There is a nice experiment you can do, illustrating this: Hold a glass of water in the palm of your hand. Not too full, especially not until you get the hang of

There is a nice experiment you can do, illustrating this: Hold a glass of water in the palm of your hand. Not too full, especially not until you get the hang of the following move: Assuming you use your right hand, rotate your hand counterclockwise. At first, your hand goes under your arm, until it has rotated about 270 degrees or so. You can now continue that rotation, but you have to lift your hand up, so it is now above the arm. Keep going, and you end up where you started, but the glass has done two full revolutions. Hopefully without spilling an water (takes practice). The fun thing is, after just one revolution, your arm is twisted into a really uncomfortable configuration. The mathematical explanation is that SO(3) is doubly connected, whereas the unit quaternions, like any sphere, is simply connected. At any time, each part of your arm has undergone some rotation from the orientation at rest. Halfway through the move, as you travel from the shoulder down to the hand, this rotation starts at the identity, and changes continuously through one rotation, back to the identity once more. But in the unit quaternions, that path takes you from one pole to the opposite pole. This explains why you can’t untwist your arm. Sorry if that made no sense. Edit: Have a look at “Your palm is a spinor” [1], and then at this Phillipine* tradtional dance [2] about 40 seconds in. I knew I had written about this before [3]. [1] https://www.youtube.com/watch?v=fTlbVLGBm3Q https://www.youtube.com/watch?v=fTlbVLGBm3Q [2] https://www.youtube.com/watch?v=mOO_IQznZCQ https://www.youtube.com/watch?v=mOO_IQznZCQ [3] https://math.stackexchange.com/a/383549 https://math.stackexchange.com/a/383549 * I wrote Thai originally.