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 7 months agoI 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
MBM@lemmings.world 7 months ago
GnomeKat@lemmy.blahaj.zone 7 months ago
Oh no, you were right on the money. In G^2^ you have two basis vectors
e1
ande2
. The geometric product of vectors specifically is equivalent touv = 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, thedot
goes to 0 and you are just left withu wedge v
. Soe1e2
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 soe2e1 = e1e2
. Meaning you can rewrite the above ase1*(-e1e2)*e2 = -(e1e1)*(e2e2) = -(e1 dot e1)*(e2 dot e2) = -(1)*(1) = -1
… Thuse1e2
squares to -1 and is the same asi
. And now you can think of the geometric product of two vectors asuv = 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 bivectorse1e2, e2e3, e3e1
. You can flip them to bee2e1, e3e2, e1e3
without any issues its just convention and then its the same as quaternions. They all square to -1 ande2e1*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.
someacnt_@lemmy.world 7 months ago
Then you can extend to arbitrary algebra
ouRKaoS@lemmy.today 7 months ago
You lost me at “arbitrarily complicated,” sorry.
TexasDrunk@lemmy.world 7 months ago
Lost me at “I think”. I don’t, apparently.
embed_me@programming.dev 7 months ago
I don’t think therefore I never was
Asafum@feddit.nl 7 months ago
I wish it was that easy! I stopped thinking and I’m still stuck here… Stupid Descartes