In even dimensions () there are up to planes of rotation span the space, so a general rotation rotates all points except the origin which is the only fixed point. In odd dimensions () there are planes and angles of rotation, the same as the even dimension one lower. These do not span the space, but leave a line which does not rotate – like the axis of rotation in three dimensions, except rotations do not take place about this line but in multiple planes orthogonal to it.
