A solid hangs at the centre of a cage, tied to it by a belt from every face. Turn it once, a full 360°, and it is back exactly where it started - but its belts are wound round it, and no amount of sliding them about will straighten them unless you turn the solid back, cut a belt, or pass one through another. Keep turning the same way, though, and during the second turn the belts slip free: after 720° everything is as it began.
One turn and no turn look the same; the belts can tell them apart. That is a fact about the geometry of rotation itself, and it is the same fact that lets an electron's quantum state change sign when the electron is turned through 360° - something measured, in neutrons, in 1975.
Drag to turn the view. Space plays and pauses, the arrow keys step, and [ ] change the speed. At 1× one full cycle — two turns — takes twelve seconds.
Looking at the solid alone a turn of 360° cannot be distinguished from no turn at all: every face is back in its place. The belts remember how it got there - not how many times it has turned, only whether that number is odd or even. Roger Penrose describes this very set-up, a book tied to a pile of books by a belt: the belt “keeps track of the parity of the number of 2π rotations … rather than totting up the entire number.”
A mathematical way to say it: follow the solid from its starting position, through its turning, back to where it began, and you have traced a loop of rotations. A loop that turns once cannot be shrunk, by any gradual change, to the loop that does nothing. A loop that turns twice can - and the belts are that shrinking, laid out in space. Watch a single belt through the whole cycle: its end at the solid makes the full double turn, its end at the cage does nothing, and every point between makes a loop part-way from one to the other. The mathematical way to write this is π1(SO(3)) = ℤ/2: rotations carry a hidden two-valued label, every orientation of the solid exists in two versions, and the belts show which one you are in. A rotation matrix forgets the label. The unit quaternions, which cover each rotation twice, as q and −q, keep it.
In quantum mechanics an electron, proton, or neutron, any particle of spin ½ is described by a wave that turns into its own negative when the particle is rotated through 360°, and comes back to itself only after 720°. Such an object is called a spinor. It sounds impossible, because a full turn ought to change nothing, and for anything you could draw with arrows it does change nothing. The belt trick is the way out: the geometry of rotation has room for exactly one bit of memory, even or odd, and a spinor is something that carries it.
The sign of a single wave cannot be seen; the flip shows only when a turned wave is compared with one that was not turned. In 1975 two groups did exactly that. Samuel Werner's group in the United States and Helmut Rauch's in Vienna split a beam of neutrons in a silicon-crystal interferometer, passed one half through a magnetic field that made the neutrons' spins precess, and let the halves meet again. As the field was raised the interference swung back and forth, and it repeated not when the spins had turned through 360° but through 720° - two full turns, just as the belts need.
The idea reaches further. Richard Feynman, in his 1986 lecture in Dirac's memory and following David Finkelstein, pointed out that if two particles are joined by a ribbon, swapping them puts a full turn into the ribbon. So exchanging two spin-½ particles should flip the sign too - the root of Pauli's exclusion principle, that no two electrons can share a state. Feynman offered it as an insight rather than a proof, but the principle it points to is why atoms have shells and solid matter does not collapse.
The belts are not a picture of an electron. Nothing is tied to one, and Penrose calls the belt an “imaginary flexible attachment.” What the trick shows is that a spin-½ particle is possible: that 360° is not quite the same as doing nothing, while 720° is. That electrons, protons and neutrons really are such particles is a fact about nature, and it is the neutron experiments, not the belts, that establish it.
Paul Dirac used the trick for years in his lectures, to make the predicted behaviour of spin ½ seem less implausible. His own model was not a belt but a pair of scissors, with strings threaded through the handles and tied to a chair; Max Newman published the proof in 1942 that an odd number of turns leaves a tangle and an even number does not. The same trick turns up as a plate or a cup balanced on the palm - turn it once and your arm is twisted, twice and it is straight again - which Feynman performed in that 1986 lecture, and in Indonesian and Filipino candle and plate dances. Engineers use it too: a cable can feed a platform that spins for ever without ever winding up.