# Can somebody explain the plate trick to me?

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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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. –  yasmar Nov 19 '10 at 3:44
@yasmar: Perhaps it was this question you were thinking of. –  Rahul Nov 19 '10 at 17:09
@Rahul Thanks. That is the one. –  yasmar Nov 19 '10 at 18:13
There is a decent video at youtube.com/watch?v=Rzt_byhgujg –  Jeff Mar 18 '14 at 18:36

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.