So I have been heavily trying to understand rotations. Rotations as i understand is a planar phenomenon. You need at least two dimensions. That is why rotations cannot work in dim 1. With 2 dimensions, rotations happen in the only plane that exists: XY. However the axis of rotation cannot actually be described in ℝ2 you have to go to ℝ3to define the primary axis of rotation. In ℝ3 is different because there are 3 planes of rotation: XY, XZ, YZ. The problem comes from properly and effectively interpolating and combining rotations. So just like we did in 2D we can define an axis or plane in ℝ4 all X,Y, and Z vectors rotate around. Or the planes XY, XZ, YZ rotate around. This is the w axis/component. So now if my intuition is correct then can you please explain how exactly quaternions fit into this and why we do not quaternions defined for 2D rotation like 3D rotation. A complex number is sufficient for a rotation in 2d because there is only one plane, and a combination of planes are not require, right? So should there not be a way to define a 2d rotation in terms of an 3d axis like 4-vector quaternion does not 3d? Thank you so much!