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I learned of the plate trick via Wikipedia, which states that this is a demonstration of the fact that SU(2) double-covers SO(3). It also offers a link to an animation of the "belt trick" which is apparently equivalent to the plate trick. Since I've thought most about the belt version, I'll phrase my question in terms of the belt trick.

I am not clear on how the plate/belt trick relates to the double covering. Specifically, I am looking for a sort of translation of each step of the belt trick into the Lie group setting. For example, am I correct in interpreting the initial twisting of the belt as corresponding to the action of a point in SU(2)? Which point? Do I have the group right?

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  • $\begingroup$ I thought somebody talked about the plate trick in an answer or a comment on this question, but I don't see it now. Anyway, the answers there are worth looking over. $\endgroup$ – yasmar Nov 19 '10 at 3:44
  • $\begingroup$ @yasmar: Perhaps it was this question you were thinking of. $\endgroup$ – Rahul Nov 19 '10 at 17:09
  • $\begingroup$ @Rahul Thanks. That is the one. $\endgroup$ – yasmar Nov 19 '10 at 18:13
  • $\begingroup$ There is a decent video at youtube.com/watch?v=Rzt_byhgujg $\endgroup$ – Jeff Mar 18 '14 at 18:36
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A 360 degree rotation of the arm results in a very sore arm unless you unwind it or wind it a second time. Think about the rotation of the hand as being a path in the space of all rotations of 3-space. Or if you really want to rotate something, tape a belt to a soccer ball and tape the other end of the belt to the wall. Just rotate the ball a full rotation, the belt gets a twist. Now rotate again; the belt has two full twists. The twists in the belt can be undone by looping the ball back through the belt as in the animation.

The diagram in the lower right shows the paths in the space $\operatorname{SO}(3)$ which is the $3$-ball with its antipodal points on its boundary identified. That is why the doubled path appears to be broken; it is passing through antipodal points.

You might play with a Möbius band and the pencil loop which maps twice around to gain more intuition. The point of thinking of the Möbius band is that it is embedded in the space of rotations.

It is really worth thinking this through while going to sleep or while riding the bus!

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  • $\begingroup$ Thank you - I especially like the idea of taping a soccer ball to a belt. So the rotation of the belt corresponds to a path in SO(3)? How then is the double-covering by SU(2) reflected in the subsequent looping motion? $\endgroup$ – NKS Nov 19 '10 at 16:54
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    $\begingroup$ @NKS: SU(2) is simply connected, so the fact that it double-covers SO(3) is equivalent to the fact that SO(3) has a fundamental group of order 2. The plate trick exhibits the nontrivial element of this group. $\endgroup$ – Qiaochu Yuan Nov 19 '10 at 16:58

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