at.algebraic topology - How does $pi_1(SO(3))$ relate exactly to the waiters trick?

(This is a bit long for a comment but isn't intended as an answer.)

I've broken glasses in the past demonstrating this trick so I strongly advise doing it with empty plastic mugs until you have it down pat. For those who can't quite work it out, here's a simpler way to do it: take an elastic band and two rods that have distinguishable "up" and "down". Loop the band around the two rods and keep it fairly taut (just so it doesn't fall off). Ascii picture:

   |     |
 /-|-----|-   
/  |     |     
  |     |  /   
 ---------/   
   |     |   

Now turn one of the rods upside-down. By moving the rod through space but without turning it (or the system falling apart), try to untangle the elastic band.

Can't do it? Good. Can? Whoops! Either you've done something wrong or Whitehead did.

Start again from the beginning. Now turn one of the rods through a full twist (so it's right-side-up again). Now try to untangle the elastic band as before.

Can't do it? Try again! It's possible.

So in mathematical terms, what you are looking for is the position of the second rod plus it's "up-down"ness. In terms of the waiter's arm, you want the position of the hand and the expression of agony on his face: if his face is in agony, his arm is twisted; if his face is calm, his arm isn't twisted.

Mnemonic: twisting someone's arm twice gets you nowhere.