct my proof, but I think the other proves are not strictly obvious and therefore needed. — Preceding unsigned comment added by Iantierney (talk • contribs) 22:19, 17 June 2015 (UTC)

The first case given is intended to do that. I don't see how a second example is going to help. The negative sign is the least of the problem – in your case you've just added i and j in the middle; you cannot do that. However, the teaching basic rules of noncommutative algebra (keeping track of multiplying on the left or right) is not the function of an encyclopaedia article. I don;t agree that anything more should be included. —Quondum 23:00, 17 June 2015 (UTC)

In that case, put in a reference to the "basic rules of noncommutative algebra". I agree that something more is needed, because I didn't get it, and I'm smarter than your average bear (seriously). — Preceding unsigned comment added by JohnL4 27709 (talk • contribs) 18:09, 16 July 2015 (UTC)

The section mentioned seems to introduce the ideas at an appropriate level. Perhaps a specific change should be suggested? The argument above makes the obvious mistake of incorrectly assuming that ijij=iijj, which is not allowed due to noncommutativity. Mark MacD (talk) 07:56, 20 July 2015 (UTC)

Is an algebra invented or discovered?

In the summary it says "In fact, the quaternions were the first noncommutative division algebra to be discovered". Is it not an invention?

No, the quaternions have been invented, but the fact that they form a division algebra is a discovery. D.Lazard (talk) 08:32, 13 January 2016 (UTC)

"isomorphic as a set"

The article includes the following sentence:

- As a set, the quaternions H are isomorphic to R⁴, a four-dimensional vector space over the real numbers.

While this is technically true, I don't think this sentence means what it is supposed mean: being "isomorphic as a set" is another way of saying "has the same cardinality as", as isomorphisms in Set are arbitrary bijections. One might as well state that "as a set, the quaternions H are isomorphic to the Cantor set. --Letkhfan (talk) 19:09, 28 April 2016 (UTC)

Good point, that's clearly not the intention. I changed it; do you think the article now conveys the intended meaning? Ozob (talk) 23:22, 28 April 2016 (UTC)

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Reduced form?

Given a quaternion a + bi + cj + dk, dividing by a yields the quaternion 1 + b/ai + c/aj + d/ak, or equivalently 1 + pi + qj + rk (where p=b/a, q=c/a, and r=d/a). What is this called? A "normalized" or "reduced" quaternion, perhaps? — Loadmaster (talk) 17:29, 4 August 2016 (UTC)

Not come across it and I struggle to think of an application. Normalised is usually reserved for dividing by the norm, i.e. by √(a² + b² + c² + d²), which is is useful for e.g. using quaternions to rotate.--JohnBlackburne^words_deeds 17:48, 4 August 2016 (UTC)

Why searching for a specific name for quaternions for which the real part is one, when complex numbers for which the real part is one do not have any specific name? D.Lazard (talk) 17:54, 4 August 2016 (UTC)

I was probably thinking of the unit quaternion, i.e., a quaternion with a norm of 1, which is exactly what JohnBlackburne mentions above. So while dividing the quaternion components by a to get a real part of 1 might be useful, it's obviously far more useful if the whole quaternion has a magnitude/norm of 1. — Loadmaster (talk) 16:37, 8 August 2016 (UTC)

Squaring Quaternions

@Wiki user wiki: This discussion belongs at Wikipedia:Reference desk/Mathematics, per WP:NOTAFORUM.--Jasper Deng (talk) 19:00, 17 May 2017 (UTC)

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Correct me if I'm wrong, but what happens when you square both sides of the fundamental quaternion equation where ijk = -1? You get ijk = -1; (ijk)^2 = 1; (i^2)(j^2)(k^2) = 1; — Maybe I made the mistake here. Is distribution possible in quaternions? (-1)(-1)(-1) = 1; -1 = 1; which is obviously false. Wiki user wiki (talk) 18:07, 17 May 2017 (UTC)

You are mistakenly treating the i, j and k as commutative. Here’s how it works out if you respect the anticommutative nature of them.

{\begin{aligned}(ijk)^{2}&=(ijk)(ijk)\\&=ijkijk\\&=-ijikjk\\&=iijkjk\\&=-iijjkk\\&=-(-1)(-1)(-1)\\&=1\end{aligned}}

Each of the intermediate step swaps two terms. And as that happens the sign changes, as every pair of i, j and k anti-commutes: see #Multiplication of basis elements. This leads quickly to the correct result.--JohnBlackburne^words_deeds 18:37, 17 May 2017 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Skew-Symmetric?

The section labeled "Matrix Representations" in the main topic refers to the representations as Skew-Symmetric, but that's true iff the Quaternion is pure imaginary, that is the Real part (and thus the diagonal of the matrix) is zero so that it satisfies the definition −A = A^T. I suggest that we just delete the reference to Skew-Symmetric and say "However, the representation of quaternions as a matrix is not unique." Sudleyplace (talk) 19:23, 29 July 2017 (UTC)

Yes, only pure quaternions represent as such matrices. Thank you for the suggestion. — Rgdboer (talk) 22:16, 29 July 2017 (UTC)

Summary should be easier to understand

The summary of this article should be re-written to be drastically simpler per wikipedia guidelines on summaries: https://en.wikipedia.org/wiki/Wikipedia:Summary_style#Levels_of_detail — Preceding unsigned comment added by 24.148.30.174 (talk) 02:10, 6 November 2017 (UTC)

You mean the first section ? To me it seems pretty good, for this sort of article. The first three paragraphs are very approachable even to a non-mathematician, certainly to anyone with any maths training. At the end it delves into more advanced mathematics, but that is fine, as that too can be found in the article. It’s hard to write a good mathematics lead section for longer articles like this, but I think this one does pretty well. How would you improve it?--JohnBlackburne^words_deeds 02:18, 6 November 2017 (UTC)

Historical claims of unpublished discovery should be just that

The categorical assertion, in any scientific field, that someone (someone as highly published as Gauss, no less) developed a ground-breaking theory or principle, decades earlier (again, by Gauss, in this article) than its accepted and published discovery date of record (in this case, by Hamilton), but that this someone had merely neglected to publish it at any time; such assertions should be regarded with healthy skepticism in all cases, not un-questioning acceptance, as in the present article. In my view, merely referencing published, post-humous, claims to this effect is not sufficient basis to render this practice non-controversial. Wikibearwithme (talk) 19:53, 13 January 2018 (UTC)

I see no controversy here. Gauss was well known for doing this kind of thing and in this particular case it is not just hearsay that Gauss had obtained the results; they were published (in 1900 though).--Bill Cherowitzo (talk) 05:19, 14 January 2018 (UTC)

I do like very much to read about mathematical gossip (salt in an otherwise boring soup) from the history of math (What educational settings/genes are responsible for Ramanujan's remarkable math competence?). I gave up, a long time ago, to believe in an essential veracity of any history, it's always a history of the victorious. So the rich de l'Hospital managed to get his name engraved in all High School(?) kids' brains, and, just luckily, the true(?) discoverer is famous, too. Gauß, the princeps mathematicorum, definitely is a winner, and yes, really "ß", but "ss" is a winner, too. I'm always in deep despair, when seeing edits about those poor, neglected ancient and precedent discoverers from non-European regions, deprived of all fame in WP, sooo reliably sourced in ... ooops. Purgy (talk) 08:31, 14 January 2018 (UTC)

Importance of the DOW

Looking at the previous state of the verted paragraph I enjoyed the flowery abundance of details: Hamilton, with his wife, on the way to some boring meeting, enjoying his stroll along a river, suddenly grasping his knive, and, in some violent moment ... wait for it, here it comes ..., noticing that it is MONDAY, sets an equation in stone!

Up to my measures, this detailed paragraph requests for mentioning the DOW, but, ... go on, carve another notch for reverting me in your mouse. Purgy (talk) 20:29, 27 March 2018 (UTC)

Please see WP:Wikipedia is not a battleground and WP:POINT. When you take every revert personally and start off so confrontationally, it's difficult to respond without turning the whole thing into an argument. Your indirect and roundabout (and difficulty with English) language can often make it difficult to figure out exactly what your point is; I'm not entirely sure what it is here. But I'll say that the IP's removal of "Monday" was a good edit, and you shouldn't have reverted it. It's so rarely included that even MOS:DATEFORMAT doesn't use it in any examples. So unless there's some very specific reason to include the day of the week, it shouldn't really be there. –Deacon Vorbis (carbon • videos) 20:46, 27 March 2018 (UTC)

Focusing on the DOW, it has been there at least since 11 February 2011‎. It completes a paragraph which, to my measures, is per se quite rich in non-mathematical detail ("flowery"). I restored this stable state after a non-vandal IP-editor removed the DOW under the summary "not important that it was Monday" with the summary "just a long format with weekday. What is important?", addressing the many paraphernalia described, which still remain within the paragraph. I consider the DOW in this detailed description as a nice tessera, making the naked date more accessible, like when you were born on a Sunday. (Are you?) OTOH, a more stringent description of Hamilton's stroll is unacceptable, also. Shouldn't this be discussed, instead of harshly(?) reverted?

I admit that I wrote the ultimate clause about a notch in your mouse in referring to our last "personal" encounter at Cesaro summation, where my edits, repairing twice a categorical error but keeping some strange notation, were reverted to re-include that same errors. However, I have not the slightest clue, where I gave reasons to point me to WP:BATTLE and WP:POINT, and additionally, to accuse me of starting off so confrontationally. I think we had a rather reasonable contact about \phantom, and I do not understand the perceived aggression from your side, lately. Purgy (talk) 13:49, 28 March 2018 (UTC)

Since there are essentially no sound arguments given, beyond "was a good edit", I restored the DOW as perfectly fitting to the given details. Furthermore, this information serves the convenience of the History-minded reader and is provided at the negligible luxuriousness of below 10 bytes. My edit reestablishes a wording, which was stable for at least more than half a decennium, and therefore should not be simply reverted again without further arguments.

Valuing the lacking standardisation in WP:MOS to an extent that justifies the removal of a DOW appears to me as a rather threatening perspective. Purgy (talk) 09:48, 31 March 2018 (UTC)

@Purgy Purgatorio: I've listed this at WP:3O. –Deacon Vorbis (carbon • videos) 13:16, 31 March 2018 (UTC)

And just to add for the record, the day of the week is practically never used in dates on Wikipedia. The MOS has so many little nitpicky guidelines about date formats and such, the fact that there's nothing there about the day of the week is rather telling – it's just not used unless there's some overriding reason, and there isn't one here. –Deacon Vorbis (carbon • videos) 13:48, 31 March 2018 (UTC)

@Deacon Vorbis: I pity that my pettiness required you to request a third opinion, instead of simply passing an argument beyond the "DOW not explicitly mentioned as a DATEFORMAT" (obviously, giving in to petitesses is no option). I deposit this regret here, dedicated to anyone who offers his judging opinion. Since I try to stay amenable always to graspable arguments, and even to decision by majority, I promise not to interfere any further in this matter. Of course, I won't change my personal conviction that keeping the DOW in this specific setting is better than to remove it. Nevertheless (mind the date!) slightly amused, Purgy (talk) 09:22, 1 April 2018 (UTC)

@Purgy Purgatorio: What ~~the fuck~~ are you talking about? Why are you saying your pettiness is requiring me to request a third opinion? Do you really think you're being petty, or are you just trying to take another passive-aggressive snipe at me? Because if it's the latter, this is really pushing into WP:CIVILity territory. I've been nothing but matter-of-fact about the day of the week issue. I only commented on your behavior briefly, I only did so because I thought you were starting off in a way that was going to make things difficult, and I did so as civilly as I could muster. Did you even read the WP:3O page? It's just there as in informal way to get a third opinion when two editors have come to an impasse over a content dispute. There's nothing binding as a result of it, but it can be a useful way to help get some further input, which we seem to need here. It's not about achieving a majority, but just getting another set of eyes on something from someone who may have something new to add. There's nothing personal about any of this. Please just stick to the content dispute here and knock off your passive-aggressive bullshit already. –Deacon Vorbis (carbon • videos) 16:33, 1 April 2018 (UTC)

Having a walk with his wife emphasises that it came to him while not explicitly working on it (eg not working on it at a desk with pen and paper and no other distractions). I cannot see any importance for it being a Monday.

Purgy, I find your manner of discussion hard to work with. You make inflammatory, personal remarks in your opening comment and then act surprised when somebody reacts to it. You go off on hyperboles with flowery language that has little to do with the original issue. (Does anybody actual say 'decennium' in real speech? Even the article it redirects to doesn't mention the word.) And when others try to stick to the actual subject you make ad hominem attacks on them about being petty (violating Wikipedia:No personal attacks) instead of providing rational reasons. Stepho talk 00:26, 2 April 2018 (UTC)

It doesn't matter one way or another. Everyone should just drop it. Sławomir Biały (talk) 11:34, 2 April 2018 (UTC)

Octonions "in some sense" normed division algebra?

What does the "in some sense" mean when it says that the octonions are "the last normed division algebra"? (Sounds like something a bunch of math/theatre majors would make?) Jimw338 (talk) 23:25, 8 June 2018 (UTC)

You’re right; from what I’ve researched, sedenions aren’t division algebras because they have zero divisors. From what I could tell, Normed was just a fancy way of saying from the Cayley Dickson construction. “In some sense” is just dumb IntegralPython (talk) 11:36, 9 June 2018 (UTC)

I don't think Imaginary quaternions is correct

Hamilton described a quaternion as consisting of a scalar and a vector part. The only things he called imaginary were the units i, j and k. He continually refers to quaternions with no scalar part as vectors. Today vector has a different meaning so we should call quaternions with no scalar part vector quaternions, not imaginary quaternions. Prof McCarthy (talk) 04:30, 5 August 2018 (UTC)

I cannot follow your your implication from "meaning of today's vectors" to not calling quaternions without real parts "imaginary".

To my taste the "imaginary property" of "three imaginary units", all squaring to

-1

still more explicit per 10:55, 6 August 2018 (UTC), is more important within the algebraic structure of quaternions than the isomorphism of their imaginary parts to 3-dim real "vector" spaces (all real finite dimensional vector spaces of equal dimension are isomorphic). I rather perceive the attribute "scalar" as alien to quaternions, I think "real" is more appropriate, since the reals also constitute the multiples of the imaginary units.

I am quite unsure, if your recent edits are an improvement, beyond fitting more to your preferences. Purgy (talk) 21:01, 5 August 2018 (UTC)

Hamilton coined the term vector in the context of quaternions, and since his time the term vector has become any element of a linear space, resolving this conflict is important for clear communication. The term imaginary quaternion is unambiguous and reflects the structure of ℍ as a union of complex planes sharing the same real line. The term vector quaternion might be read as quaternion vector and mean any quaternion in the 4-space. If we were communicating with classical Hamiltonian quaternions, the term vector would mean something 3-dimensional, but here in 2018 students know about vector spaces before entering abstract algebra and encountering quaternions, so the classical terminology would cause confusion. For clarity, the imaginary part of a quaternion was traditionally called the vector part. See Krishnaswami & Sachdev (2016) Algebra and Geometry of Hamilton’s quaternions for use of imaginary part.— Rgdboer (talk) 22:37, 5 August 2018 (UTC)

Here is what Hamilton said: On the other hand, the algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion; and may be denoted by prefixing the characteristic Vect., or V. We may therefore say that a quaternion is in general the sum of its own scalar and vector parts, R. W. Hamilton, On Quaternions, or On A New System of Imaginaries in Algebra. Philosophy Magazine. This seems unambiguous to me. I am not sure where the preference for imaginary quaternion arises because it clearly conflicts with what Hamilton called a biquaternion which had real and imaginary quaternion parts. Prof McCarthy (talk) 00:33, 6 August 2018 (UTC)

Considering the paper's title On A New System of Imaginaries in Algebra and Hamiltons o-tone in the citation ... the algebraically imaginary part, being ..., together with the various phrasings of may be called/denoted and we may say, instead of we call/denote/say, one might even assume unambiguity to the contrary of your primary preference, of course, not rendering your picture false thereby. Certainly, I agree to the vector notion being extremely useful and intuitive, but I still oppose to inflicting this name as a primary one in the whole algebraic object, just because some substructure of the quaternions may be conveniently associated to vectors in the familiar 3D-space (which, btw, would need some algebraic pimping to hold the candle to "vector quaternions").

I stated already above where my preference for "imaginary" comes from, and I deny that it would "conflict clearly" with any other notion (I did not use the term "complexification", nor did I mention Cayley-Dickson). I still cannot see, however, how the implication (blah, so we should blabla) in the last sentence of the starting comment is to be understood.

I would enjoy to contribute by arguing, but I certainly won't start edit warring, or continue to comment on simply repeated, but unreasoned or incoherent implications. Purgy (talk) 10:55, 6 August 2018 (UTC)

Right quaternion: Re-reading the article, the word is not vector or imaginary, but right. By way of explanation, every non-zero quaternion that is not on the real axis has a polar decomposition

q=\rho \exp(\theta r),\quad r>0,\ \ 0<\theta <\pi ,\ \ r^{2}=-1.

By Euler's formula, θ = π/2 implies the real part of q is zero. In Hamilton’s terminology, q = (T q)(U q) where T gives the tensor of q (Tq = ρ) and where Uq gives the versor of q (Uq = exp(θ r) ). The imaginary units r in S² are the right versors (See unit vector#Right versor.) — Rgdboer (talk) 22:52, 7 August 2018 (UTC)

Conjugation confusion

On 2018-10-09 the section on "Conjugation, the norm, and reciprocal" included the comment that, "Unlike the situation in the complex plane, the conjugation of a quaternion can be expressed entirely with multiplication and addition ... ."

How is this "unlike the situation in the complex plane"?

I will delete the "unlike" phrase. If you think it belongs there, please explain here what I'm missing or find a less ambiguous way of saying whatever you think it says.

Thanks, DavidMCEddy (talk) 02:49, 10 October 2018 (UTC)

Probably you missed the necessity of the non-holomorphic functions "Real/Imaginary part" for constructing

{\bar {z}}

from

z

. There is no chance to do the conjugation only with multiplication, addition, and the resp. constructions of inverses from the field of complex numbers. I'll try to be more explicative in the article. Purgy (talk) 06:14, 10 October 2018 (UTC)

Quaternions in physics

Shouldn’t the section of quaternions in physics be taken out of the history of quaternions? It seems to me as if the section doesn’t have much to do with history in the first place, and it could be greatly expanded on if taken out. Should I move it? IntegralPython (talk) 22:13, 8 June 2018 (UTC)

"Be Bold" 192.91.171.34 (talk) 14:56, 10 February 2019 (UTC)

non-commutativity and solutions to polynomial

I find the remark on the number of solutions and non-commutativity rather mysterious. Isn't the fact that the whole unit ball solves the root of unity equation a mere result of the fact that the quaternions are defined by four scalars, rather then anything about commutativity? Especially since z commutes with z anyway? --193.81.148.197 (talk) 04:54, 28 October 2018 (UTC)

Where is this remark? Anyway, the fundamental theorem of algebra implies that any polynomial equation over the reals (and more generally over any field (mathematics)) has no more solutions than its degree. Quaternions are the most common example of a division algebra. As division algebras differ from field only by their non-commutativity, it is interesting to know that commutativity is essential for the above property on the number of solutions, that is that the equation

x^{2}-1=0

has much more than two solutions. D.Lazard (talk) 08:58, 28 October 2018 (UTC)

In case you're interested, I suppose it's about here. It might be interesting how one could could blame failed commutativity for this in more succinct detail. Purgy (talk) 11:15, 28 October 2018 (UTC)

@User:D.Lazard: "Where is this remark?" In the section "Definition" the subsection "Center" and in there the last paragraph. I don't doubt its correct its just as @User:Purgy Purgatorio points out not all clear from the statement there how the two arguments possibly could be connected. Even in your explanation now, you mention that the fundamental theorem of algebra does not hold for that division algebra, why? And even if it doesn't hold it still could be that in this particular division algebra we have less than "much more" solutions. My point is the sentence in question makes one expect some short logical argument rather than something deeply hidden somewhere. --193.81.148.197 (talk) 14:39, 28 October 2018 (UTC)

Clarified. D.Lazard (talk) 15:15, 10 February 2019 (UTC)

Computational complexity of quaternion multiplication

There needs to be some analysis in this article of the comparative computational complexity of quaternion multiplication for rotation (in terms of the number of multiplications and additions needed) vs. the computational complexity of the equivalent multiplication by a rotation matrix, for both quaternion times vector ( $\mathbf {p'} =\mathbf {q} \mathbf {p} \mathbf {q} ^{-1}$ ) (c.f. matrix times vector) and quaternion times quaternion (c.f. matrix times matrix) multiplication. I also found this quaternion multiplication mechanism that claims to be faster than standard quaternion multiplication (presumably meaning requiring fewer multiplications), although the derivation pages are both broken, so maybe the Wayback machine can help. I don't know whether this faster formulation is as well-conditioned as standard quaternion multiplication. If someone wants to do the analysis, it would be a good contribution! — Preceding unsigned comment added by 66.60.126.246 (talk) 09:51, 22 August 2019 (UTC)

Rotation matrices would be a blatant off-topic here, but probably on-topic in rotation formalisms in three dimensions. Incnis Mrsi (talk) 07:07, 23 August 2019 (UTC)

Boldface or not for unit quaternion?

In the article, the unit quaternion are denoted either $i, j, k$ (mainly at the veginning of the article) or $i, j, k$ (in most of the article); one sees also $i, j, k$ .

AFAIK, $i, j, k$ is the most common notation, and it is the only one that is coherent with the usual notation for the imaginary unit. Thus, I'll be bold, and convert bf to italics. D.Lazard (talk) 16:27, 21 January 2020 (UTC)

Exponential, logarithm, and power functions

In absence of any argument in favor of θ⁄2 simply $θ$ is obviously preferable. No collaborative attitude ⇒ no respect for the Australian IP’s preferences. Incnis Mrsi (talk) 22:18, 11 August 2019 (UTC)

Not a single book uses $θ$ , everyone uses θ⁄2. Not fixing this mistake long time ago is a shame. I want to see where the original contributer got his information from.193.116.118.102 (talk) 05:01, 12 August 2019 (UTC)

^[1] 193.116.118.102 (talk) 05:04, 12 August 2019 (UTC)

References

^ Corke, P. (2017). Robotics, Vision and Control Fundamental Algorithms In MATLAB® Second, Completely Revised, Extended And Updated Edition (Second edition.). https://doi.org/10.1007/978-3-319-54413-7

Where s/he got “information”? The exponentiation formula for

| q | = 1

is a generalization of the Euler's formula, whereas generalization for an arbitrary

| q | > 0

is an obvious factorization. It is pretty trivial. In absence of objections I replace the angle variable with

φ

, assuming that the IP insists that

θ

is a well-known notation in this context. This article—let alone this specific section—is not about 3D rotations and halving the angle only brings an unwarranted sense of mystery. Incnis Mrsi (talk) 06:16, 12 August 2019 (UTC)

Also note that the Australian caused a notation conflict with the section immediately after. Incnis Mrsi (talk) 11:29, 12 August 2019 (UTC)

Why half the angle? Maybe this tradition comes from spinors in 3-dim space. (Just guessing.) When one rotates a spinor by θ⁄2 (in the 2-dim complex space of spinors), the corresponding vector in the 3-dim real space rotates by

θ

. This way, two orthogolal spinor state vectors correspond to "spin up" and "spin down". Boris Tsirelson (talk) 15:55, 12 August 2019 (UTC)

But I see, this agrument is already voiced in the edit summary of this edit. Boris Tsirelson (talk) 16:00, 12 August 2019 (UTC)

@Tsirel: half the angle because angles in the vector representation (

λ = 1

) a.k.a. “conjugation” are twice angles in spinor representations (

λ = ½

). Robot people namely learn vectors but not spinors – it’s pretty foreseeable. But I like the latter, dammit. Incnis Mrsi (talk) 16:19, 12 August 2019 (UTC)

Check the page Quaternions_and_spatial_rotation. it uses θ⁄2 and explains why. 193.116.118.102 (talk) 16:25, 12 August 2019 (UTC)

A mapping from rotation matrices to quaternions is two-valued (not uniquely defined). If we rotate vectors by

θ = \pm180°

, then what is θ⁄2: +90° or −90°? The same

diag(1, -1, -1)

rotation matrix, but different quaternionic versors. Perhaps Peter Corke knows about it, but likely not most readers of his book. We, mathematicians, won’t tolerate multivalued functions in a formula which describes as simple thing as exponentiation. Incnis Mrsi (talk) 08:40, 13 August 2019 (UTC)

@Arthur Rubin: ideas about letter for the variable: is $φ$ nice, or restore $θ$ , or something else? May we suspect that “ $θ$ ” brings a strong connotation namely with vector angles (as opposed to quaternion/spinor angles)? Incnis Mrsi (talk) 20:00, 12 August 2019 (UTC)

I suspect

θ

does connect to vector angles in this context. Perhaps

φ

would be better. — Arthur Rubin (talk) 20:43, 12 August 2019 (UTC)

It turned out that Corke actually used θ⁄2, with the letter theta, in the formula (2.22). May I deem that the

θ = 2 φ

thing satisfied everybody except the Australian user? Incnis Mrsi (talk) 08:40, 13 August 2019 (UTC)

I'm fairly certain the conversion to log quaternion is incorrect. You couldn't just add the angles if you divide by the square normal, the normalization of $\mathbf {v}$ in $\ln(q)=\ln \|q\|+{\frac {\mathbf {v} }{\|\mathbf {v} \|}}\arccos {\frac {a}{\|q\|}}~~$ should be $\ln(q)=\ln \|q\|+{\frac {\mathbf {v} }{\sum _{i=1}^{n}\left|v_{i}\right|}}\arccos {\frac {a}{\|q\|}}~~$ which is the [Manhattan normal], which then allows log quaternions to be added together directly. I've been working on implementing a system using log quaternions^[1], especially that have a `0` real part, so $\exp(0)=1$ and is a unit quaternion for rotation. ^[2] ^[3]

Rotation of log quaternions around any axis can be done with partial Rodrigues Formula... given a axis of rotation (normalized with a square normal - $sqrt(xx+yy+zz)$ ), and an angle of rotation to apply to the rotation; compute the square normal of the log-quaternion, again $v \over {\sqrt {(}}xx+yy+zz)$

   $q_{nX}={q_{X} \over {\sqrt {(}}q_{X}*q_{X}+q_{Y}*q_{Y}+q_{Z}*q_{Z})}$ 
   $q_{nY}={q_{Y} \over {\sqrt {(}}q_{X}*q_{X}+q_{Y}*q_{Y}+q_{Z}*q_{Z})}$ 
   $q_{nZ}={q_{Z} \over {\sqrt {(}}q_{X}*q_{X}+q_{Y}*q_{Y}+q_{Z}*q_{Z})}$ 
   $logQuat_{normal}=(q_{nX},q_{nY},q_{nZ})$ 
   $q_{sin}=sin({(|q_{X}|+|q_{Y}|+|q_{Z}|) \over 2})$ 
   $q_{cos}=cos({(|q_{X}|+|q_{Y}|+|q_{Z}|) \over 2})$

   $rotation_{axis}=(rot_{X},rot_{Y},rot_{Z})$ 
   $a_{x}=rotation_{axis_{X}}$ 
   $a_{y}=rotation_{axis_{Y}}$ 
   $a_{z}=rotation_{axis_{Z}}$ 
   $a_{sin}=sin(rotation_{angle})$ 
   $a_{cos}=cos(rotation_{angle})$

    $cos_{C_{over2}}=q_{cos}*a_{cos}-q_{sin}*a_{sin}*(logQuat_{normal}\cdot rotation_{axis})$ 
    $result_{angle}=cos^{-}1(cos_{C_{over2}})*2$ 
   this is A X q times cos(a+b)  added with sin(a+b) scaling the two axles...
    $result_{axis_{X}}=a_{sin}*q_{cos}*a_{X}+q_{sin}*a_{cos}*q_{nX}+q_{sin}*a_{sin}*(ay*q_{nZ}-a_{Z}*q_{nY})$ 
    $result_{axis_{Y}}=a_{sin}*q_{cos}*a_{Y}+q_{sin}*a_{cos}*q_{nY}+q_{sin}*a_{sin}*(az*q_{nX}-a_{X}*q_{nZ})$ 
    $result_{axis_{Z}}=a_{sin}*q_{cos}*a_{Z}+q_{sin}*a_{cos}*q_{nZ}+q_{sin}*a_{sin}*(ax*q_{nY}-a_{Y}*q_{nX})$ 
    $sin_{angle}=sin(result_{angle}/2);$  // same as sqrt(xx+yy+zz) of the result axis (x,y,z)
    $result_{normalizer}=sin_{angle}*(|result_{axis_{X}}/sin_{angle}|+|result_{axis_{Y}}/sin_{angle}|+|result_{axis_{Z}}/sin_{angle}|)$

and finally the resulting log quaternion:

   $q_{w}=0$ 
   $q_{x}=result_{axis_{X}}/result_{normalizer}*result_{a}ngle$ 
   $q_{y}=result_{axis_{Y}}/result_{normalizer}*result_{a}ngle$ 
   $q_{z}=result_{axis_{Z}}/result_{normalizer}*result_{a}ngle$

And I also have to disagree that the /2 is irrelevant.

D3x0r (talk) —Preceding undated comment added 11:45, 26 July 2020 (UTC)

References

Theorem Regarding Functions over the quaternions

Essentially, for any analytic/holomorphic function, $F$ , over $\mathbb {H}$ , there is a mathematical theorem saying that:

if $F(x+y\ \mathbf {i} )=u+v\ \mathbf {i}$ ,

then $F(x+y\ \mathbf {l} )=u+v\ \mathbf {l}$ , with $l$ being a quaternion satisfying $l^{2}=-1$ .

This is equivalent to saying that $x+y\ \mathbf {l}$ creates a copy of $\mathbb {C}$ in $\mathbb {H}$ .

Are there any sources we could cite for this, so we can add this to the article?— Preceding unsigned comment added by NegativeZ (talk • contribs) 16:50, 6 July 2022 (UTC)

See quaternion analysis for the topic of quaternion functions. Rgdboer (talk) 04:35, 7 July 2022 (UTC)

Written in pocketbook, not carved in stone

From page 441 of Lester Ward’s 1903 Pure Sociology:

To-morrow will be the fifteenth birthday of the Quaternions. They started into life, or light, full-grown, on the 16th of October, 1813, as I was walking with Lady Ilamilton to Dublin, and came up to Brougham Bridge, which my boys have since called Quaternion Bridge. That is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k; eractly such as I have used them ever since. I pulled out, on the spot, a pocketbook which still exists, and made an entry, on which, at the rery moment, I felt that it might be worth my while to expend the labor of at least ten (or it might be fifteen) years to come. But then, it is fair to say that this was because I felt a problem to have been at that moment solved, - an intellectual want relieved – which had haunted me for at least fifteen years before. [1]

1. North British Rerier, Vol. XLV (N.S., Vol. VI), September-December, 1NB, p. 57. Extract from a letter dated Oct. 15, 1858, giving an account of the discovery; in an article on Sir William Rowan Hamilton. — Preceding unsigned comment added by 2601:249:8E00:540:109C:CAC7:3E79:CAD (talk) 04:48, 10 August 2022 (UTC)

[1] Corke, P. (2017). Robotics, Vision and Control Fundamental Algorithms In MATLAB® Second, Completely Revised, Extended And Updated Edition (Second edition.). https://doi.org/10.1007/978-3-319-54413-7

[2] ttps://github.com/d3x0r/STFRPhysics

[3] ttps://arxiv.org/pdf/1810.08871.pdf

[4] ttp://www.acsel-lab.com/arithmetic/arith20/papers/ARITH20_Arnold.pdf

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[2]

[3]