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xkforce@lemmy.world ⁨4⁩ ⁨weeks⁩ ago

• GnomeKat@lemmy.blahaj.zone ⁨4⁩ ⁨weeks⁩ ago

  I think you can make arbitrarily complicated roots if you move over to G^n^ which includes the R and C roots…

  For example (3e1e2e3e4)^2 = 9 in G^4^, complex roots are covered because e1e2^2 = -1 making it identical to i so G^n^ (n>=2) includes C.

  G^n^ also includes all the vectors so any vector with length 3 will square to 9 because u^2 = u dot u = |u|^2

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  • ouRKaoS@lemmy.today ⁨4⁩ ⁨weeks⁩ ago

    You lost me at “arbitrarily complicated,” sorry.

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    • TexasDrunk@lemmy.world ⁨4⁩ ⁨weeks⁩ ago

      Lost me at “I think”. I don’t, apparently.

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      • embed_me@programming.dev ⁨4⁩ ⁨weeks⁩ ago

        I don’t think therefore I never was

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  • MBM@lemmings.world ⁨4⁩ ⁨weeks⁩ ago

    I thought this would be related to quaternions, octonions etc. but no, it’s multivectors and wedge products. Very neat, I didn’t know you could use them like that.

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    • GnomeKat@lemmy.blahaj.zone ⁨4⁩ ⁨weeks⁩ ago

      Oh no, you were right on the money. In G^2^ you have two basis vectors e1 and e2. The geometric product of vectors specifically is equivalent to uv = u dot v + u wedge v… the dot returns a scalar, the wedge returns a bivector. When you have two vectors be orthonormal like the basis vectors, the dot goes to 0 and you are just left with u wedge v. So e1e2 returns a bivector with norm 1, its the only basis bivector for G^2^.

      e1e2^2 = (e1e2)*(e1e2) = e1e2e1e2

      A nice thing about the geometric product is its associative so you can rewrite as e1*(e2e1)*e2… again that middle product is still just a wedge but the wedge product is anti commutative so e2e1 = e1e2. Meaning you can rewrite the above as e1*(-e1e2)*e2 = -(e1e1)*(e2e2) = -(e1 dot e1)*(e2 dot e2) = -(1)*(1) = -1… Thus e1e2 squares to -1 and is the same as i. And now you can think of the geometric product of two vectors as uv = u dot v + u wedge v = a + bi` which is just a complex number.

      In G^3^ you can do the same but now you have 3 basis vectors to work with, e1, e2, e3. Meaning you can construct 3 new basis bivectors e1e2, e2e3, e3e1. You can flip them to be e2e1, e3e2, e1e3 without any issues its just convention and then its the same as quaternions. They all square to -1 and e2e1*e3e2*e1e3 = -e2e1e2e3e1e3 = e2e1e2e1e3e3 = e2e1e2e1 = -1 which is the same as i,j,k of quaternions. So just like in G^2^ the bivectors + scalars form C you get the quaternions in G^3^. Both of them are just bivectors and they work the same way. Octonions and beyond can be made in higher dimensions. Geometric algebra is truly some cool shit.

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  • someacnt_@lemmy.world ⁨4⁩ ⁨weeks⁩ ago

    Then you can extend to arbitrary algebra

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• Beetschnapps@lemmy.world ⁨4⁩ ⁨weeks⁩ ago

  To translate: As a child learning math this means “ignore math, the explanations don’t explain anything real, they only explain math. So instead focus on language and the arts.”

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  • overcast5348@lemmy.world ⁨4⁩ ⁨weeks⁩ ago

    I’m guessing that you were one of those “I won’t ever use all this math” kind of students?

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    • Aussiemandeus@aussie.zone ⁨4⁩ ⁨weeks⁩ ago

      Boy do I ever use maths at work all the damn time.

      And I’m a mechanic

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    • Beetschnapps@lemmy.world ⁨4⁩ ⁨weeks⁩ ago

      I was one of those students who asked how it would be used, the teachers didn’t do the whole real world application part, and I never needed to go past trig.

      I work with engineers and use math like any other human on the planet but really wish mathematics was taught differently to make it more interesting. You hear a PHD candidate talk about the hairy ball problem and the math is interesting. Math class never was.

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  • Maalus@lemmy.world ⁨4⁩ ⁨weeks⁩ ago

    Or you were just shit at maths and don’t have any idea how useful it is because you avoid it like the plague.

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