I am studying the rotation in the space. I am wondering there is any relation between the rotation and the dimension of the space?
For example, in the case of odd dimensions the rotation is around the origin while in the even dimensions the rotations is around one axes!
Or maybe in spaces with odd dimensions we have rotation around some plane of odd dimensions and in spaces with even dimensions we have rotation around planes of even dimensions!
Indeed, my goal is to see which are the curves $\gamma(t)$ where $\gamma(t)=\alpha(t)Rot(a_1t,...,a_kt)$. Here $\alpha(t)$ is a straight line in $\mathbb{R}^n$ and \begin{equation} Rot(a_1t,...,a_kt)=\left\lbrace \begin{aligned} R(a_1t)\oplus ...\oplus R(a_kt)\oplus 1 &\hspace{4mm} \text{if} & n=2k\\ R(a_1t)\oplus ...\oplus R(a_kt) \hspace{7mm} &\hspace{4mm} \text{if} &n=2k-1, \end{aligned}\right. \end{equation} and \begin{equation} R(a_it)= \left( {\begin{array}{cc} \cos a_it & \sin a_it \\ -\sin a_it & \cos a_it \\ \end{array} } \right). \end{equation}