Spin-½ Lab: The Belt Trick, 720° and the Dirac String
A free interactive spin-1/2 lab: Dirac's belt trick in 3D, where a ball tied to a wall by ribbons is tangled after 360 degrees but can be untangled after 720; five pictures of why a spinor needs two turns; a torus coloured by phase; and an elastic sheet that lies flat again only after 720 degrees. With quaternions, live readouts and the maths.
A ball (the "particle") hangs in the middle, tied to a far wall by ribbons. Turn the ball once, 360°: it looks exactly as it did, but the ribbons are tangled, and no amount of moving the ribbons will undo it while you hold the ball still. Turn it a second time, 720°, and something surprising happens: now the ribbons can be untangled, without turning the ball at all. Press Untangle to watch. Drag the picture to look around it, scroll or pinch to zoom.
What you are seeing
Think of the space around the ball as nested shells, like the layers of an onion. The wall (the outer shell) never moves; the shell touching the ball turns with the ball; each shell in between turns by some amount, and the ribbons simply ride along with their shells. In Twist mode every shell turns by a share of the ball's angle (θ at the ball, 0 at the wall), so the ribbons wind up.
After 360° the ball looks the same, yet the ribbons hold one full twist. The readout q shows why this cannot be undone: follow the shells from the wall (q = +1) in to the ball, keeping track of the sign that rotations normally forget, and you arrive at q = −1. No smooth change of the ribbons can flip that sign while the ball and the wall are held still. After 720° you arrive at q = +1 again, and the ribbons can be lifted round the ball (that is what Untangle does) back to where they started. Try Try at 360°: the same move only works there if the ball swings through a full turn of its own.
Keep turning, never tangle is the famous version where the ball spins on and on (think of a waiter carrying a tray, or the "Balinese cup trick"): the ribbons sway round the ball and are back home after every 720°. This is not just a trick of ribbons: an electron's or a neutron's wave function, a spinor, gets the same −1 after one turn and only comes back after two. The Spin Simulator shows the 1975 neutron experiment that measured it, and the Quaternions page shows why q = cos(θ/2) + sin(θ/2)·n is the natural way to write a turn.
Five pictures of the same idea: something at the centre (the core) turns by θ while the stuff around it (the medium) only turns by θ / 2. When the core has made one full turn, the medium is upside down; only after two turns is everything as it was. Pick a picture, press Play, or drag the slider.
A doughnut (torus) coloured by a phase: every colour is one angle. The big circle stands for the core's turn, the small circle (the tube) for the medium's. The bright line follows one colour: for spin ½ it goes once round the tube while going twice round the big circle, so it only closes after the core has turned 720°. Drag to look around, scroll or pinch to zoom.
Why a torus?
Two angles together (the core's angle and the medium's) make a torus: going once round the hole changes the first angle by 360°, going once round the tube changes the second. A line on the torus is a rule linking the two. A spin-1 object (an ordinary arrow) turns with its core, one-to-one: the line closes after one lap. A spin-½ object's medium turns half as fast, so its line is a (2, 1) torus knot: two laps round the hole for one round the tube. Colour the torus by the phase 2v − u (v = tube angle, u = core angle): it is smooth everywhere, and every line of one colour is such a spin-½ path. After one core turn you are on the far side of the tube, with the opposite colour beside you; after two you are home.
A flat elastic sheet with a ball in the middle. The ball turns about an axis lying in the sheet, and drags the sheet near it; far away (beyond the influence radius) the sheet is held fixed. After 360° the sheet is still crumpled; after 720° it lies flat again while the ball has turned twice. Tick "naive twist" to see the same turn without the untangling move: then even 720° leaves the sheet twisted.
An elastic medium (a model, not settled physics)
Chantal Roth's idea is to picture a particle as a knot of motion in an elastic, space-filling medium: the core turns, the medium around it is dragged along and has to stay connected to the still medium far away. That is a hypothesis, a mechanical way to imagine spin; standard quantum mechanics does not say space is such a material. What is established mathematics is the topology underneath: rotations in 3D form a space in which a 360° loop cannot be shrunk to nothing but a 720° loop can (the rotation group SO(3) is doubly covered by the unit quaternions, SU(2)). The sheet here is deformed by exactly the same shell rotations as the belt trick, with the shell number set by how strongly each point is dragged: so the sheet must come back flat at 720°, and cannot at 360°.
How the Spin-½ Lab works
Everything is drawn in your browser from a few formulas; nothing is uploaded. The idea behind every tab is one fact about turning things in three dimensions: a turn of 360° is not always the same as no turn at all, but a turn of 720° is. Objects that notice the difference are called spin-½: electrons, protons, neutrons and quarks all are.
The belt trick (Dirac's string trick)
Paul Dirac liked to show it with scissors tied to a chair by strings: give the scissors a full turn and the strings are tangled; give them a second turn and the strings can be untangled by passing them round the scissors. The page draws it with shells: the turn of the shell at distance r is a rotation F(r), the wall shell is never turned and the innermost shell turns with the ball. Untangling means changing the whole family of shell rotations smoothly while keeping both ends fixed. Joining the shells from the wall to the ball draws a path in the space of rotations; a 360° path is a loop that cannot be shrunk to a point there, a 720° path can. The family G(r, s) above is one way to shrink it: the 720° twist is written as two 360° twists about n, and as s goes from 0 to 1 the axis of the first one swings round from n to −n, until the two cancel shell by shell. The ball stays still throughout, because each factor is a full 360° turn at the ball and (−1)(−1) = +1. Each frame of the animation is checked in the page's tests: the wall shell stays at the identity, the ball shell stays put, neighbouring shells differ by a small turn, and the lifted quaternion at the ball is −1 after 360° and +1 after 720°.
Five ways back after 720°
The same 2 : 1 ratio appears in many pictures: a core tied to an elastic medium (the medium halfway out turns half as much), a wave on a ring whose crest takes two laps to come back, a direction carried round a Möbius band (one lap flips it, two restore it), a ring of beads whose arrows turn at half the ring's speed and come in ± pairs, and the rotation field of an elastic solid around a turning core. The dials compare the core's cos θ (back to 1 every 360°) with the medium's cos θ/2 (back to 1 only every 720°). Pictures 1 and 5, a particle as a core in an elastic medium, follow Chantal Roth's mechanical model; the other three are standard geometry.
Questions
- Is the electron really a little ball with ribbons?
- No one knows what an electron "is" inside, and standard quantum mechanics treats it as a point. The ribbons are a way to see a mathematical fact: an electron's state (a spinor) changes sign after a 360° turn and comes back after 720°, exactly as the ribbon picture does. That sign change has been measured with neutrons (see the Spin Simulator).
- If the ball looks the same after 360°, how can anything be different?
- The ball alone cannot tell, just as a rotation matrix cannot tell. What differs is how it is connected to the rest of the world: the ribbons (or the wave's phase) remember the turn. Quaternions remember it too: q and −q give the same rotation, but following the turn smoothly from q = +1 ends at −1 after one turn.
- Why exactly 720° and not 1080°?
- Because two loops of 360° join into one that can be shrunk to nothing, the "number of tangles" only counts up to one: an even number of turns is untangled, an odd number is tangled once. 1080° is tangled again, 1440° is free, and so on.
- What does "Keep turning, never tangle" show?
- A motion where the ball turns continuously and the ribbons never pile up: they swing round the ball and repeat every 720°. Your arm does the same when you turn a cup on your palm twice (the "Balinese cup trick" or "plate trick").
- Is the elastic medium real?
- It is a model. The 720° topology it shows is solid mathematics, but whether space behaves like an elastic medium around particles is an open idea explored by Chantal Roth and others, not accepted physics.
Related: Spin Simulator (neutron 720° experiment) · Quaternions · 3D Graphing Calculator · Electromagnetic Fields Simulator · MES Physics · More MES tools