# Rotation in Higher Dimensions

In a world of three spatial dimensions plus time, every atom rotates around a line, the axis of rotation.

In a world of $N$ spatial dimensions where $N$ is greater than 3, must every atom rotate, and if so does it rotate around a line, a plane, or a subspace of smaller number of dimensions?

1. One may show that a general rotation $R\in SO(N)$ in $N\geq 2$ spatial dimensions can be composed
$$R ~=~ R_1\circ \ldots\circ R_{k}$$ of at most $k=[\frac{N}{2}]$ pairwise commuting rotations $$R_1,\ldots, R_{k}~\in~ SO(N)$$ that each leaves a co-dimension-2 subspace invariant (although not necessarily the same subspace).

2. More explicitly, given a rotation $R\in SO(N)$ there exists an orthonormal basis $(e_1, \ldots, e_N)$ [which may depend on $R$] such that the rotation $R$ is represented by a block-diagonal matrix of the form $$\begin{pmatrix} \cos\theta_1 & \sin\theta_1 & \cr -\sin\theta_1 & \cos\theta_1 & \cr && \cos\theta_2 & \sin\theta_2 & \cr &&-\sin\theta_2 & \cos\theta_2 & \cr &&&& \ddots \cr &&&&&\cos\theta_k & \sin\theta_k & \cr &&&&&-\sin\theta_k & \cos\theta_k & \cr &&&&&&&1\cr &&&&&&&&1\cr &&&&&&&&& \ddots \cr &&&&&&&&&&1\end{pmatrix}.$$

3. The rotation $R$ itself is only guaranteed to leave invariant a dimension-1 subspace (=a line through the origin) if the space dimension $N$ is odd.

In 2d, a rotation matrix has the form $$r(\theta)=\left(\begin{array}{cc} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right):= \left(\begin{array}{cc} c(\theta)&-s(\theta)\\ s(\theta)&c(\theta)\end{array}\right)$$ and rotates vector in a plane.

In 3d a rotation matrix can be written as a product $$r_{12}(\psi)r_{13}(\theta)r_{12}(\varphi)$$ where \begin{align} r_{12}(\psi)&=\left(\begin{array}{ccc} c(\psi)&-s(\psi)&0\\ s(\psi)&c(\psi)&0\\ 0&0&1 \end{array}\right)\\ r_{13}(\theta)&=\left(\begin{array}{ccc} c(\psi)&0&-s(\psi)\\ 0&1&0\\ s(\psi)&0&c(\psi) \end{array}\right) \end{align} leaving one axis invariant. This axis can be identified by the row or column containing $0$s everywhere except for one entry.

In SO(4), one can write a rotation matrix as a sequence or $r_{ij}$ matrices. $r_{12}$ would have the form $$r_{12}(\psi)=\left(\begin{array}{cccc} c(\psi)&-s(\psi)&0&0\\ s(\psi)&c(\psi)&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{array}\right)$$ and so leaves a 2-dimensional subspace invariant. An SO(4) matrix can be written in the factored form $$r_{34}(\beta_1)r_{23}(\beta_2)r_{12}(\beta_3) r_{34}(\beta_4)r_{23}(\beta_5)r_{34}(\beta_6)$$ by restricting to real values the entries of the $SU(4)$ matrix factored as as done here. This is not by any means the only possible factorization.

Obviously, an SO(5) rotation can be written in terms of matrices leaving a 3-dimensional subspace invariant etc.

• What about double rotations in 4D? The simultaneous rotations in 2 orthogonal planes intercepting in a point at the origin (replacing units in your matrix with sines and cosines). Dec 1 '17 at 13:33
• @safesphere I’m not sure I understand your question. It’s still a 4d rotation, but clearly it can be realized as a sequence of two commuting SO(2) rotations: $r_{12} r_{34}$. Take $r_{23}(0)=1$ etc. Dec 1 '17 at 13:38
• As a clarification of what I mean, please see this math.stackexchange.com/q/2543122 and also see Double Rottions under Geometry here en.wikipedia.org/wiki/… Dec 1 '17 at 13:58
• @safesphere I really need more coffee... I still don’t get what you ask, but the link I will read with interest later. Dec 1 '17 at 14:04
• No problem :) To clarify again, in 3D, an object can rotate only around one axis. If we try rotating an object around two axes at the same time, this only turns the rotation axis, but it remains a single one and the rotation still has a single rate. In 4D however an object can rotate in one plane (single rotation) and also in another orthogonal plane at the same time (dual rotation) independently and with a different rate. Dec 1 '17 at 14:12