alt Hacker NewsI 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).
> 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
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/