ladder_chain_belt_gold_BT0002

Dirac’s belt trick as a homotopy

Paint one of the “rungs” of the belt.

(When the belt is worn by someone who is standing up, a rung will be a vertical strip.)

Take a video of the performance of the belt trick. Notice how the painted rung is rotating.
The homotopy is $H:X\times [0,1]\to Y$ where $X$ is the interval $[0,1]$ and $Y$ is $\mathrm{SO}(3)$.
$x\mapsto H(x,0)$ is the twisted belt and $x\mapsto H(x,1)$ is the straightened belt.

Each $x$ is a rung of the belt.

$H(x,t)$ is the rotation of rung $x$ at time $t$ in the video.

The rungs do not merely rotate, they are also translated in space during the belt trick. However, we can ignore the translation and focus on the rotation.
For instance, the top rung (where the buckle is) is translated but does not rotate.

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