  The Dirac belt trick is often employed in physics classrooms to show that a
$2\pi$ rotation is not topologically equivalent to the absence of rotation
whereas a $4\pi$ rotation is, mirroring a key property of quaternions and their
isomorphic cousins, spinors. The belt trick can leave the student wondering if
a real understanding of quaternions and spinors has been achieved, or if the
trick is just an amusing analogy. The goal of this paper is to demystify the
belt trick and to show that it implies an underlying \emph{four-dimensional}
parameter space for rotations that is simply connected. An investigation into
the geometry of this four-dimensional space leads directly to the system of
quaternions, and to an interpretation of three-dimensional vectors as the
generators of rotations in this larger four-dimensional world. The paper also
shows why quaternions are the natural extension of complex numbers to four
dimensions. The level of the paper is suitable for undergraduate students of
physics.
