1. Introduction:
The 'Belt-Trick' visualizes the nature of the
group of rotations (or the Group SO(3)) in three-dimensional
space. One of the unique and interesting properties of this group is
that a belt with two twists and fixed ends can be continuously deformed
into a straight and untwisted belt. The same, however, is not true for
a belt with only one twist.
This striking phenomena can be demonstrated by using
any ordinary belt. However, the physical demonstration lacks sufficient
information which is necessary to completely characterize the behavior
of the belt. This shortcoming of the physical demonstration is overcome
by using graphical visualization of the phenomena. The visualization
allows not only the manipulation of the belt-deformation, but also displays
the relevant information from a unique perspective that is entirely
missing in the physical counterpart of the trick.
2. Using Quaternions to Describe Belt-Trick:
The unique perspective is provided by employing Quaternions
to visualize the intriguing phenomena of the belt-trick. Quaternions
are vectors in four-dimensional Euclidean space (R4)
and unit-quaternions represent orientations in the lower, three-dimensional
space. An orientation can be thought of as a triad of three, mutually
perpendicular axes, with one axis along the fixed axis of rotation.
Quaternions are used in this manner to represent orientations
of the segments on the belt, which lie adjacent to one another along
the length of the belt. Since quaternions are four-dimensional vectors
(think quaternion, q = (w, x, y, z), where w, x, y, z ε R,
real-numbers), they are visualized by projecting the three out-of-the-four
components of each quaternion into the three-dimensional subspace. A
visualization of the quaternions is called the quaternion-map
(more on quaternion-map here).
Once each segment-orientation has been captured by
a corresponding quaternion, the series of quaternions are connected
to obtain a connected curve in the four-dimensional space, which we
call the quaternion-curve. The behavior of this quaternion-curve,
during belt deformation, then provides an insight into why only a doubly-twisted
belt straightens out.
3. Properties of Quaternion-Curve:
The quaternions for the belt-segments are displayed
as beads on the continuous quaternion-curve. Each bead corresponds to
the orientation of a single belt-segment. Interestingly, the correspondence
between quaternions and orientations in three-dimensional space is a
2:1 mapping. This mapping implies that each orientation in the
three-dimensional space is represented by two quaternions in the quaternion-space.
This pair of quaternions lie diametrically opposite on the unit four-dimensional
sphere (or the hyper-sphere). The two-to-one mapping, also know as the
'double cover', has significant implications on how a belt can
deform during manipulation.
The double-cover mapping governs the nature of the
quaternion-curve for a singly-twisted (or 360°)
and a doubly-twisted (or 720°) belt. In the case
of a single or 360°-twist, the quaternion curve
is an open curve. Since the ends of the belt are fixed and not allowed
to move during the belt-deformations, the corresponding quaternions
always remain at the opposite poles on the four-dimensional sphere.
For a double or 720°-twist, the quaternion-curve
forms a loop and quaternions corresponding to the end segments are identical.
After the belt ends are fixed and the intermediate
segments are moved in space, the quaternion-curve also undergoes a deformation.
It is observed that in the case of 360°-twist,
the quaternion-curve flips from one hemisphere of the hyper-sphere to
another. Consequently, only the direction of the twist in the belt changes
and it still remains twisted. See the snapshots below.
On the other hand, for the 720°-twist,
the quaternion-curve continuously shrinks until it collapses into a
single quaternion. This implies that all the quaternions on the quaternion-curve
become identical and therefore all the orientations of the belt-segments
become the same. Consequently, the belt is returned to its initial untwisted
state. See the snapshots below.