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Quaternion Algebras
- bnolsen 9mo agoQuaternions are left handed spinors. Please use the correct unified math models (clidfford algebra) rather than these 3d only hacks.
- koolala 9mo agoDescribing those algebras in quaternion terms simplifies understanding most of them. Like biquaternions and dual quaternions. A 2d quaternion just has no j or k term and works for 2d math.
- zhug3 9mo ago[dead]
- conformist 9mo agoI like quaternions as much as the next guy (I’ve used them in numerical computations etc), but what is it about them that makes them show up on the front page every few weeks?
- ajkjk 9mo agoLots of engineers are lapsed physicists who have hobbyist interest in this stuff. Also lots of HN readers are actual physicists or mathematicians. It's not all techies. Also, lots of engineers have at some point learned some computer graphics and so been exposed to quaternions in that setting. Since they're mysterious and hard to wrap your head around most people don't really 'get it', leaving a sort of standing curiosity that articles like this tap into.
- socalgal2 9mo agoThat people haven't gotten used to geometry algebra yet.
- K0balt 9mo agoThey sound shiny and mysterious?
- Scene_Cast2 9mo agoThen why not geometric / Clifford algebra?
- woopsn 9mo agoWhy anything except universal algebra? I mean, granted, I myself dove right into topos theory...
- esafak 9mo agoThey get mentioned too. Things people would like to know more about.
- Sharlin 9mo ago"Quaternions" definitely sounds shinier and more mysterious than "geometric algebra". Indeed I can’t immediately come up with any math term more shiny and mysterious, except maybe "transcendental", but as a concept transcendentals are much more familiar to most than quaternions.
- RobotToaster 9mo agonon-Euclidean geometry is a disappointingly boring subject.
- bhadass 9mo agotrue on the naming, but i think geometric/clifford algebra has its own mysterious aura precisely because it can be framed as "suppressed" or "overlooked".. plus it genuinely does have elegant mathematical structure backing up the hype funny thing is quaternions had that exact same energy in the computer graphics community for years. after ken shoemake introduced them to CG in 1985, there was a long period of "why are we using euler angles like cavemen when this exists??". now quaternions are well known tooling for people in graphics and the mystique has worn off at least in that community.
- oceansky 9mo agoThe word "quaternion" just rolls of the tongue. I always upvote it.
- jesuslop 9mo agoBaez wrote some ideas in [1], one I'm liking connects Lorentz group in dimensions 3,4,6 and 10 with the modular group SL(2,Z) that is at a crossroads of several hardcore math themes. For Lie algebras: sl(2, R) ≅ so(2,1) sl(2, C) ≅ so(3,1) sl(2, H) ≅ so(5,1) sl(2, O) ≅ so(9,1) Dirac equation is the C case, the other cases have their uses. [1] https://arxiv.org/abs/math/0105155 https://arxiv.org/abs/math/0105155
- westurner 9mo agoThat quaternions also solve for what we normally have 3D+time for. And Lewis Carroll (Oxford (Math)), preferred Euclidean geometry over quaternions, for "Alice's Adventures in Wonderland" (1865). Quaternions: q = a + bi + cj + dk -1 = i^2 = j^2 = k^2 Summarized by a model: > In a quaternion, if you lose the scalar (a) — the "real" or "time" component — you are left with only the three imaginary components (i, j, k) rotating endlessly in a circle. (An exercise for learning about Lorentzian mechanics, then undefined: Rotate a cube about a point other than its origin. Then, rotate the camera about the origin.) 4D Quaternions (a + bi + cj + dk) are more efficient for computers than 3D+t Euclideans. Quaternions do not have the Gimbal Lock problem that Euclidean vectors have. Quaternions interpolate more smoothly and efficiently, which is valuable for interpolating between keyframes in a physical simulation. Why are rotations and a scalar a better fit? Quaternions were published by William Rowan Hamilton (Trinity,) in 1843, in application to classical mechanics and Lagrangian mechanics. Maxwell's (1861,1862) original ~20 equations are also quaternionic; things are related with complex rotations in EM field theory too. Oliver Heaviside then "simplified" those quaternionic expressions into accessible vectors. Is there Gimbal Lock in the Heaviside-Hertz vector field reinterpretation of Maxwell's quaternionic EM field theory? Maxwell's has U(1) gauge symmetry. And then quantum has complex vectors and some unitarity, too History of quaternions: https://en.wikipedia.org/wiki/History_of_quaternions https://en.wikipedia.org/wiki/History_of_quaternions
- westurner 9mo agoShouldn't there be symmetry and unitarity given energy conservation? And quaternions express this with rotations in SO(3), but is there a better model than quaternions for EM field theory since 1861? -1 = i^2 = j^2 = k^2 q = a + bi + cj + dk q = a + xi + yj + zk (Edit) Quaternions and spatial rotation: https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati... QED: Quantum electrodynamics: https://en.wikipedia.org/wiki/Quantum_electrodynamics https://en.wikipedia.org/wiki/Quantum_electrodynamics : > Mathematically, QED is an abelian gauge theory with the symmetry group U(1), defined on Minkowski space (flat spacetime). The gauge field, which mediates the interaction between the charged spin-1/2 fields, is the electromagnetic field. The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field in natural units gives rise to the action [...] And QED is the basis for the Standard Model of particle physics and for some theories of n-body quantum gravity.
- ajross 9mo agoBecause in 1985 Ken Shoemake dropped the idea like a bomb on the computer graphics industry and it changed the way hackers thought about rotations forever. https://www.ljll.fr/~frey/papers/scientific%20visualisation/Shoemake%20K.,%20Quaternions.pdf https://www.ljll.fr/~frey/papers/scientific%20visualisation/... I mean, there are practical reasons too (which are mostly just isomorphic to the stuff in the paper). But really that's why. It's part of our cultural history in ways that more esoteric math isn't.
- aardvarkdriver 9mo agoThank you for the reference, this is one of the more approachable sources for learning the nuts and bolts of quaternions that I've seen!
- phkahler 9mo agoBecause people like me use quaternions but have never attained a full understanding like 3x3 rotation matricies. I will be reading the above link since its only 12 pages and someone indicated its an easier read.
- stared 9mo agoIn short: scaling -> real numbers 1d rotations and scaling -> complex numbers 2d rotations and scaling -> quaternions In the case of quaternions, there is called double-covering, which turns out (rather than being an artefact), play fundamental role in particle physics.
- conformist 9mo agoYeah sure SU(2) up to sign is isomorphic to SO(3) and whatnot… I think it’s probably mostly the computer graphics history and the cool name that gets people excited about quaternions?
- deleted 9mo ago[deleted]
- the_fall 9mo agoHonestly, with all my love for the HN community, I think we have a couple of topics that just get upvoted without reading because they signal that you're in the ingroup. Few years back, another reliably upvoted thing was anything with "Bayesian" in the name. In the past couple of years, "busy beavers" would also get upvotes even though they have no practical use, their mathematical significance is dubious, and few people understand them in the first place.
- stared 9mo agoOne of perks of PhD of quantum physics is that quaternions get mundane, vide https://en.wikipedia.org/wiki/Pauli_matrices https://en.wikipedia.org/wiki/Pauli_matrices. SU(2) is everywhere (normalized quaternions). These are are more common than regular rotations of 3D space, O(3). Because SU(2) we get a lot of interesting phenomena, including that there are two types of particles, bosons and fermions. We get some interesting phenomena that only rotating by 720deg (two full rotations) bring back to the initial state. And I am not talking only about USB-A, but about spinors (https://en.wikipedia.org/wiki/Spinor https://en.wikipedia.org/wiki/Spinor) - there are some party tricks around that (vide https://www.reddit.com/r/physicsmemes/comments/181oldw/a_german_demonstrating_720_degrees_symmetry_for/ https://www.reddit.com/r/physicsmemes/comments/181oldw/a_ger...).
- colechristensen 9mo agoNeat, I didn't realize the SU(2) spinor stuff worked through quaternions. That's one of those things I've vaguely learned but really want to spend the time to learn academically so I can actually do the math instead of just hearing about it.
- wolfi1 9mo agothe late Doug Sweetser had in the 90s a website where he used quaternions (and quaternion analysis) for describing physics from his viewpoint. it was quit interesting. he had several github repositories [0] where he described his ideas. worth a look [0] https://github.com/dougsweetser?tab=repositories https://github.com/dougsweetser?tab=repositories
- carabiner 9mo agoaverage period between quaternion posts what about kalman filters?
- atultw 9mo agoKalman filters so real. I think part of it is that LLMs suggest it as a solution all the time