I apologize if this is a bit simplistic, but how can there be rotational degrees, or in simpler terms more than one "direction," even if we are speaking about a single plane? Is this just a supposition which is required in order for geometry to correspond to the world insofar as it is (for normal people) practically a Euclidean space, or instead is its soundness derived from a prior proposition? Similarly, how can there be multiple planes -- which, as I understand, are at right angles to each other? And to this questions about planes, I also have the same question regarding its logical soundness.
I am not trained in mathematics, so if the answer is complicated, it may need to be simplified for me. I cannot explain it, perhaps because I lack the terms to be more precise, but I find it very strange. That there is more than one direction and dimension is obvious enough. But as to how... I am very much left speechless.
(As a side note, this question reaches for me even to the level of the distinction between points on a line. Unless, I suppose, there is somehow an absolute and fixed point [which, on second thought, would seem to push the question back a step further], how can there be any distinction between one point and another without recourse to brute factuality?)
Rotation as such takes place within a 2D subspace. Everything what is orthogonal to that subspace serves as "axis" to rotate around. So in a 2D space you just can rotate around a point. Within 3D you can rotate around a linear axis. Within 4D you can rotate around an orthogonal 2D subspace. And therefore you can have there 2 independend rotations at the same time, one within the one suspace and one within the orthogonal subspace. This then is what is called a Clifford rotation. Within 5D you even can rotate around a 3D subspace. Etc.
--- rk