A visualization of Dirac's Belt Trick and Feynman's Plate Trick by Bob Palais
Each time the applet plays, each cube starts and ends at the same reference configuration-"the identity".
The leftmost cube goes through two full rotations about a fixed axis, the rightmost stays fixed at the identity.
At all times, every cube is in a configuration close to that of its neighbors.
If the leftmost cube only went through one rotation, we could not deform it to
a stationary cube!
You cannot eliminate an odd number of full twists from a belt without moving
its ends, but you
can eliminate any even number of full twists by passing the belt "around one end" without moving it,
because "half-twisted points" can only appear or disappear in pairs, like particles. Try it!
A twice twisted belt represents the leftmost cube, and an untwisted belt represents the rightmost cube
Viewing all of the cubes at once represents "Feynman's plate trick", an orthogonal view of the same homotopy.
The leftmost cube is the waiter's hand going through two full rotations about the vertical axis
to not drop your food! The rightmost cube is the waiter's fixed shoulder. If you try it you will feel
as "twisted" as the cubes after one rotation. Hint: Do one rotation above your elbow and another below!
If one rotation could be deformed smoothly to none, you would be able to turn the plate once and end up untwisted.