Quaternion Algebras

2026-01-275:068937jvoight.github.io

The official version on Springer's website is open access. [Link] Quaternion algebras, official version Errata and addenda [PDF] Errata and addenda for first edition (2021 printing) Current version…

The official version on Springer's website is open access.

[Link] Quaternion algebras, official version

Errata and addenda

[PDF] Errata and addenda for first edition (2021 printing)

Current version

The errata and addenda have been incorporated into the following (up to date) version.

[PDF] Quaternion algebras, post-publication version (v.1.0.6u, October 6, 2025)

The stable version kept for those who need a version that is updated less often.

[PDF] Quaternion algebras, stable post-publication version (v.1.0.5, January 10, 2024)

Supplementary material

[PDF] Quaternion algebras companion

Archived versions

The following drafts are just for archival purposes--you will almost surely want one of the versions above.

[PDF]v.0.9.28, May 20, 2021
[PDF]v.0.9.23, August 2, 2020 [PDF]v.0.9.15, May 26, 2019 [PDF]v.0.9.14, July 7, 2018 [PDF]v.0.9.2, April 18, 2017


Read the original article

Comments

  • By stared 2026-01-3114:401 reply

    One of perks of PhD of quantum physics is that quaternions get mundane, vide 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) - there are some party tricks around that (vide https://www.reddit.com/r/physicsmemes/comments/181oldw/a_ger...).

    • By colechristensen 2026-01-3116:41

      Neat, 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.

  • By conformist 2026-01-3113:379 reply

    I 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?

    • By ajkjk 2026-01-3118:371 reply

      Lots 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.

      • By bee_rider 2026-01-3119:121 reply

        I wonder if it is also some engineers who are nostalgic for the feeling of hard things becoming easier that complex numbers provided.

        • By fragmede 2026-01-3119:48

          omg yes! The absolutely ridiculous algebra that gets avoided by "simply" introducing i is astonishing!

    • By jesuslop 2026-01-3114:411 reply

      Baez 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

      • By AlecBG 2026-01-3116:341 reply

        You can describe spinning particles in flat space and AdS space really nicely with twistors using this

        • By jesuslop 2026-02-043:27

          Wish I knew better. Any reading suggestion for the concrete connection you made?

    • By oceansky 2026-01-3114:37

      The word "quaternion" just rolls of the tongue. I always upvote it.

    • By ajross 2026-01-3114:532 reply

      Because 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/...

      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.

      • By phkahler 2026-01-3117:20

        Because 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.

      • By aardvarkdriver 2026-01-3115:291 reply

        Thank you for the reference, this is one of the more approachable sources for learning the nuts and bolts of quaternions that I've seen!

    • By westurner 2026-01-3114:492 reply

      That 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

      • By westurner 2026-01-3118:21

        Shouldn'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_rotati...

        QED: 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.

      • By koolala 2026-01-3120:04

        Hamilton came up with the term vector for the 3 component part of the quaternion.

    • By K0balt 2026-01-3113:412 reply

      They sound shiny and mysterious?

      • By chpatrick 2026-01-3114:191 reply

        I know you know, just practical intuition for 3D graphics in case someone finds it useful:

        There's a 1-1 mapping between complex numbers and 2D rotation matrices that only do rotation and scaling. The benefit is that the complex number only has two coefficients, not four like the matrix. Multiplying these complex numbers is the same as multiplying the equivalent matrices. Quaternions are the same idea just in 3 dimensions (so with 3 imaginary units i j k, not just i, one per plane).

        • By Frotag 2026-01-3115:29

          > Quaternions are the same idea just in 3 dimensions (so with 3 imaginary units i j k, not just i, one per plane).

          I justify quaternions to myself with the intuition from [1]. In essence quaternions represent rotations in 4D, where multiplying by a "unit" (i,j,k), rotates two distinct planes by 90 degrees. The reason introducing a single unit j doesn't work is the same reason this rotation-is-multiplication trick doesn't work in 1D (or really any odd-number of dimensions). Anyways if we call this 4th axis w and pick a simple rule like ij = k then we get some nice properties like

              - multiplying by i rotates xy + zw planes by 90 degrees
              - j rotates xz + yw
              - k rotates xw + yz
              - 1 rotates nothing
          
          Notably this definition covers all 6 unique planes. But if we want to rotate only a single plane, we have to make up a new property, something that lets us rotate say xz by 90 and yw by -90. So we make up another rule that multiplying by a unit on the right does this, which algebraically looks like ij = -ji. This is incidentally why the rotation formulas have 1/2 everywhere, because if we want to rotate xy by 90, we multiply on the left by i/2 then on the right by -i/2.

          [1] https://www.reedbeta.com/blog/why-quaternions-double-cover/

      • By Scene_Cast2 2026-01-3113:543 reply

        Then why not geometric / Clifford algebra?

        • By esafak 2026-01-3119:23

          They get mentioned too. Things people would like to know more about.

        • By woopsn 2026-01-3118:53

          Why anything except universal algebra? I mean, granted, I myself dove right into topos theory...

        • By Sharlin 2026-01-3114:102 reply

          "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.

          • By bhadass 2026-01-3117:20

            true 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.

          • By RobotToaster 2026-01-3115:14

            non-Euclidean geometry is a disappointingly boring subject.

    • By stared 2026-01-3114:571 reply

      In 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.

      • By conformist 2026-01-3115:161 reply

        Yeah 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?

    • By the_fall 2026-01-3116:43

      Honestly, 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.

    • By socalgal2 2026-01-3119:50

      That people haven't gotten used to geometry algebra yet.

  • By wolfi1 2026-01-3115:02

    the 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

HackerNews