mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, the plate trick, also known as Dirac's string trick (after Paul Dirac, who introduced and popularized it), the belt trick, or the Balinese cup trick (it appears in the Balinese candle dance), is any of several demonstrations of the idea that rotating an object with strings attached to it by 360 degrees does not return the system to its original state, while a second rotation of 360 degrees, a total rotation of 720 degrees, does. Mathematically, it is a demonstration of the theorem that

SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...

(which double-covers

SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a ...

) is

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...

. To say that SU(2) double-covers SO(3) essentially means that the unit

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

s represent the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

of rotations twice over. A detailed, intuitive, yet semi-formal articulation can be found in the article on tangloids.

Demonstrations

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 passing over the elbow, twisting the arm, and then another rotation passing under the elbow untwists it. In

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

, the trick illustrates the quaternionic mathematics behind the spin of

spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...

s. As with the plate trick, these particles' spins return to their original state only after two full rotations, not after one.

The belt trick

The same phenomenon can be demonstrated using a leather belt with an ordinary frame buckle, 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. 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)) 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. The belt trick has been theoretically constructed in 1-d Classical Heisenberg model as a breather solution.

References

* * {{Cite journal, last1=Pengelley, first1=David, last2=Ramras, first2=Daniel, date=2017-02-21, title=How Efficiently Can One Untangle a Double-Twist? Waving is Believing!, journal=The Mathematical Intelligencer, volume=39, language=en, pages=27–40, doi=10.1007/s00283-016-9690-x, issn=0343-6993, arxiv=1610.04680, s2cid=119577398

External links

Animation of the Dirac belt trick, including the path through SU(2)Animation of the Dirac belt trick, with a double beltAnimation 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 onceMechanical linkage implementing the belt trick
* ttps://www.youtube.com/watch?v=Rzt_byhgujg Video of Balinese cup trickbr>
Rotation in three dimensions Spinors Topology of Lie groups Science demonstrations

Demonstrations

The belt trick

See also

References

External links