Quaternions and spherical trigonometry | Hacker News Reader

Quaternions and spherical trigonometry

terrytao.wordpress.com

124 pointst5548 comments

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.

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.

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

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!!!

I just want to say I'm rooting for you, and hope you enjoy the book and learn a lot from it.

I had a bad experience with complex analysis as a teen (took a grad class that was a bit over my head). Many years later, I got Tristan Needham's "Visual Complex Analysis" and the whole thing clicked for me - I'm a visual person and do a lot of geometry. I hope your experience is similar.

haha thank you!

Awesome :) Yes, I saw that book this morning on my Amazon travels... though I'll put it in my wishlist because I think this morning blew out my yearly book allocation lol.

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 of 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.

Woah. Ok, that model totally blew my mind. I never understood what angular momentum had anything to do with it, but the gyroscope analogy makes total sense. Thank you!

Sounds interesting - would you have a good intro book on spinors you could recommend?

Unfortunately i can't. I've been thinking about spinors and related concepts as a hobby for maybe two years now. But i never found an explanation like mine above anywhere. Good geometric intuition about these things does not seem to be highly valued in physics education so people don't seek it. It turns out you can represent a dirac spinor (as the sum of left and right weyl spinors) extremely well just with your two hands. the dirac matrices simply tell you how to rotate your hands or mirror them (which you can do because you have two!). I should make a video about this actually...

> I should make a video about this actually...

sounds good, I've had a look at a few books but would be good to see something like that

> Oh… dont tell me Quantum Spin just called Spin because it’s a Spinor rather than something actually metaphorically spinning?!

That's a great way to understand them if you are already comfortable with the algebra of spinors and spin groups, but it doesn't short-circuit the history—spinors were so called after quantum spin (https://en.wikipedia.org/wiki/Spinor#History), and I believe that was so called because, yes, it was envisioned as something at least conceptually spinning.

Interesting.

I've tried many times to go through Modern Algebra texts, and so I on-the-surface get that it's an algebra of sorts.

When I have enough time I'm going to finally go through mathacademy.com, because I think it really does suck not knowing advanced maths but want to do the hard sciences

Quantum spin is an intrinsic angular momentum of a particle. It's angular momentum that is 'just there' as a key component of the particle.

In early days it was hypothesised that particles were spinning about their own axes, but this isn't accurate.

All the interesting stuff of Spin from its quantizable nature, the non-commutatability of spin measurements along orthogonal directions, the very different fundamental behavior of particles with half-integer spin (Fermions, eg Electrons, Protons) vs integer spin (Bosons eg Photons), how Spins interact (eg spins of say two electrons with half-integer spin interacting as a Spin-0 Boson in a Cooper Pair of a superconductor), or spin interacting with orbital angular momentum eg electron spin interacting with it's orbit around proton in an atom.

At the end of the day Spin isn't a terrible name for it.

I believe Spinor vectors are merely named after the eigenvectors used to represent spin itself, not the other way around as you suggested.

I think we maybe saying the same thing (unless I'm not reading that right) - Spin is not named because it's physically spinning (ok intrinsically perhaps, but that still isn't intuitive to me) but because the way we measure its interactions it's easier to describe using Spinors?

You can certainly use quaternions (as Hamilton demonstrated) or a Clifford algebra to recover the spherical trigonometry laws, but plain vectors work too in a short derivation. It is actually one of the simple exercises introducing reciprocal bases in Louis Brand's book Vector and Tensor Analysis (https://archive.org/details/vectortensoranal00branrich) or its abridged version, Vector Calculus.

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?

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...

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/

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.

> The video doesn't really explain how it extends to higher dimensions tho, that I could discern.

The neat thing is that it "extends" automatically. The math is exactly the same. You literally just apply the same fundamental rules with an additional basis vector and it all just works.

MacDonald's book [1] proves this more formally. Another neat thing is there are two ways to prove it. The first is the geometric two-reflections-is-a-rotation trick given in the linked article. The second is straightforward algebraic manipulation of terms via properties of the geometric product. It's in the book and I can try to regurgitate it here if there's interest; I personally found this formulation easier to follow.

If you really want your mind blown, look into the GA formulation of Maxwell's laws and the associated extension to the spacetime (4d) algebra, which actually makes them simpler. That's derived in MacDonald's book on "Geometric Calculus" [2]. There's all kinds of other cool ideas in that book like a GA formulation of the fundamental law of calculus from which you can derive a lot of the "lesser" theorems like Green's law.

Take all of this with a grain of salt. I'm merely an enthusiast and fan, not an expert. And GA unfortunately has (from what I can tell) some standardization and nomenclature issues (e.g. disagreement over the true "dot product" among various similar but technically distinct formulations)

> I wonder how/if any of this can be applied to LLMs 'Semantic Space'.

Yeah, an interesting point. Geometric and linear algebra are two sides of the same coin; there's a reason why MacDonald's first book is called _Linear and_ Geometric Algebra. In that sense, Geometric Algebra is another way of looking at common Linear Algebra concepts where algebraic operations often have a sensible geometric meaning.

1. https://www.faculty.luther.edu/~macdonal/laga/ 2. https://www.faculty.luther.edu/~macdonal/vagc/

Interesting ideas there thanks. I do know about that Maxwell derivation that involves Minkowski space, Lorentz transform consistency, etc, although I haven't fully memorized how it works, so that I can conjure up how it works from memory. I don't really think in equations, I think in visualizations, so I know a lot more than I can prove with math. You're right it's mind-blowing stuff for people like us that are interested in it.

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...

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!

> 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

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).

From your post I can tell you're way better at math/geometry than me, but I understood 80% of that. :)

Cayley-Dickson is interesting especially for Physics of course, because it brings in the concept of 'variable dimensions'. I think the flattening of objects, and the stopping of clocks (in Relativity), due to Lorentz effects in Minkowski space both on Black Hole Event Horizons and for objects approaching light speed (anywhere Lorentz holds) is, at the limits, ultimately the loss of a dimension, which would be my overall interpretation of what Cayley-Dickson is about too, in very broad terms.

So if Minkowski space is 4 dimensional, there would be some geometry for a 5-Dim Minkowski and it would use Octonians maybe, and that would be the geometry of the universe our universe is "embedded in"...I mean assuming of course you believe our universe is a Black Hole and we are all on an Event Horizon embedded in a 5D universe. Ya know, as one does. lol.

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.

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.

Indeed, there is a polar coordinates representation of the complex numbers with the nice property that when you multiply you multiply lengths and add angles.

Sniffy nailed it (above) when he mentioned Cayley-Dickson. That's exactly what I was getting at, and had never researched.

You can just start with N orthogonal directions which is R^N. There's no need to use the Cayley-Dickson construction to get orthogonality.

Algebraically, and also in group theory, it's exactly that. A complex plane is re-interpreted as a half-way mirror operation on an orthogonal axis or quadrature plane (eg the imaginary axis in the case of the 2D Argand plane). Which is what you you get when you multiply by i in complex numbers or i/j/k in quaternions - a 90 degree rotation. The correct way to do arbitrary rotations in this case is to use an exponential process, and then you get Euler's equation, the quaternion symmetry operation or, in general, the exponential map of an infinitesimal transformation in a Lie group.

In higher dimensions you get other type of (sometimes weird) operations, related to the Cartan–Dieudonné theorem.

https://eater.net/quaternions

wow love the fact that you can interact with the videos

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.

What a beautiful and high quality presentations. All the praises to the author.

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)

I still gotta grasp this for IMUs

In general, the better sensor fusion algorithms incorporate kalman filtering.

https://www.olliw.eu/2013/imu-data-fusing/#chapter23

Best of luck =3

Yeah something else I need to do. lazy question, if I have an IMU that is swaying around as I try to move in a linear direction (e.g. Forward) is that something this kind of filtering would be used for? Regarding displacement estimation.

Edit: I get fusion is regarding multiple sensors

Normally, there are several types of sensors available, but most use 9DOF packages (3-axis gyroscope, accelerometer, and magnetometer)

gyroscope: fast over-sampled low-pass filter, but slowly drifts compounding heading errors

accelerometer: relatively stable, but dead-reckoning errors compound quickly

magnetometer: best stability, but low-sample rate and vulnerable to metal/magnets fooling/blinding the sensors

The fusion algorithms usually weights which data is consistent with the motion path, and attenuates the estimated pose errors.

Notably, not all sensors are equal quality, but there are probably better options now. =3

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.

FYI - if you're like me, you just looked at the page thinking it had some links to 3blue1brown videos. Not obvious that those are thumbnails link to interactive video apps. Very cool.

like drawing triangles on a beach ball!

Reply on news.ycombinator.com