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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 homotopy equivalent to a null rotation, but a path that produces a double rotation (720°) is null equivalent.
Belt trick has been witnessed in 1-d
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
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and physics
Physics is the natural science that studies matter, its 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 which r ...
, the plate trick, also known as Dirac
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety o ...
's string trick, the belt trick, or the Balinese cup trick, 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 more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
(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 tr ...
) 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 between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
. 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. Hamilton defined a quatern ...
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 ide ...
of rotations twice over. A detailed, intuitive, yet semi-formal articulation can be found in the article on tangloids
Tangloids is a mathematical game for two players created by Piet Hein to model the calculus of spinors.
A description of the game appeared in the book ''"Martin Gardner's New Mathematical Diversions from Scientific American"'' by Martin Gardner ...
.
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. Inmathematical physics
Mathematical physics refers to 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 t ...
, the trick illustrates the quaternionic mathematics behind the spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
of spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
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 homotopy equivalent to a null rotation, but a path that produces a double rotation (720°) is null equivalent.
Belt trick has been witnessed in 1-d Classical Heisenberg model The Classical Heisenberg model, developed by Werner Heisenberg, is the n = 3 case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena.
Definition
It can be formulated as follows: take a ...
as a breather solution.
See also
*Anti-twister mechanism
The anti-twister or antitwister mechanism is a method of connecting a flexible link between two objects, one of which is rotating with respect to the other, in a way that prevents the link from becoming twisted. The link could be an electrical cabl ...
*Spin–statistics theorem
In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles that ...
*Orientation entanglement
In mathematics and physics, the notion of orientation entanglement is sometimes used to develop intuition relating to the geometry of spinors or alternatively as a concrete realization of the failure of the special orthogonal groups to be sim ...
*Tangloids
Tangloids is a mathematical game for two players created by Piet Hein to model the calculus of spinors.
A description of the game appeared in the book ''"Martin Gardner's New Mathematical Diversions from Scientific American"'' by Martin Gardner ...
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=119577398External links
Animation of the Dirac belt trick, including the path through SU(2)
* ttps://www.youtube.com/watch?v=Rzt_byhgujg Video of Balinese cup trickbr>
Rotation in three dimensions Spinors Topology of Lie groups Science demonstrations