This is a practical demonstration that the Lie group SO(n,R) has a fundamental group of order 2 for large n.

"You hold the coffee cup with your right hand underneath it, straight out in front of you. Now bring it left, under your underarm, awkwardly around front with your elbow straight up in the air. That's 360 degrees, and you're a pretzel. Keep going around counterclockwise, this time swinging your arm around over your head. At 720 degrees the coffee cup is back where it started, unspilled, and your arm is straight once more. Keep going round and round until you believe it."

Try it. It's blowing my mind, man.