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When you twist your arm or a belt by 360 degrees, the hand or endpoint is back to where it started but the rest of your arm or belt is still twisted. But if you do a 720 degree twist, you can manage to untwist your arm or belt! This is known as Dirac's Belt Trick or the Balinese Cup Trick. This crazy fact is even connected to physics with spin 1/2 particles, so let's try and figure out why! We will study rotations in 2 and 3 dimensions, and specifically study them topologically as opposed to algebraically as you might have seen before with rotation matrices. For a 2D rotation this is identified with points on a circle S^1. For a 3D rotation we need both an axis or rotation and an angle of rotation and we identify this with the solid ball of radius pi where a point in the ball gives a vector from the origin to the point that is our axis of rotation and the length of this vector is the angle. There is a catch: we have a double counting along the boundary so we have to identify antipodal points as the same. If you eliminate the origin (ie no rotation) this is sometimes called the Special Orthogonal Group SO(3) which is topologically the same as 3D Real Projective Space RP(3). A belt is then a path and I show an explicit way I can continuously deform the 4pi rotation path back to the identity.

0:00 Dirac's Belt Trick
1:37 2D rotations and the Circle
2:35 Axis and angle of 3D rotations
3:24 Modelling rotations using a solid ball
5:31 The double counting problem and identifying antipodes
6:43 Paths of Rotations
9:49 Deforming the 4π path
12:49 Thanks to Brilliant.org/TreforBazett

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