Thus, a simple way of visualizing the Hopf fibration is as follows. Any point on the -sphere is equivalent to a quaternion, which in turn is equivalent to a particular rotation of a Cartesian coordinate frame in three dimensions. The set of all possible quaternions produces the set of all possible rotations, which moves the tip of one unit vector of such a coordinate frame (say, the vector) to all possible points on a unit -sphere. However, fixing the tip of the vector does not specify the rotation fully; a further rotation is possible about the axis. Thus, the -sphere is mapped onto the -sphere, plus a single rotation.