Post making contest 6.0

I've finally taken the time to really dig in and understand SU(2) and SO(3), and I have to say, it's fucking awesome. This is definitely one of my new favorite things. I mean look at this shit, somehow numbers like this:

[a + bi, -c+di]
[c + di, a - bi]

Are the same thing as numbers like this (!):

w + xi + yj + kz

Which are the same thing as good ol' ordinary rotation matrices (but twice!):

[angles, numbers, sin()]
[theta, stuff, 0]
[cos(-x), -1, 3] (rotation matrices are a pain to write)

Honestly even just the fact that the first type stays as the first type is really cool, let alone that it's a valid interpretation of quaternions. From their I knew that quaternions could represent positions/rotations in 3D space, but I had no idea that there was that extra half, or that that extra sign was what led to spinors, and in turn, the entire concept of half-integer spin in general, and from there, the concepts of mass and the pauli exclusion principle!!!!!!!

and since the whole SU(2)/SO(3) thing only really happens in 2 & 3 dimensions, I think that means that half-integer spin, and therefore massive particles, might only be possible in universes with three dimensions! or vice versa, their very existence is what the dimensions come from in the first place


fuck me this is so cool