Dirac Belt TrickTie one end of a belt down and spin the other end a full turn: the belt is now visibly twisted, and no amount of sliding or looping removes the twist without turning it back. Spin it a second full turn, same direction, and — astonishingly — the belt can now be worked completely flat again, without ever untwisting the handle. Watch the little marker's colour: it only completes one full cycle every two turns.Turning an object by 360° is not, topologically, the same as not turning it at all — only 720° gets back to a path that can shrink away to nothing. The rotation group SO(3) is double-covered by the unit quaternions SU(2): a rotation by angle φ lifts to q(φ)=(cos φ/2, sin φ/2·axis), and q(2π)=−1 while q(4π)=+1. That extra minus sign is exactly the phase an electron's wavefunction picks up after one full turn — the reason spin-½ particles need two full rotations to look the same again.