5 ms·
Possibly off-topic: why does practically any description of quaternions include the anecdote about somebody carving something into some bridge in Ireland? I’ve
by umanwizard 8y ago
Possibly off-topic: why does practically any description of quaternions include the anecdote about somebody carving something into some bridge in Ireland?
I’ve learned about plenty of mathematical concepts while having no idea who discovered them or under what circumstances. Why are quaternions the exception?
- brandonmenc 8y ago> somebody
- umanwizard 8y agoWhat exactly is your point? I said "somebody" for effect, not because I don't know who Hamilton was.
- twtw 8y agoI don't know, but here are my ideas: - quaternions maybe have a more interesting backstory than other constructs. Not everything was carved into stone. - quaternions never really became part of mainstream mathematical education. This makes them more niche and strange, and worth telling a story about. It's not as interesting to tell a story about something commonplace.
- Entalpi 8y agoPeople should read up on Galois theory and the story behind it. It involves a french youth, the french revolution, groundbreaking maths, and tangentially Poisson, Fourier and Cauchy, Gauss, Jacobi. His life should be a movie. :-)
- GlenTheMachine 8y agoMy doctoral dissertation used quaternions to describe spacecraft attitudes. My advisor was a stickler for not only appropriately citing references, but citing the most original (in the sense of oldest) references. This was something of a problem because the original references to some of the mathematical techniques I used were several hundred years old and not in English, and under the theory that a big part of the purpose of a citation is to help the reader understand the backgroud material this seemed a bit excessive. Nobody is going to learn French in order to read about Lagrange multipliers from Lagrange's original papers. But for quaternions, it was easy: I actually cited the Brougham Bridge inscription. One cannot, of course, check the bridge out of the engineering library to check the citation, but clearly this was the original “publication” of quaternion multiplication. My advisor finally got the point.
- ModernMech 8y ago> Nobody is going to learn French in order to read about Lagrange multipliers from Lagrange's original papers. You'd think, but I had a professor in physics who learned German just so he could read Boltzmann's original works.
- twtw 8y agoIt used to be common in hard sciences and mathematics for English-speaking doctoral candidates to be required to show some aptitude for German or Russian. Maybe still is, dunno.
- joeberon 8y agoIt’s definitely not
- dboreham 8y agoI believe German was required in order to study Chemistry back in the day.
- hprotagonist 8y ago
- nj65537 8y agoI think the story is told as a sort of cultural marker of what a Big F-ing Deal this discovery was/is. Mathematics, collectively, struggled for a long time to find a way to make 3-dimensional numbers into an algebra in a way that extends the algebra of complex numbers. (The cross and dot products are unsatisfying, because they don't have division.) The shock that this can be done in four, -- not three -- dimensions is still sort of reverberating, and that's what I think this story marks. It's a short stand-in for the longer story I've just summarized, and it evokes (or is meant to evoke) the mind-shattering thrill of discovery. I don't necessarily think the story accomplishes this -- your question is but one piece of evidence that it doesn't -- but I think for those who spend a good amount of time with these kinds of algebra questions, it comes to take on that role, and that's why I think it's repeated. (Teaser -- if you want to know more about these kinds of questions, Google for "real division algebras". There are not very many, and they way they are organized is not, I think, something one would expect.)
- jf- 8y ago> To find a way to make 3-dimensional numbers into an algebra in a way that extends the algebra of complex numbers What does that mean? My understanding was that Hamilton was searching for a way to make the manipulation of points in space easier, such as rotation, and noticed that the imaginary part of the complex numbers could be manipulated in the way he wanted. He then created a rather artificial tool in the form of the quaternions that allowed this.
- xelxebar 8y agoBy "extends" here we formally mean "normed division algebra". Basically, we want any 2D slice of our space to be equivalent to the complex numbers. This is analogous to how the reals embed into the complex plane or how slices of vector spaces are still vector spaces. We don't want things to depend on a particular basis (i.e. implementation). Anyway, it's pretty easy to make up some multiplication on 3D vectors, like multiplying their components. However, in general, it won't play nicely with such arbitrary 2D slices. As it turns out, this slicing property is equivalent to having multiplication play nicely with vector norms: |ab| = |a| |b|. that is, multiplication of vectors multiplies their lengths. Getting a multiplication with this property is the hard part, per se, and is only possible in dimensions 1, 2, 4, and 8.
- jf- 8y agoPeople think it’s a cute story and there’s a plaque on the bridge. It doesn’t need to be anything more than that.
- samontar 8y agoIt wasn’t when we learned it in high school, but it’s just that it’s a story that’s easy to tell. Just like the Königsberg reference for Eulerian graphs; Kekule and the Benzene ring; Lorenz, the weather, the butterfly, and deterministic chaos.
- umanwizard 8y agoThanks - out of all the replies to my comment, I think this is one of the most plausible hypotheses. I indeed think the Konigsberg and benzene stories are as commonly trotted out as the Hamilton one.
- garmaine 8y agoSurely you read about Galois fatal dual? I think it's just that these are instances where there is an interesting story attached.
- KboPAacDA3 8y agoAnd why do descriptions of quaternions include a rigorous proof? I trust quaternions work, so skip the proof and show me how to use them in a practical, efficient way.
- umanwizard 8y agoIn mathematics often proofs lead to understanding why something works (on the other hand, many proofs aren’t written in a way that make the main ideas clear or easy to tease out from details)
- joppy 8y agoThe fact that the quaternions “work” (are a finite-dimensional real vector space with an associative real-bilinear multiplication, and a division operation) is quite special. In fact there are only three objects satisfying this criteria: the real numbers themselves, the complex numbers, and the quaternions.
- j1vms 8y ago> there are only three objects satisfying this criteria: the real numbers themselves, the complex numbers, and the quaternions. Does a proof exist that these are the only three of such objects?
- billfruit 8y agoVery curious indeed why it is so, esp since there is normally zero historical background explained about vectors and matrices.
- beautifulfreak 8y agoMaybe you haven't heard of Oliver Heaviside. https://en.wikipedia.org/wiki/Oliver_Heaviside https://en.wikipedia.org/wiki/Oliver_Heaviside
- Retra 8y agoThere are millions of abstractions in mathematics, and the truth is that the human brain has a much better time remembering stories than it does math. You teach the stories so people have some context to associate complex ideas with.