Dirac’s Belt Trick: Why a 2π rotation twists space but a 4π rotation fixes it

Visit ► to help you learn STEM topics for free, and the first 200 people will get 20% off an annual premium subscription. Check out my MATH MERCH line in collaboration with Beautiful Equations ► 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 rot