wa wa wa

  • 0 Posts
  • 64 Comments
Joined 1 year ago
cake
Cake day: July 12th, 2023

help-circle

  • That’s low key the charm of it. I use to only use reddit and it confused me too at first. I never used Twitter but on mastodon you can actually make friends with people. Friends who will discuss the topics you are interested in. I have queer friends on there and gamedev friend and math friends. We are interested in the same things, we boost posts that we all are interested in. Make general often vague looking posts because we know who will see it, like a long drawn out conversation. Its so much cozier than lemmy or reddit, here everyone is trying to correct each other or yell at each other or be the cleverest comment in the thread. On mastodon people are alot more interested in being authentic, thats been my experience of mastodon at least.













  • Its the algebraic properties that are important, not all vectors are n-tuples, eg the set of polynomials of degree less than n.

    You need a basis to coordinate a vector, you can work with vectors without doing that and just deal with the algebraic properties. The coordinate representation is dependent on the basis chosen and isn’t fundamental to the vector. So calling them n-tuples isn’t technically correct.

    You can turn them into a set of coordinates if you have a basis, but the fact that you can do that is because of the algebraic properties so it’s those properties which define what a vector is.





  • Gnome Kat@lemmy.blahaj.zonetoScience Memes@mander.xyzteachings
    link
    fedilink
    English
    arrow-up
    2
    ·
    edit-2
    7 months ago

    Oh no, you were right on the money. In G2 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 G2.

    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 G3 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 G2 the bivectors + scalars form C you get the quaternions in G3. 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.



  • Gnome Kat@lemmy.blahaj.zonetoScience Memes@mander.xyzteachings
    link
    fedilink
    English
    arrow-up
    5
    arrow-down
    1
    ·
    edit-2
    7 months ago

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

    For example the grade 4 blade (3e1e2e3e4)^2 = 9 in G4

    Complex roots are covered because the grade 2 blade (e1e2)^2 = -1 making it identical to i so Gn (n>=2) includes C.

    Gn also includes all the scalars (grade 0 blades) so all the real roots are included.

    Gn also includes all the vectors (grade 1 blades) so any vector with length 3 will square to 9 because u^2 = u dot u = |u|^2 where u is a vector.

    All blades will square to a scalar but blades are not the only thing in Gn so things get weird with the multivectors(sums of different grades). Any blade with grade n%4 < 2 will square to a positive scalar and the other grades will square to a negative, with the abs of the scalar equal to the norm2 of the blade. Can pretty much just make as many roots as you want if you are willing to move into higher dimensional spaces and use a way cooler product.