[itishappy]

It's no lie, quaternions do actually have 4 dimensions. The part I take issue with is that rotors also require 4 dimensions to represent 3d rotations, they're just labeled slightly more intuitively.

    quaterions:
    0*1 + b*i + c*j + d*k

    rotors:
    0*1 + b*xy + c*yz + d*zx

I've included real components, but when representing rotations they'll always be zero. (They'll be non-zero during intermediate calculations though, so you need to consider them!)

Now... rotors do have some unique powers in that they're incredibly general. You don't need to hop from complex numbers to quaternions when you move between spaces and beyond, you can just use rotors for everything:

    2d:
    complex numbers
    rotors

    3d:
    quaternions
    rotors

    4d:
    octonions
    rotors

    Minkowski spacetime:
    ???
    rotors

[koolala]

I don't consider variables to equal dimensions so I'm glad you put them in 3D too. People take the term 'dimension' very literally when thinking about things.

What's the difference between doing:

rotors: 01 + bxy + cyz + dzx

and

quaterions: 01 + bjk + cki + d*ij

?

edit: https://api.lib.kyushu-u.ac.jp/opac_download_md/410895/178c.... this seems to explain the difference

[itishappy]

I'm wrong. Too late to edit, correction below:

    quaterions:
    a*1 + b*i + c*j + d*k

    rotors:
    a*1 + b*xy + c*yz + d*zx

The representations I shared previously with zero real component are for the points under rotation, not the rotors themselves which have real components in the general form. Apologies for misinformation!

The exceptions are 0 degrees and 180 degree rotations (and 360, 540, etc...), which will have one and zero as the real components, respectively.