Can we rotate about a plane or other multi-dimensional "axis"?

Question

I was examining the nature of rotating objects and realized a pattern that could be extended to higher dimensions. If you have a line (1 dimensional) and rotate it along a point on the line (1 dimensional), the line rotates into the 2nd dimension. Similarly, if you have a square (2 dimensional) and rotate it along a line on the square, it rotates into the 3rd dimension.

My question is whether or not the trend would continue to where, if you rotate a cube (3 dimensional) about the plane of a square, it would rotate into the 4th dimension. Rather, are axes greater than one dimension possible?

Answer 1

In four-dimensional space, objects do indeed rotate around a plane. In fact, they can rotate around two planes at the same time. When you model rotations using quaternions, you essentially get two rotations around two separate planes; however, rotation is performed using a conjugation operation, and you can view the second product as undoing the unwanted rotation and doubling up the desired rotation. (That's not a particularly insightful way of thinking about it, but it's basically right, I think.)

Answer 2

A rotation (in according orientation) is being described by the following matrix $$\left(\begin{array}{ccccc}cos(\phi)&sin(\phi)&0&\cdots&0\\ -sin(\phi)&cos(\phi)&0&\cdots&0\\ 0&0&1&0&\vdots\\ \vdots&\vdots&0&\ddots&0\\ 0&0&\cdots&0&1\end{array}\right)$$ i.e. it is the identity within a subspace of codimension 2 (searving as "axis"), while the rotation moves (everything else) along a 2-dimensional path around it.

--- rk