This is sometimes called Dirac's belt trick. There is a Wikipedia article under the name the plate trick. As that article says, it demonstrates the theorem that "SU(2) (which double-covers SO(3)) is simply connected."

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          [![Dirac Belt Trick illustrated by George Francis.][1]][1]
          ^{Dirac Belt Trick illustrated by George Francis. From "Torus Immersions and Transformations."}

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This is sometimes called Dirac's belt trick. There is a Wikipedia article under the name the plate trick. As that article says, it demonstrates the theorem that "SU(2) (which double-covers SO(3)) is simply connected."

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          [![Dirac Belt Trick illustrated by George Francis.][1]][1]
^{Dirac Belt Trick illustrated by George Francis. From Carlo H. Séquin, "Torus Immersions and Transformations," 2013.}

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