[HN Gopher] Quaternions and spherical trigonometry
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Quaternions and spherical trigonometry
Author : t55
Score : 68 points
Date : 2025-01-30 17:57 UTC (5 hours ago)
(HTM) web link (terrytao.wordpress.com)
(TXT) w3m dump (terrytao.wordpress.com)
| aap_ wrote:
| Interestingly showing spherical trig identities was how Hamilton
| demonstrated his quaternions to the royal irish academy when he
| found them. See (D) through (K):
| https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quater...
|
| When i first read this i found it very hard to understand,
| because i was unfamiliar with spherical trigonometry, but there's
| quite some beauty to be found there.
| incognito124 wrote:
| Related:
|
| https://eater.net/quaternions
| t55 wrote:
| wow love the fact that you can interact with the videos
| incognito124 wrote:
| That's what's most valuable to me too. If I remember
| correctly, Ben Eater (8bit CPU from scratch guy) and Grant
| Sanderson (3b1b guy) tried to do a little experiment on
| education with this guided-yet-interactive learning material
| back in 2018 (!!). As someone working in EDU sector of tech,
| I'm constantly amazed by what people come up with in order to
| explain things to other people (or even themselves). Whatever
| the experiment was, I'd say it was a success, and I hope we
| get to see more such material on other topics in future.
| FpUser wrote:
| What a beautiful and high quality presentations. All the
| praises to the author.
| dang wrote:
| Related:
|
| _Visualizing quaternions (2018)_ -
| https://news.ycombinator.com/item?id=38043644 - Oct 2023 (42
| comments)
|
| _Visualizing quaternions: an explorable video series (2018)_ -
| https://news.ycombinator.com/item?id=31083042 - April 2022 (15
| comments)
|
| _Visualizing quaternions: An explorable video series_ -
| https://news.ycombinator.com/item?id=18310788 - Oct 2018 (32
| comments)
| ge96 wrote:
| I still gotta grasp this for IMUs
| jdranczewski wrote:
| This was an invaluable resource 5 years ago when I was working
| on a summer research project making a ray tracing-based optical
| levitation simulator - initially it felt a bit insane to try to
| deeply understand this obscure bit of maths to implement
| rotations, but once it clicked it _clicked_. Quaternions ended
| up being a super neat formalism for writing and computing
| rotational equations of motion.
|
| https://github.com/jdranczewski/optical-levitation-raytracin...
| for my repo, and
| https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5165 for
| the rotational dynamics with quaternions.
| andrewfromx wrote:
| like drawing triangles on a beach ball!
| quantadev wrote:
| Just a tiny rant: In my view complex numbers are really about the
| concept of "orthogonality". The complex 'dimension' is orthogonal
| to the 'real' dimension, but anything in reality that's a
| continuum of values can be seen as a dimension, and therefore
| each one must have an orthogonal. That is, whenever you have a
| direction in a higher dimensional space (regardless of
| dimensionality) any vector will have a normal direction
| (perpendicular direction).
|
| What basic complex numbers represent is a way of doing rotations
| where something moves from one direction towards it's orthogonal.
| That's what Euler's Formula is about also, which shows the
| relationship of 'e' and 'i' in this of course.
|
| Now what Quaternions represents is the realization that if
| complex numbers have two components (real, imaginary) then we can
| treat each of those as a base vector and find a sort of 'next
| level up' orthogonality to each one individually.
|
| I'm not good enough at math/geometry to know if this kind of
| 'next level up' bifurcation of dimensionality extends up past
| Quaternions or not (like something called Octernions, 16ions,
| 32ions, 64ions, etc), but it seems like is would?
| hgomersall wrote:
| Geometric algebra would be what you're looking for here. This
| is a great intro to the topic:
| https://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-ar...
| AnIrishDuck wrote:
| Also the (provocatively titled) "Let's Remove Quaternions
| from every 3d Engine" [1]
|
| Spoiler alert: rotors are mechanically identical to
| quaternions, while being easier to understand. If you
| understand rotors, you understand quaternions. You can fit
| the laws you need to understand rotors on a business card.
|
| Plus, rotors abstract to higher and lower (well, there's only
| one plane and its two respective orientations in 2d, but
| still) dimensions.
|
| Complex numbers as planes (bivectors in GA parlance) has been
| the most mind-opening mathematical concept I've been exposed
| to in the last decade. The associated geometric product has
| helped me better understand concepts (like "handedness") that
| troubled me during undergrad engineering.
|
| 1. https://marctenbosch.com/quaternions/
| quantadev wrote:
| I had never even heard of rotors! Thanks for this. I
| watched that video. The video doesn't really explain how it
| extends to higher dimensions tho, that I could discern.
|
| I wonder how/if any of this can be applied to LLMs
| 'Semantic Space'. As you might know, Vector Databases are
| used a lot (especially with RAG - Retrieval Augmented
| Generation) mainly for Cosine Similarity, but there is a
| 'directionality' in Semantic Space, and so in some sense we
| can treat this space as if it's real geometry. I know a TON
| of research is done in this space, especially around what
| they call 'Mechanistic Interpretability' of LLMs.
| wanderingmoose wrote:
| I really like the welch labs series on imaginary numbers which
| covers the first part of what you talk about -- leveling up the
| notion of what a complex number is. Though his focus was more
| on solving simple equations with no real roots, but really
| detailing how/what is really going on.
|
| It is a great precursor to then thinking about quaternions
|
| https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX...
| quantadev wrote:
| That exact video is the one that always comes to my mind when
| thinking of YT videos on this! I've seen it years ago.
| Definitely worth a watch for anyone who hasn't see it!
| VyseofArcadia wrote:
| > I'm not good enough at math/geometry to know if this kind of
| 'next level up' bifurcation of dimensionality extends up past
| Quaternions or not (like something called Octernions, 16ions,
| 32ions, 64ions, etc), but it seems like is would?
|
| Octonions and up (more generally known as hypercomplex numbers)
| exist, but every time you pull the "double dimensions by adding
| more imaginary components" trick[0], you lose another useful
| property.
|
| Real to complex loses total ordering. Complex to quaternion
| loses commutativity. Quaternion to octonion loses associativity
| (but they are at least alternative). The sedenions aren't even
| alternative, and they have zero divisors to boot.
|
| You can also generalize hypercomplex numbers to the study of
| Clifford algebras.
|
| [0] The Cayley-Dickson construction
| Sniffnoy wrote:
| Octonions and so on up are indeed a thing, but I don't think
| they do what you want. Even aside from the fact that they're
| restricted to power-of-2 dimensions, their algebraic properties
| get worse as you iterate the Cayley-Dickson process. The
| octonions aren't even an associative algebra, although they do
| have some weaker associativity properties which I'll skip
| detailing here. The quaternions are as far as most
| mathematicians are willing to go -- non-commutativity is
| commonplace, but who wants to deal with non-associativity?
|
| But while the octonions at least have some mathematical
| relevance (they're actually connected to various exceptional
| objects, such as the exception Lie group G_2!), the sedenions
| and beyond basically don't. They have a _tiny_ bit of
| associativity but not enough that they connect to any things or
| that hardly anyone wants to study them -- and worse yet, there
| are zero divisors so cancellation (ab=ac = > b=c for nonzero a)
| doesn't even hold. (Inverses exist, yes, but without
| associativity, inverses don't imply cancellation! And therefore
| aren't much use.)
|
| As another commenter mentioned, what you might be looking for
| instead if it's orthogonality you're focused on is Clifford
| algebras (aka geometric algebra). However, if you want to get
| the complex numbers or quaternions out of it, you'd need to use
| a _negative_ -definite quadratic form -- if you use a positive-
| definite one, you'd instead get the split-complex numbers,
| which are much less interesting (and you'd get something
| similar instead of the quaternions).
| nh23423fefe wrote:
| I don't agree. Complex numbers are the algebraic closure of the
| reals. Or the quotient of the real polynomial ring by
| (x^2+1=0). Or whatever other construction. The multiplication
| rule is the essence of C.
|
| Orthogonality is captured linear algebra over R^2, but R^2
| isn't a field or an algebra.
| markisus wrote:
| I think there is still a geometric viewpoint you can bring to
| the multiplicative structure of C. For example there is the
| extremely natural homeomorphism between unit C and SO(2). And
| C minus origin to (R+, SO(2)). It's completely intuitive for
| mathematicians to say that 1 and i are separated by 90
| degrees.
| alfiedotwtf wrote:
| Hold up! Omg, can someone who's done physics chime in please...
| whenever I've looked at GUT etc, I've always seen U(n), SU(n),
| but never knew what they were - are they what's referred to in
| this article? Is that just the Unitary Group and Special Unitary
| Group??! All that time I thought it was all impenetrable but it's
| just algebra?
|
| Omg wow... the theoretical physics I'm talking about is just
| quaternions and Lie Algebra isn't it? Oh... dont tell me Quantum
| Spin just called Spin because it's a Spinor rather than something
| actually metaphorically spinning?!
|
| Please chime in if you know what I'm talking about and can
| confirm this or shoot it down.
| particleguy wrote:
| Yes, Quantum Field Theory can be explained through Lie groups.
| SU(2) is isomorphic to the quaternions of norm 1, and SU(2) is
| important if you want to understand the Lorentz group and
| Poincare group, which represent the symmetries of spacetime and
| special relativity. Check out the text book Physics From
| Symmetry by Jakob Schwichtenberg if you would like an approach
| that derives modern physics primarily from algebra
| alfiedotwtf wrote:
| OMG thank you. Purchasing right now!
|
| You DO NOT understand how happy I am right now. Truely!
|
| I did general physics for a year at uni as part of my
| Computer Engineering course, then switching to Computer
| Science where I picked up a year of quantum mechanics. Since
| then whenever I lay in bed and thought about physics I would
| end up awake for hours. So damn interesting but the maths
| always held me back, so sadly gave up.
|
| I don't know what's changed (maybe maturity or maybe Vyvanse
| lol) but I'm slowly putting the pieces together. It's always
| been in my outer periphery but still out of reach. Your
| confirmation has and will change my life. Maybe not career
| wise or life altering seen from the outside, but hot damn you
| have at least cleared my constant nagging guilt for not
| perusing maths and physics because you've just made it
| slightly closer within reach. Can't wait for the book to
| arrive. Thank you!!!
| aap_ wrote:
| To understand spin it's good to consider a gyroscope. when it
| has a lot of angular momentum there are two stable states for
| it in a gravitational field: aligned or anti-aligned with
| gravitation, up or down. in all other cases the gyroscope
| precesses. a spinor doesn't spin quite like a gyroscope but it
| is spinning in a sense (after all spin is angular momentum).
| but just like the gyroscope, you can think of it as having two
| stable states: in alignment with a magnetic field, or in anti-
| alignment. and because the magnetic field is a measure for some
| kind of rotation it can add to or subtract from the angular
| momentum of a spinor. this difference is "felt" as a negative
| or positive potential difference this you can think of as two
| opposite forces on the spinor that split it apart into up and
| down components. the interesting thing is that a spinor with an
| arbitrary axis can _always_ be written as the sum
| /superposition of an up and down spinor for some chosen
| direction. turns out quaternions have precisely the properties
| that you need to model this. i hope this was intelligible, it's
| a bit hard to put the geometry into words.
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