140:, whose prong serves as a pointer. The end opposite the buckle is clamped so it cannot move. The belt is extended without a twist and the buckle is kept horizontal while being turned clockwise one complete turn (360°), as evidenced by watching the prong. The belt will then appear twisted, and no maneuvering of the buckle that keeps it horizontal and pointed in the same direction can undo the twist. Obviously a 360° turn counterclockwise would undo the twist. The surprise element of the trick is that a second 360° turn in the clockwise direction, while apparently making the belt even more twisted, does allow the belt to be returned to its untwisted state by maneuvering the buckle under the clamped end while always keeping the buckle horizontal and pointed in the same direction.
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Mathematically, the belt serves as a record, as one moves along it, of how the buckle was transformed from its original position, with the belt untwisted, to its final rotated position. The clamped end always represents the null rotation. The trick demonstrates that a path in rotation space (SO(3))
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Resting a small plate flat on the palm, it is possible to perform two rotations of one's hand while keeping the plate upright. After the first rotation of the hand, the arm will be twisted, but after the second rotation it will end in the original position. To do this, the hand makes one rotation
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Animation of the extended Dirac belt trick, showing that spin 1/2 particles are fermions: they can be untangled after switching particle positions twice, but not once
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that produces a 360 degree rotation is not homotopic to a null rotation, but a path that produces a double rotation (720°) is null-homotopic.
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Air on the Dirac
Strings, showing the belt trick with several belts attached to a spherical particle, by Louis Kauffman and colleagues
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112:. As with the plate trick, these particles' spins return to their original state only after two full rotations, not after one.
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Pengelley, David; Ramras, Daniel (2017-02-21). "How
Efficiently Can One Untangle a Double-Twist? Waving is Believing!".
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392:"Rogue breather modes: Topological sectors, and the 'belt-trick', in a one-dimensional ferromagnetic spin chain"
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of rotations twice over. A detailed, intuitive, yet semi-formal articulation can be found in the article on
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passing over the elbow, twisting the arm, and then another rotation passing under the elbow untwists it.
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Staley, Mark (2010-01-12). "Understanding
Quaternions and the Dirac Belt Trick".
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Staley, Mark (May 2010). "Understanding
Quaternions and the Dirac Belt Trick".
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The same phenomenon can be demonstrated using a leather belt with an ordinary
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76:. To say that SU(2) double-covers SO(3) essentially means that the unit
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Animation of the Dirac belt trick, including the path through SU(2)
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49:(after Paul Dirac, who introduced and popularized it), the
218:"Testing a conjecture on the origin of the standard model"
315:"Advanced Quantum Mechanics, Lecture 5, time point 51:53"
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Bolker, Ethan D. (November 1973). "The Spinor
Spanner".
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Animation of the Dirac belt trick, with a double belt
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Belt trick has been theoretically constructed in 1-d
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Mathematic demonstration of rotations in 3-dimensions
344:"The Strange Numbers That Birthed Modern Algebra"
544:Mechanical linkage implementing the belt trick
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390:Rahul, O. R.; Murugesh, S. (2019-05-01).
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222:The European Physical Journal Plus
216:Schiller, Christoph (2021-01-13).
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448:The American Mathematical Monthly
329:"Actor Performs the Plate Trick"
564:The double-tipping nullhomotopy
234:10.1140/epjp/s13360-020-01046-8
477:The Mathematical Intelligencer
396:Chaos, Solitons & Fractals
124:Leather belt with frame buckle
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579:Rotation in three dimensions
57:(it appears in the Balinese
554:Video of Balinese cup trick
418:10.1016/j.chaos.2019.02.012
342:Charlie Wood (6 Sep 2018).
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132:Dirac belt trick simulation
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366:"Dirac Belt Trick"
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40:plate trick
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182:References
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18:Belt trick
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176:Tangloids
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110:spinors
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63:SU(2)
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