tag:blogger.com,1999:blog-7068528325708136131.post5643242368507140281..comments2018-03-06T11:42:40.325-08:00Comments on Structural insight: The great vectors-versus-quaternions debateJohn Shutthttp://www.blogger.com/profile/00041398073010099077noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-7068528325708136131.post-31105700710095961592017-12-23T14:34:41.414-08:002017-12-23T14:34:41.414-08:00Few people have realized (and probably none of tho...Few people have realized (and probably none of those involved in the quaternions-vectors debate) that quaternions are the "square root" of an algebraic number identity: Leonhard Euler's four-squares identity.<br />The quaternion multiplication rule:<br />(a0+ia1+ja2+ka3)* (b0+ib1+jb2+kb3) =<br /> (a0b0-a1b1-a2b2-a3b3)+ <br />i(a0b1+a1b0+a2b3-a3b2)+ <br />j(a0b2-a1b3+a2b0+a3b1)+ <br />k(a0b3+a1b2-a2b1+a3b0). <br />if squared, considering that <br />(a*b)*(a*b)" = a*(b*b")*a" and<br />q*q"= q"*q = (q0^2 + q1^2 + q2^2 + q3^2) <br />yields EULER’s 4-square-identity:<br />(v0^2+v1^2+v2^2+v3^2)(w0^2+w1^2+w2^2+w3^2) = <br />(v0w0 - v1w1 - v2w2 - v3w3)^2 + <br />(v0w1 + v1w0 + v2w3 - v3w2)^2 + <br />(v0w2 - v1w3 + v2w0 + v3w1)^2 + <br />(v0w3 + v1w2 - v2w1 + v3w0)^2 . <br />(Proof of the identity by algebraic evaluation)<br /><br />Hence, working with quaternions, rather than with <br />vectors, provides the metaphysical security of being<br />backed by an algebraic number identity.<br /><br />Quaternions are isoclinic double-rotation operators, performing an isoclinic double-rotation-stretch operation around a point in 4-dimensional space. A four-dimensional double-rotation has a curl and a source part; the curl being a 3-dimensional left- or right-rotation, and the source being an in-out-, or an out-in-rotation involving the fourth dimension.<br /><br />Quaternion electrodynamics is straightgforward in 4-dimensional space; e.g. the gradient δ of the four-potential A yields directly the electromagnetic field in terms of source (E) and curl (B):<br />(δ0 + iδ1 + jδ2 + kδ3)*(A0 + iA1 + jA2 + kA3) =<br />((δ0A0 - δ1A1 - δ2A2 - δ3A3) + <br />i(δ0A1 + δ1A0 + δ2A3 - δ3A2) + <br />j(δ0A2 + δ2A0 - δ1A3 + δ3A1) + <br />k(δ0A3 + δ3A0 + δ1A2 - δ2A1)). <br />The first line vanishes identically under <br />Lorenz gauge, and the remaining 3 lines are the<br />components of E/c and B:<br />δ*A = i(E1/c + B1) +j(E2/c + B2) +k(E3/c + B3)<br />The square, (δ*A)*(δ*A)" equals (E⁄c)^2 + B^2<br />i.e. the mixed terms of E and B vanish identically.<br /><br />Edgar Mueller<br /><br /><br /><br /><br />Edgar Mullerhttps://www.blogger.com/profile/07863646381225436867noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-32306304949571936332017-06-23T01:59:46.274-07:002017-06-23T01:59:46.274-07:00see Hestines "Geometric Algebra", modern...see Hestines "Geometric Algebra", modern promotion of Clifford/Grassman ideas to see how Gibbs style Vectors can play nice with Rotors, Versors, "Quaternions"<br /> Vq isomorphic with Grassman Bivectors, the even subalgebra of 3D Geometric Algebra isomorphic to full Quaternions (up to a handedness convention) fredhttps://www.blogger.com/profile/15335139136937996010noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-44662230148184913782017-06-23T01:55:45.604-07:002017-06-23T01:55:45.604-07:00This comment has been removed by the author.fredhttps://www.blogger.com/profile/15335139136937996010noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-12629970958717838502016-04-03T21:08:23.734-07:002016-04-03T21:08:23.734-07:00/Simple/, but perhaps not so much /trivial/. See C.../Simple/, but perhaps not so much /trivial/. See <a href="https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction" rel="nofollow">Cayley–Dickson construction</a>.John Shutthttps://www.blogger.com/profile/00041398073010099077noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-23985169983565089262016-04-03T21:05:55.315-07:002016-04-03T21:05:55.315-07:00This comment has been removed by the author.John Shutthttps://www.blogger.com/profile/00041398073010099077noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-48009461691568349742016-04-03T19:06:07.371-07:002016-04-03T19:06:07.371-07:00My favorite insight about quaternions is a trivial...My favorite insight about quaternions is a trivial one (but then I'm a trivial sort of fellow) which justifies their alternate name of hypercomplex numbers: that an arbitrary rectangular quaternion a + bi + cj + dk can be rewritten as (a + bi) + (c + di)j, thus showing how to compose a hypercomplex number as the sum of a hyperreal (i.e. complex) number and a hyperimaginary number, the latter being a hyperreal number multiplied by j.John Cowanhttps://www.blogger.com/profile/11452247999156925669noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-63827837469589899652014-05-04T22:24:03.412-07:002014-05-04T22:24:03.412-07:00I really enjoyed this story! I know we used quart...I really enjoyed this story! I know we used quarternions in a 3D geospatial project I worked on in 2005, so they are seeing some use in industry. And the guy assigned to that part of the project seemed excited to get to spend some time studying them. Richard Toddhttps://www.blogger.com/profile/12940402476622678832noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-61156048467985814192014-03-29T17:50:53.235-07:002014-03-29T17:50:53.235-07:00It seems someone else, quite recently, also used q...It seems someone else, quite recently, also used quaternions as a topic for an academic paper assignment. <a href="http://thequaternionsdebate.blogspot.com/p/the-quaternion-debate.html" rel="nofollow">here</a>.John Shutthttps://www.blogger.com/profile/00041398073010099077noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-21541374179912926952014-03-22T05:46:11.071-07:002014-03-22T05:46:11.071-07:00Ah! Thanks. :-)
Hm, the url didn't come out a...Ah! Thanks. :-)<br />Hm, the url didn't come out as a link; let's see...<br /><a href="https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation" rel="nofollow">link</a>. :-)John Shutthttps://www.blogger.com/profile/00041398073010099077noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-58265900627874972472014-03-21T23:49:07.382-07:002014-03-21T23:49:07.382-07:00There's some discussion of the use of unit qua...There's some discussion of the use of unit quaternions to represent 3d rotations here: http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotationMarc Coramhttps://www.blogger.com/profile/03880982688942937603noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-27767597079670345082014-03-20T08:35:29.126-07:002014-03-20T08:35:29.126-07:00I've heard in recent years (but don't have...I've heard in recent years (but don't have a reference at my fingertips) some movie-quality CGI software uses quaternions because quaternion rotations don't produce the weird artifacts that can result from using Cartesian coordinates and trig functions.<br /><br />I've done enough pencil-and-paper algebra with quaternions to realize that it's very error-prone. For writing stuff out by hand, I can see the practical attraction of lining up your numbers in neat rows and columns (matrix algebra). I therefore got it into my head that, in order to make quaternions easy to work with, what's needed is software that makes back-of-the-envelope algebra easy to do flawlessly on a computer. My limited once-upon-a-time experience with Mathematica suggested it wasn't even in the right ballpark for the level of facility I wanted. I've struggled for years just to <i>describe</i> to others what my vision of this software is, but really I've long concluded there'd be no way I could possibly describe it clearly enough for a software team (or crowd) to implement, so the only way to realize it would be to write the software myself — and that would be such a monumental task that the only way a single person could possibly accomplish it would be to use a programming language with vastly more powerful abstraction support than any language I've seen or heard of. You may notice that for nearly all of the 28 years since I wrote this paper on quaternions, I've been dedicated to <a href="http://fexpr.blogspot.com/2013/12/abstractive-power.html" rel="nofollow">understanding abstraction</a> and <a href="http://fexpr.blogspot.com/2011/06/primacy-of-syntax.html" rel="nofollow">finding ways to vastly increase it</a>.John Shutthttps://www.blogger.com/profile/00041398073010099077noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-27255553325206435142014-03-20T07:43:08.194-07:002014-03-20T07:43:08.194-07:00So where is the high-quality quaternion library, p...So where is the high-quality quaternion library, preferably written in Scheme, that we all need to return to the ways of our beloved ancestors?<br /><br />After Grassmann gave up for good and all on getting recognized by mathematicians, he switched to historical linguistics, the "hardest" of all the historical sciences. He is still remembered for <a href="http://en.wikipedia.org/wiki/Grassmann%27s_law" rel="nofollow">Grassmann's Law</a>, which explains why we write the Enlightened One's name as "Buddha" and not "Bhuddha", and why the technical term for the compulsion to pull out one's own hair is <i>trichotillomania</i> and not <i>thrichotillomania</i> even though the Greek word for hair is <i>thrix</i>. See also Kevin Wald's song <a href="http://math.uchicago.edu/~wald/lit/laws.txt" rel="nofollow">"Bartholomae, Grassmann, and Grimm"</a>.John Cowanhttps://www.blogger.com/profile/11452247999156925669noreply@blogger.com